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This term would "fix" the circuit problem we have in Figure 4, and would make Farday's Law and Ampere's Law more symmetric. This was Maxwell's great contribution. And you might think it is a weak contribution. But the existance of this term unified the equations and led to understanding the propagation of electromagnetic waves, and the proof that all waves travel at the same speed (the speed of light)! And it was this unification of the equations that Maxwell presented, that led the collective set to be known as Maxwell's Equations. So, if we add the displacement current to Ampere's Law as written in Equation [6], then we have the final form of Ampere's Law:

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Maxwell knew that the Electric Field (and Electric Flux Density (D) was changing within the capacitor. And he knew that a time-varying magnetic field gave rise to a solenoidal Electric Field (i.e. this is Farday's Law - the curl of E equals the time derivative of B). So, why is not that a time varying D field would give rise to a solenoidal H field (i.e. gives rise to the curl of H). The universe loves symmetry, so why not introduce this term? And so Maxwell did, and he called this term the displacement current density:

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However, a capacitor is basically two parallel conductive plates separated by air. Hence, there is no conductive path for the current to flow through. This means that no electric current can flow through the air of the capacitor. This is a problem if we think about Equation [8]. To show it more clearly, let's take a volume that goes through the capacitor, and see if the divergence of J is zero:

n Figure 4, we have drawn an imaginary volume in red, and we want to check if the divergence of the current density is zero. The volume we've chosen, has one end (labeled side 1) where the current enters the volume via the black wire. The other end of our volume (labeled side 2) splits the capacitor in half.

We know that the current flows in the loop. So current enters through Side 1 of our red volume. However, there is no electric current that exits side 2. No current flows within the air of the capacitor. This means that current enters the volume, but nothing leaves it - so the divergence of J is not zero. We have just violated our Equation [8], which means the theory does not hold. And this was the state of things, until Maxwell came along.

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Displacement Current Density (Ampere's Law)-Part 3

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Displacement Current Density (Ampere's Law)-Part 3

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This seems somewhat reasonable, as electric current in circuits flows in a loop. But let's look what happens if we put a capacitor in the circuit:
Now, we know from electric circuit theory that if the voltage is not constant (for example, any periodic wave, such as the 60 Hz voltage that comes out of your power outlets) then current will flow through the capacitor. That is, we have I not equal to zero in Figure 3.

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Displacement Current Density (Ampere's Law)-Part 2

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Displacement Current Density (Ampere's Law)-Part 2

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So let's take the divergence of Ampere's Law as written in Equation [6]:
If the divergence of J is always zero, this means that the electric current flowing into any region is always equal to the electric current flowing out of the region (no divergence).

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Displacement Current Density ( Ampere's Law)-Part 1

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Displacement Current Density ( Ampere's Law)-Part 1

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the divergence of the curl of any vector field is always zero:

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Mathematical Expression for Ampere's Law- Part 3

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Mathematical Expression for Ampere's Law- Part 3

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So now we have the original Ampere's Law (Equation [2]) rewritten in terms of surface integrals (Equations [4] and [5]). Hence, we can substitute them together and get a new form for Ampere's Law:
Now, we have a new form of Ampere's Law: the curl of the magnetic field is equal to the Electric Current Density.

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Mathematical Expression For Ampere's Law (Part 2)

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Mathematical Expression For Ampere's Law (Part 2)

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We can also rewrite the total current () as the surface integral of the Current Density (J):

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Mathematical Expression for Ampere's Law

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Mathematical Expression for Ampere's Law

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On the right side equality in Equation [4], we have used Stokes' Theorem to change a line integral around a closed loop into the curl of the same field through the surface enclosed by the loop (S).

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If we are adding up a constant value for the magnetic field (we'll call it H), then the left side of Equation [2] becomes simple:
Hence, we have figured out what the magnitude of the H field is. And since r was arbitrary, we know what the H-field is everywhere. Equation [3] states that the Magnetic Field decreases in magnitude as you move farther from the wire (due to the 1/r term).

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Ampere's Law, which relates a electric current flowing and a magnetic field wrapping around it:

Equation [2] can be explained: Suppose you have a conductor (wire) carrying a current, I. Then this current produces a Magnetic Field which circles the wire.

The left side of Equation [2] means: If you take any imaginary path that encircles the wire, and you add up the Magnetic Field at each point along that path, then it will numerically equal the amount of current that is encircled by this path (which is why we write  for encircled or enclosed current).

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The meaning of the last of Maxwell's Equations, Ampere's Law, which is given in Equation [1]:

Ampere was a scientist experimenting with forces on wires carrying electric current. He was doing these experiments back in the 1820s, about the same time that Farday was working on Faraday's Law. Ampere and Farday didn't know that there work would be unified by Maxwell himself, about 4 decades later.

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If we replace Farday's Law of Equation [2], with the terms we found in Equation [3] and Equation [7], then we get:

In Equation [8], we note that if we have two integrals over surfaces, and the surfaces can be however we choose, then the quantities we integrate must also be the same. And this is how we obtained Faraday's Law in final form, as listed on Maxwell's Equations.

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Now, a mathematician named Stokes figured out that integrating (averaging) of a field around a loop is exactly equivalent to integrating the curl of the field within the loop. The curl is the measure of the rotation of a field, so the curl of a vector field within a surface should be related to the integral of a field around a loop that encloses the surface.

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Hence, Equations [4] and [5] tell us that the differential amount of EMF at any point along the circuit (dEMF in [4]) is equal to the E field at that location. Therefore:

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Now, recall that the Electric Field is directly related to force from electric charges. And Voltage is also defined as the sum (integral) of the Electric Field across a path [recall that the E-field is measured in Volts/meter]. Hence, the E-field is actually the spatial-derivative of voltage (E-field is equal to the rate of change of the voltage with respect to distance).

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To find the total EMF induced around the whole circuit, we sum up over the length of the wire the EMF produced at each point. This is known as a line integral. This is written as:

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We know that the rate of change of the total magnetic flux is equal to the opposite of the EMF, or the electric force within the wire. The total magnetic flux is simply the integral (or sum) of the B field over the area enclosed by the wire:

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Faraday figured out that a changing Magnetic Flux within a circuit (or closed loop of wire) produced an induced EMF, or voltage within the circuit. In Equation [2],  is the Magnetic Flux within a circuit, and EMF is the electro-motive force, which is basically a voltage source. Equation [2] then says that the induced voltage in a circuit is the opposite of the time-rate-of-change of the magnetic flux.

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Faraday was a scientist experimenting with circuits and magnetic coils way back in the 1830s. His experiment setup, which led to Farday's Law, is shown in Figure 1:
The experiment itself is somewhat simple. When the battery is disconnected, we have no electric current flowing through the wire. Hence there is no magnetic flux induced within the Iron (Magnetic Core). The Iron is like a highway for Magnetic Fields - they flow very easily through magnetic material. So the purpose of the core is to create a path for the Magnetic Flux to flow.

When the switch is closed, the electric current will flow within the wire attached to the battery. When this current flows, it has an associated magnetic field (or magnetic flux) with it. When the wire wraps around the left side of the magnetic core (as shown in Figure 1), a magnetic field (magnetic flux) is induced within the core. This flux travels around the core. So the Magnetic Flux produced by the wired coil on the left exists within the wired coil on the right, which is connected to the ammeter.

Now, a funny thing happens, which Faraday observed. When he closed the switch, then current would begin flowing and the ammeter would spike one way (say measuring +10 Amps on the other side). But this was very brief, and the current on the right coil would go to zero. When the switch was opened, the measured current would spike to the other side (say -10 Amps would be measured), and then the measured current on the right side would again be zero.

Faraday figured out what was happening. When the switch was initially changed from open to closed, the magnetic flux within the magnetic core increased from zero to some maximum number (which was a constant value, versus time). When the flux was increasing, there existed an induced current on the opposite side.

Similarly, when the switch was opened, the magnetic flux in the core would decrease from it's constant value back to zero. Hence, a decreasing flux within the core induced an opposite current on the right side.

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The meaning of the 3rd of Maxwell's Equations, Faraday's Law, which is given in Equation [1]:

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First, observe both of Gauss' Laws, written in Equation [1]:
You see that both of these equations specify the divergence of the field in question. For the top equation, we know that Gauss' Law for Electric Fields states that the divergence of the Electric Flux Density D is equal to the volume electric charge density. But the second equation, Gauss' Magnetism law states that the divergence of the Magnetic Flux Density (B) is zero.

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Equation [2] states that the amount of charge inside a volume V (=) is equal to the total amount of Electric Flux (D) exiting the surface S. That is, to determine the Electric Flux leaving the region V, we only need to know how much electric charge is within the volume. We rewrite Equation [2] with more of the terms defined in Equation [3]:

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Equation [1] is known as Gauss' Law in point form. That is, Equation [1] is true at any point in space. That is, if there exists electric charge somewhere, then the divergence of D at that point is nonzero, otherwise it is equal to zero.

To get some more intuition on Gauss' Law, let's look at Gauss' Law in integral form. To do this, we assume some arbitrary volume (we'll call it V) which has a boundary (which is written S). Then integrating Equation [1] over the volume V gives Gauss' Law in integral form:

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Gauss' Law is the first of Maxwell's Equations which dictates how the Electric Field behaves around electric charges. Gauss' Law can be written in terms of the Electric Flux Density and the Electric Charge Density

The symbol delta is the divergence operator.

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Maxwell's Equations are a set of 4 complicated equations that describe the world of electromagnetics. These equations describe how electric and magnetic fields propagate, interact, and how they are influenced by objects.

James Clerk Maxwell [1831-1879] was an Einstein/Newton-level genius who took a set of known experimental laws (Faraday's Law, Ampere's Law) and unified them into a symmetric coherent set of Equations known as Maxwell's Equations. Maxwell was one of the first to determine the speed of propagation of electromagnetic (EM) waves was the same as the speed of light - and hence to conclude that EM waves and visible light were really the same thing.

Maxwell's Equations are critical in understanding Antennas and Electromagnetics. They are formidable to look at - so complicated that most electrical engineers and physicists don't even really know what they mean. Shrouded in complex math (which is likely so "intellectual" people can feel superior in discussing them), true understanding of these equations is hard to come by.

This leads to the reason for this website - an intuitive tutorial of Maxwell's Equations. I will avoid if at all possible the mathematical difficulties that arise, and instead describe what the equations mean. And don't be afraid - the math is so complicated that those who do understand complex vector calculus still cannot apply Maxwell's Equations in anything but the simplest scenarios. For this reason, intuitive knowledge of Maxwell's Equations is far superior than mathematical manipulation-based knowledge. To understand the world, you must understand what equations mean, and not just know mathematical constructs. I believe the accepted methods of teaching electromagnetics and Maxwell's Equations do not produce understanding. And with that, let's say something about these equations.

Maxwell's Equations are laws - just like the law of gravity. These equations are rules the universe uses to govern the behavior of electric and magnetic fields. A flow of electric current will produce a magnetic field. If the current flow varies with time (as in any wave or periodic signal), the magnetic field will also give rise to an electric field. Maxwell's Equations shows that separated charge (positive and negative) gives rise to an electric field - and if this is varying in time as well will give rise to a propagating electric field, further giving rise to a propgating magnetic field.

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In the centre-tap circuit shown in Fig. 2, the diodes are assumed to be ideal
i.e. having zero internal resistance. Find :(i) d.c. output voltage(ii) peak inverse voltage (iii) rectification efficiency.

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A full-wave rectifier uses two diodes, the internal resistance of each diode may
be assumed constant at 20 Ω. The transformer r.m.s. secondary voltage from centre tap to each end of secondary is 50 V and load resistance is 980 Ω. Find : (i) the mean load current (ii) the r.m.s. value of load current.

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A half-wave rectifier is used to supply 50V d.c. to a resistive load of 800 Ω. The
diode has a resistance of 25 Ω. Calculate a.c. voltage required.

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A crystal diode having internal resistance rf = 20Ω is used for half-wave rectification. If the applied voltage v = 50 sin ω t and load resistance RL= 800 Ω, find :
(i) Im, Idc, Irms (ii) a.c. power input and d.c. power output (iii) d.c. output voltage (iv) efficiency of rectification.

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An a.c. supply of 230 V is applied to a half-wave rectifier circuit through a
transformer of turn ratio 10 : 1. Find (i) the output d.c. voltage and (ii) the peak inverse voltage. Assume the diode to be ideal.

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The applied input a.c. power to a half-wave rectifier is 100 watts. The d.c. output power obtained is 40 watts.
(i) What is the rectification efficiency ?
(ii) What happens to remaining 60 watts?

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full-wave rectifier converts the whole of the input waveform to one of constant polarity (positive or negative) at its output. Mathematically, this corresponds to the absolute value function. Full-wave rectification converts both polarities of the input waveform to pulsating DC (direct current), and yields a higher average output voltage. Two diodes and a center tapped transformer, or four diodes in a bridge configuration and any AC source (including a transformer without center tap),  Single semiconductor diodes, double diodes with a common cathode or common anode, and four- or six-diode bridges are manufactured as single components.

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In half-wave rectification of a single-phase supply, either the positive or negative half of the AC wave is passed, while the other half is blocked. Mathematically, it is a step function (for positive pass, negative block): passing positive corresponds to the ramp function being the identity on positive inputs, blocking negative corresponds to being zero on negative inputs. Because only one half of the input waveform reaches the output, mean voltage is lower. Half-wave rectification requires a single diode in a single-phase supply, or three in a three-phase supply. Rectifiers yield a unidirectional but pulsating direct current; half-wave rectifiers produce far more ripple than full-wave rectifiers, and much more filtering is needed to eliminate harmonics of the AC frequency from the output. 

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A rectifier is an electrical device that converts alternating current (AC), which periodically reverses direction, to direct current (DC), which flows in only one direction. The reverse operation is performed by the inverter.

The process is known as rectification, since it "straightens" the direction of current. Physically, rectifiers take a number of forms, including vacuum tube diodes, wet chemical cells, mercury-arc valves, stacks of copper and selenium oxide plates, semiconductor diodes, silicon-controlled rectifiers and other silicon-based semiconductor switches. Historically, even synchronous electromechanical switches and motors have been used. Early radio receivers, called crystal radios, used a "cat's whisker" of fine wire pressing on a crystal of galena (lead sulfide) to serve as a point-contact rectifier or "crystal detector".

Rectifiers have many uses, but are often found serving as components of DC power supplies and high-voltage direct current power transmission systems. Rectification may serve in roles other than to generate direct current for use as a source of power. As noted, detectors of radio signals serve as rectifiers. In gas heating systems flame rectification is used to detect presence of a flame.

Depending on the type of alternating current supply and the arrangement of the rectifier circuit, the output voltage may require additional smoothing to produce a uniform steady voltage. Many applications of rectifiers, such as power supplies for radio, television and computer equipment, require a steady constant DC voltage (as would be produced by a battery). In these applications the output of the rectifier is smoothed by an electronic filter, which may be a capacitor, choke, or set of capacitors, chokes and resistors, possibly followed by a voltage regulator to produce a steady voltage.

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1) Determine the energy stored in an inductor with L = 70 mH, if a current of 3A flows through it.

2) A 45 Ohm resistor is connected to a 0.35 mH inductor & to a 50 Hz, 100 V r.m.s source.Determine the power factor & the average power delivered to the circuit.

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A 45 Ohm resistor is connected to a 25 micro Farad capacitor & to a 50 Hz, 100 V r.m.s source. Determine the power factor & the average power delivered to the circuit.

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Real Power in AC Circuits

Real power (P), also known as true or active power, performs the “real work” within an electrical circuit. Real power, measured in watts, defines the power consumed by the resistive part of a circuit. Then real power, (P) in an AC circuit is the same as power, P in a DC circuit. So just like DC circuits, it is always calculated as I2*R, where R is the total resistive component of the circuit.
resistance phasor

As resistances do not produce any phasor difference (phase shift) between voltage and current waveforms, all the useful power is delivered directly to the resistance and converted to heat, light and work. Then the power consumed by a resistance is real power which is fundamentally the circuits average power.

To find the corresponding value of the real power the rms voltage and current values are multiplied by the cosine of the phase angle, Φ as shown.
 

Real Power   P = I2R = V*I*cos(Φ)  Watts, (W)
 

But as their is no phase difference between the voltage and the current in a resistive circuit, the phase shift between the two waveforms will be zero (0). 

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In an AC circuit, the voltage and current waveforms are sinusoidal so their amplitudes are constantly changing over time. Since we know that power is voltage times the current (P = V*I), maximum power will occur when the two voltage and current waveforms are lined up with each other. That is, their peaks and zero crossover points occur at the same time. When this happens the two waveforms are said to be “in-phase”.

The three main components in an AC circuit which can affect the relationship between the voltage and current waveforms, and therefore their phase difference, by defining the total impedance of the circuit are the resistor, the capacitor and the inductor.

The impedance, (Z) of an AC circuit is equivalent to the resistance calculated in DC circuits, with impedance given in ohms. For AC circuits, impedance is generally defined as the ratio of the voltage and current phasor’s produced by a circuit component. Phasor’s are straight lines drawn in such a way as to represents a voltage or current amplitude by its length and its phase difference with respect to other phasor lines by its angular position relative to the other phasor’s.

AC circuits contain both resistance and reactance that are combined together to give a total impedance (Z) that limits current flow around the circuit. But an AC circuits impedance is not equal to the algebraic sum of the resistive and reactive ohmic values as a pure resistance and pure reactance are 90o out-of-phase with each other. But we can use this 90o phase difference as the sides of a right angled triangle, called an impedance triangle, with the impedance being the hypotenuse as determined by Pythagoras theorem.

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Exercise: Series Resonance & damping Oscillation.

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Exercise: Series Resonance & damping Oscillation.

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A series RCL circuit has R = 180 Ohm, L = 25 mH, V = 30 V & 2(pi)f = 5.5 x 10^3 /s. Determine the capacitance value for a maximum r.m.s current. Determine also the maximum r.m.s current in the circuit.

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Inductive Reactance and Capacitive Reactance Vs Frequency.

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Inductive Reactance and Capacitive Reactance Vs Frequency.

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it can be concluded that the inductive reactance is directly proportional to frequency and capacitive reactance is inversely proportional to frequency, i.e at low frequency XL is low and XC is high but there must be a frequency, where the value of inductive reactance becomes equal to capacitive reactance. Now if we plot a single graph of inductive reactance vs frequency and capacitive reactance vs frequency, then there must occur a point where these two graphs cut each other. At that point of intersection, the inductive and capacitive reactance becomes equal and the frequency at which these two reactances become equal, is called resonant frequency, fr.

At resonant frequency, XL = XC

PLEASE FIND THE RELATED EQUATIONS FOR XL = XC !

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Variation of Capacitive Reactance Vs Frequency.

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Variation of Capacitive Reactance Vs Frequency.

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From the formula of capacitive reactance XC = 1 / 2πfC that, frequency and capacitive reactance are inversely proportional to each other. In case of DC or when frequency is zero, capacitive reactance becomes infinity and circuit behaves as open circuit and when frequency increases and becomes infinite, capacitive reactance decreases and becomes zero at infinite frequency, at that point the circuit acts as short circuit, so the capacitive reactance increases with decease in frequency and if we plot a graph between capacitive reactance and frequency, it is an hyperbolic curve as shown in figure above.

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Variation in Inductive Reactance and Capacitive Reactance with Frequency

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Variation in Inductive Reactance and Capacitive Reactance with Frequency

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The inductive reactance XL = 2πfL means inductive reactance is directly proportional to frequency (XL and prop ƒ). When the frequency is zero or in case of DC, inductive reactance is also zero, the circuit acts as a short circuit; but when frequency increases; inductive reactance also increases. At infinite frequency, inductive reactance becomes infinity and circuit behaves as open circuit. It means that, when frequency increases inductive reactance also increases and when frequency decreases, inductive reactance also decreases. So, if we plot a graph between inductive reactance and frequency, it is a straight line linear curve passing through origin as shown in the figure above.

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Consider a RLC circuit in which resistor, inductor and capacitor are connected in series across a voltage supply. This series RLC circuit has a distinguishing property of resonating at a specific frequency called resonant frequency.

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Resonance in Series RLC circuit (part 2)

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Resonance in Series RLC circuit (part 2)

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1. When a current flows in an inductor, energy gets stored in magnetic field.

2. When a capacitor is charged, energy gets stored in static electric field.

The magnetic field in the inductor is built by the current, which is provided by the discharging capacitor. Similarly, the capacitor is charged by the current produced by collapsing magnetic field of inductor and this process continues on and on, causing electrical energy to oscillate between the magnetic field and the electric field. In some cases, at certain frequency called resonant frequency, the inductive reactance of the circuit becomes equal to capacitive reactance which causes the electrical energy to oscillate between the electric field of the capacitor and magnetic field of the inductor. This forms a harmonic oscillator for current. In RLC circuit, the presence of resistor causes these oscillation to die out over period of time and is called damping effect of resistor.

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Determine the impedance, r.m.s current & phase angle between the current & voltage of a series RCL a.c. circuit which consists of a resistor with R = 200 Ohm, an inductor L = 0.8 H & a capacitor with C = 3.80 micro F. The alternating voltage supply is 150 V at 60 Hz.

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RCL : Resistance,capacitance & inductance.

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RCL : Resistance,capacitance & inductance.

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https://www.youtube.com/watch?v=6ieUk3Rm1yY

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An inductor of 3.0 H has a resistance of 70 Ohm. It is connected in series with a resistor of 450 Ohm and an alternating voltage supply of 240 V & frequency of 60 Hz.Determine the r.m.s current in the circuit & the phase between the applied voltage & the current in the circuit.

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https://www.youtube.com/watch/_A6fCgQdS2Q

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A capacitor of 1500 micro F is connected to 2.5 V, 0.45 A lamp & a 60 Hz supply in series.Determine the r.m.s voltage supply to the lamp,the voltage across the capacitor & the resistor respectively.

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Angular Frequency Formula

The angular frequency of the circuit can be found with the formula:

ω = 2πf

The angular frequency ω in radians per second is equal to 2 times π times the characteristic frequency f in hertz.

Impedance Formula

The impedance of an RC circuit can be found using a few formulas:

Z = R + 1/jωC
|Z| = √(R² + 1/(ωC)²)

Where j is the imaginary unit, Z is the impedance in ohms, R is the resistance in ohms, C is the capacitance in farads, and ω is the angular frequency in rad/s.

Capacitive Reactance Formula

The capacitive reactance of an RC circuit can be found using the formula:

X = 1/ωC

The capacitive reactance X is equal to 1 divided by the angular frequency ω times the capacitance C.

Phase Difference Formula

This formula expresses the phase difference between total voltage and total current.

φ = tan⁻¹(-1/ωCR)

φ is the phase difference, ω is the angular frequency, C is the capacitance, and R is the resistance.

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Charge Formula

The maximum charge in the resistor capacitor circuit can be found using the formula:

Q = CV

The charge Q in coulombs is equal to the capacitance C in farads times the voltage V.

Current Formula

The maximum current of the RC circuit can be found using Ohm’s Law. The formula is:

I = VR

The current I in amps is equal to the voltage V divided by the resistance R in ohms.

Characteristic Frequency Formula

The characteristic frequency, often called the ordinary or cyclic frequency, of the circuit can be found using this formula:

f = 1/2πRC

The frequency f in hertz is equal to 1 divided by 2 times π times the resistance R in ohms times the capacitance C in farads.

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A resistor-capacitor circuit, or RC circuit, is a circuit with a resistor and capacitor connected in series. The capacitor in the circuit stores energy and the resistor changes the charge and discharge rate of the capacitor. These circuits are most commonly used for filtering the waveform and are used to create low-pass, high-pass, and band-pass filters.

RC Circuit Formulas

RC circuits have several characteristics, including a time constant, energy storage, charge, impedance, capacitive reactance, characteristic frequency, and angular frequency. Calculating each of these characteristics of the circuit can be done using the following formulas.

Time Constant Formula

The time constant, expressed as tau (τ) is the time it takes in seconds for the capacitor in the RC circuit to reach 63.2% charge. The formula to calculate the time constant is:

τ = RC

The time constant τ is equal to the resistance R in ohms times the capacitance C in farads. The capacitor will reach a 63.2% charge in τ, 86.5% in 2τ, and 99.3% in 5τ.

Energy Formula

The energy stored in the fully charged capacitor in an RC circuit can be found using the formula:

E = CV²2

The energy E in joules is equal to the capacitance C in farads times the voltage V squared, divided by two.

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1) The capacitive reactance of an a.c circuit at 129 Hz is 30 Ohm. If the frequency of the a.c is changed to 15 000 Hz,determine the capacitive reactance of the capacitor.

2) In a purely capacitive a.c. circuit, the peak value of the voltage source , V0 = 150 V.If the angular frequency ,omega, of the a.c is 140 rad/s and a capacitor of C= 5 micro F is connected in series to the voltage source,determine the r.m.s current in the circuit.

3) The r.m.s voltage of 120 V is supplied to a purely inductive circuit.At a frequency of 50 Hz the r.m.s current is 15 A. Determine the inductance & also find the frequency of the circuit if the r.m.s current is reduced to 7.50 A.

4) An inductor has 60 Ohm reactance at 50 Hz. If the inductor is then connected to a 100 Hz source which has an r.m.s voltage of 120 V,calculate the peak current.

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In an a.c. circuit an inductor of 30 mH is connected to a voltage source with r.m.s value of 160 V. If the frequency of the a.c. is 60 Hz ,determine the inductive reactance and the r.m.s current in the circuit.

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A capacitor of 10.0 micro F is connected to an a.c. generator which produces an r.m.s volatage of 200 V at a frequency of 50 Hz. Determine the capacitive reactance and the r.m.s current in the circuit.

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Impedance, Resistance and Reactance - difference

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Phasor Diagram for Pure Resistive Circuits.

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Phasor Diagram for Pure Resistive Circuits.

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Phasor Diagram for Pure Resistive Circuits - Electrical Engineering

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1) A light bulb which uses an average power of 50 W is connected to a 50 Hz power source. If the r.m.s volatge of the power source is 120 V, determine the resistance of the light bulb.
2) A heating element is connected to an a.c. volatge source of an r.m.s volateg of 110 V. If an r.m.s current of 8.0 A occurs, determine the resistance & the average power input in the heating element.

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Explain the following statement: "the voltage across a capacitance lags the current by 90 degrees."

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Phasor diagram of a sinusoidal waveform.(Continue)

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Phasor diagram of a sinusoidal waveform.(Continue)

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If however, the waveforms are frozen at time, t = 30o, the corresponding phasor diagram would look like the one shown on the right. Once again the current phasor lags behind the voltage phasor as the two waveforms are of the same frequency.

However, as the current waveform is now crossing the horizontal zero axis line at this instant in time we can use the current phasor as our new reference and correctly say that the voltage phasor is “leading” the current phasor by angle, Φ. Either way, one phasor is designated as the reference phasor and all the other phasors will be either leading or lagging with respect to this reference.

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The phasor diagram is drawn corresponding to time zero ( t = 0 ) on the horizontal axis. The lengths of the phasors are proportional to the values of the voltage, ( V ) and the current, ( I ) at the instant in time that the phasor diagram is drawn. The current phasor lags the voltage phasor by the angle, Φ, as the two phasors rotate in an anticlockwise direction as stated earlier, therefore the angle, Φ is also measured in the same anticlockwise direction.

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Phasor Diagram of a Sinusoidal Waveform.

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Phasor Diagram of a Sinusoidal Waveform.

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As the single vector rotates in an anti-clockwise direction, its tip at point A will rotate one complete revolution of 360o or 2π representing one complete cycle. If the length of its moving tip is transferred at different angular intervals in time to a graph as shown above, a sinusoidal waveform would be drawn starting at the left with zero time. Each position along the horizontal axis indicates the time that has elapsed since zero time, t = 0. When the vector is horizontal the tip of the vector represents the angles at 0o, 180o and at 360o.

Likewise, when the tip of the vector is vertical it represents the positive peak value, ( +Am ) at 90o or π/2 and the negative peak value, ( -Am ) at 270o or 3π/2. Then the time axis of the waveform represents the angle either in degrees or radians through which the phasor has moved. So we can say that a phasor represent a scaled voltage or current value of a rotating vector which is “frozen” at some point in time, ( t ) and in our example above, this is at an angle of 30o.

Sometimes when we are analysing alternating waveforms we may need to know the position of the phasor, representing the Alternating Quantity at some particular instant in time especially when we want to compare two different waveforms on the same axis. For example, voltage and current. We have assumed in the waveform above that the waveform starts at time t = 0 with a corresponding phase angle in either degrees or radians.

But if a second waveform starts to the left or to the right of this zero point or we want to represent in phasor notation the relationship between the two waveforms then we will need to take into account this phase difference, Φ of the waveform.

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Phasor Diagrams are a graphical way of representing the magnitude and directional relationship between two or more alternating quantities .

Sinusoidal waveforms of the same frequency can have a Phase Difference between themselves which represents the angular difference of the two sinusoidal waveforms. Also the terms “lead” and “lag” as well as “in-phase” and “out-of-phase” are commonly used to indicate the relationship of one waveform to the other with the generalized sinusoidal expression given as: A(t) = Am sin(ωt ± Φ) representing the sinusoid in the time-domain form.

But when presented mathematically in this way it is sometimes difficult to visualise this angular or phasor difference between two or more sinusoidal waveforms. One way to overcome this problem is to represent the sinusoids graphically within the spacial or phasor-domain form by using Phasor Diagrams, and this is achieved by the rotating vector method.

Basically a rotating vector, simply called a “Phasor” is a scaled line whose length represents an AC quantity that has both magnitude (“peak amplitude”) and direction (“phase”) which is “frozen” at some point in time.

A phasor is a vector that has an arrow head at one end which signifies partly the maximum value of the vector quantity ( V or I ) and partly the end of the vector that rotates.

Generally, vectors are assumed to pivot at one end around a fixed zero point known as the “point of origin” while the arrowed end representing the quantity, freely rotates in an anti-clockwise direction at an angular velocity, ( ω ) of one full revolution for every cycle. This anti-clockwise rotation of the vector is considered to be a positive rotation. Likewise, a clockwise rotation is considered to be a negative rotation.

Although the both the terms vectors and phasors are used to describe a rotating line that itself has both magnitude and direction, the main difference between the two is that a vectors magnitude is the “peak value” of the sinusoid while a phasors magnitude is the “rms value” of the sinusoid. In both cases the phase angle and direction remains the same.

The phase of an alternating quantity at any instant in time can be represented by a phasor diagram, so phasor diagrams can be thought of as “functions of time”. A complete sine wave can be constructed by a single vector rotating at an angular velocity of ω = 2πƒ, where ƒ is the frequency of the waveform. Then a Phasor is a quantity that has both “Magnitude” and “Direction”.

Generally, when constructing a phasor diagram, angular velocity of a sine wave is always assumed to be: ω in rad/sec.

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Phasor Diagrams are a graphical way of representing the magnitude and directional relationship between two or more alternating quantities .
Sinusoidal waveforms of the same frequency can have a Phase Difference between themselves which represents the angular difference of the two sinusoidal waveforms. Also the terms “lead” and “lag” as well as “in-phase” and “out-of-phase” are commonly used to indicate the relationship of one waveform to the other with the generalized sinusoidal expression given as: A(t) = Am sin(ωt ± Φ) representing the sinusoid in the time-domain form.
But when presented mathematically in this way it is sometimes difficult to visualise this angular or phasor difference between two or more sinusoidal waveforms. One way to overcome this problem is to represent the sinusoids graphically within the spacial or phasor-domain form by using Phasor Diagrams, and this is achieved by the rotating vector method.

Basically a rotating vector, simply called a “Phasor” is a scaled line whose length represents an AC quantity that has both magnitude (“peak amplitude”) and direction (“phase”) which is “frozen” at some point in time.

A phasor is a vector that has an arrow head at one end which signifies partly the maximum value of the vector quantity ( V or I ) and partly the end of the vector that rotates.

Generally, vectors are assumed to pivot at one end around a fixed zero point known as the “point of origin” while the arrowed end representing the quantity, freely rotates in an anti-clockwise direction at an angular velocity, ( ω ) of one full revolution for every cycle. This anti-clockwise rotation of the vector is considered to be a positive rotation. Likewise, a clockwise rotation is considered to be a negative rotation.

Although the both the terms vectors and phasors are used to describe a rotating line that itself has both magnitude and direction, the main difference between the two is that a vectors magnitude is the “peak value” of the sinusoid while a phasors magnitude is the “rms value” of the sinusoid. In both cases the phase angle and direction remains the same.

The phase of an alternating quantity at any instant in time can be represented by a phasor diagram, so phasor diagrams can be thought of as “functions of time”. A complete sine wave can be constructed by a single vector rotating at an angular velocity of ω = 2πƒ, where ƒ is the frequency of the waveform. Then a Phasor is a quantity that has both “Magnitude” and “Direction”.

Generally, when constructing a phasor diagram, angular velocity of a sine wave is always assumed to be: ω in rad/sec.

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AC: An electric current that continually reverses direction.

The AC is the short form for Alternating Current, in which the electric charge reverses direction at regular intervals to produce alternate positive and negative values of the same magnitude.

The alternating current follows a sine waveform where the voltage increases steadily from zero, rising to the maximum positive peak voltage. It then reverses and drops down through zero into negative direction until it reaches the negative peak value, which is equal to the positive in magnitude and only different in polarity. The voltage reverses again and climbs toward the zero point to complete one cycle. This process repeats at the rated frequency of either the 50 HZ or 60 HZ (cycles per second).
The rate of reversing direction is quantified by the number of complete cycles per second and is known as the frequency. The two commonly used frequency standards for domestic and industrial applications are the 50 Hertz which is used in most parts of the world, and 60 Hertz used in the USA and some other regions.

The other frequency is 400 Hz this is used in aircrafts, spacecraft, marine, military and other sensitive applications where light equipment and higher motor speeds are desired.

The ac is generated using hydro, diesel, steam or wind turbines. Other sources are the renewable energy sources such as the solar; however, some of these produces the direct current and must be inverted to alternating current before being fed into the grid.

The alternating current is the common form of electrical power generated and distributed due its ease of generation and distribution. The alternating voltage is easily stepped up and down to suit any required voltage level. To minimize power losses in the conductors, the electrical power is transmitted at high voltages and low currents. This is later stepped down at the distribution and consumer level to suit the consumer’s needs.

Majority of electrical and electronic equipment use ac power at either 220-240 volts or 110-120V for domestic and office applications, and 415V for industrial. However, most of the equipment and especially all electronics use external or internal power supply units to convert the ac into the appropriate direct current (DC) required by the electronic devices and circuits.

Ac is normally supplied to equipment using three wires

The hot wire transmits the power.

Neutral provides a return path for the current in the hot wire. It is also connected to the earth.

The third wire is the ground which is also connected to the earth, this is connected to the metallic parts of the equipment to provide safety and eliminate electric shock hazards.

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The current in coils of a motor is 4 A when it is rotating at a maximum speed but its value is 11 A at the time when it is first turned on. If the volatge suppled to the motor is 120 V, determine the back e.m.f and the resistance of coils.

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The resistance of coils in a motor is 15 Ohm & the motor is supplied by a volatge of 150 V. The back e.m.f of the motor as it is operating at its maximum speed is 80 V. Determine the current of the coils when the motor is running at the maximum speed.

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1. The back emf opposes the supply voltage. The supply voltage induces the current in the coil which rotates the armature. The electrical work required by the motor for causing the current against the back emf is converted into mechanical energy. And that energy is induced in the armature of the motor. Thus, we can say that energy conversion in the DC motor is possible only because of the back emf.

The mechanical energy induced in the motor is the product of the back emf and the armature current, i.e., EbIa.

2. The back emf makes the DC motor self-regulating machine, i.e., the back emf develops the armature current according to the need of the motor. The armature current of the motor is calculated as:

Let’s understand how the back emf makes motor self-regulating.

Consider the motor is running at no-load condition. At no load, the DC motor requires small torque for controlling the friction and windage loss. The motor withdraws less current. As the back emf depends on the current their value also decreases. The magnitude of the back EMF is nearly equal to the supply voltage.

If the sudden load is applied to the motor, the motor becomes slow down. As the speed of the motor decreases, the magnitude of their back emf also falls down. The small back emf withdraw heavy current from the supply. The large armature current induces the large torque in the armature, which is the need of the motor. Thus, the motor moves continuously at a new speed.

If the load on the motor is suddenly reduced, the driving torque on the motor is more than the load torque. The driving torque increases the speed of the motor which also increases their back emf.  The high value of back emf decreases the armature current. The small magnitude of armature current develops less driving torque, which is equal to the load torque. And the motor will rotate uniformly at the new speed.

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part 3

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The magnitude of the back emf is given by the same expression shown below:

(Refer to part 3)

Where Eb is the induced emf of the motor known as Back EMF, A is the number of parallel paths through the armature between the brushes of opposite polarity. P is the number of poles, N is the speed, Z is the total number of conductors in the armature and ϕ is the useful flux per pole.

A simple conventional circuit diagram of the machine working as a motor is shown in the diagram
In this case, the magnitude of the back emf is always less than the applied voltage. The difference between the two is nearly equal when the motor runs under normal conditions.

The current induces on the motor because of the main supply. The relation between the main supply, back emf and armature current is given as Eb = V – IaRa.

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When the current-carrying conductor placed in a magnetic field, the torque induces on the conductor, the torque rotates the conductor which cuts the flux of the magnetic field. According to the Electromagnetic Induction Phenomenon “when the conductor cuts the magnetic field, EMF induces in the conductor”.

The Fleming right-hand rule determines the direction of the induced EMF.

According to Fleming Right Hand Rule, if we hold our thumb, middle finger and index finger of the right hand by an angle of 90°, then the index finger represents the direction of the magnetic field. The thumb shows the direction of motion of the conductor and the middle finger represents the emf induces on the conductor.

On applying the right-hand rule in the figure shown below, it is seen that the direction of the induced emf is opposite to the applied voltage. Thereby the emf is known as the counter emf or back emf.

The back emf is developed in series with the applied voltage, but opposite in direction, i.e., the back emf opposes the current which causes it.

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1) A ""step up"transformer increases volatge of 80 V to 180 V. What is the value of the current in the secondary coil with respect to the primary coil, assuming the transformer is 100 percent efficient.
2) The secondary coil of a transformer has 1800 turns & the primary coil of the same transformer has 120 turns. The input voltage & the output current of the transformer are 120 V & 8.0 A respectively. Calculate the volatge of the secondary coil & also the current in the primary coil.

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1) An electric generator produces 100 A at 4500 V. A "step up" transformer is used to increase the voltage to 240 000 V before it is sent on high voltage transmission line to a city. The total resistance of the transmission line is 40.0 ohm.
(i) Determine the lost in power in percentage.
(ii) If the voltage were not stepped up, calculate the lost of the original power in the transmission line.

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Assuming an ideal transformer and the phase angles:  ΦP ≡ ΦS

Note that the order of the numbers when expressing a transformers turns ratio value is very important as the turns ratio 3:1 expresses a very different transformer relationship and output voltage than one in which the turns ratio is given as: 1:3

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Where:

VP  –  is the Primary Voltage

VS  –  is the Secondary Voltage

NP  –  is the Number of Primary Windings

NS  –  is the Number of Secondary Windings

Φ (phi)  –  is the Flux Linkage

Notice that the two coil windings are not electrically connected but are only linked magnetically. A single-phase transformer can operate to either increase or decrease the voltage applied to the primary winding. When a transformer is used to “increase” the voltage on its secondary winding with respect to the primary, it is called a Step-up transformer. When it is used to “decrease” the voltage on the secondary winding with respect to the primary it is called a Step-down transformer.

However, a third condition exists in which a transformer produces the same voltage on its secondary as is applied to its primary winding. In other words, its output is identical with respect to voltage, current and power transferred. This type of transformer is called an “Impedance Transformer” and is mainly used for impedance matching or the isolation of adjoining electrical circuits.

The difference in voltage between the primary and the secondary windings is achieved by changing the number of coil turns in the primary winding ( NP ) compared to the number of coil turns on the secondary winding ( NS ).

As the transformer is basically a linear device, a ratio now exists between the number of turns of the primary coil divided by the number of turns of the secondary coil. This ratio, called the ratio of transformation, more commonly known as a transformers “turns ratio”, ( TR ). This turns ratio value dictates the operation of the transformer and the corresponding voltage available on the secondary winding.

It is necessary to know the ratio of the number of turns of wire on the primary winding compared to the secondary winding. The turns ratio, which has no units, compares the two windings in order and is written with a colon, such as 3:1 (3-to-1). This means in this example, that if there are 3 volts on the primary winding there will be 1 volt on the secondary winding, 3 volts-to-1 volt. Then we can see that if the ratio between the number of turns changes the resulting voltages must also change by the same ratio, and this is true.

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Excercise:
1)The current in a solenoid changes at a rate 4 A/s. The e.m.f. induced in the nearby coil is 10 mV. Determine the mutual inductance,M in the two coils.

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Excercise: Energy stored in an inductor.

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Excercise: Energy stored in an inductor.

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1) Calculate the energy stored in an inductor with L = 75.0 mH, if a current of 3.00 A flows through it.
2) An inductor has 300 turns with radius of 5.0 cm.If a current of 0.45 A flows through it,determine the energy stored in the inductor.
3) An inductor of L = 4.0 H is connected in series with a 25 V battery & a resistor of 10.0 Ohm. Determine the energy stored in the inductor when the current reaches its maximum value.

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Obtain an expression for an inductance L of an N turn solenoid if the length & the cross-sectional area of the solenoid is l and A respectively .Given N = 150, l = 7.0 cm, A = 0.45 cm^2 & the core is air,determine the inductance L. The same solenoid is then wound on an iron core of u = 4000(u -naught).Calculate L of the new solenoid.

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According to Lenz’s law, a conducting loop when subjected to varying magnetic field gets an emf induced into it causing flow of current in a direction opposing the change causing it. The case goes similar to when instead of a conducting closed loop, the change in magnetic field through a conducting body, say a filament or a slab of magnetic material or non-magnetic material, causes currents to flow into its cross sections at appropriate closed paths.

These currents are given the name eddy currents after the water eddies that are small swirling whirlpools observed in lakes and oceans. These eddy current loops can be both beneficial and undesirable.
While they cause undesirable high heat losses in the material such as transformer core, eddy currents find applications in various industrial processes like induction heating, metallurgy, welding, braking etc. This article deals with the theory and applications of eddy current phenomenon.

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A coil has an area of 0.15 m^2. It is rotating 70 revolutions per second with its axis perpendicular to a uniform magnetic field of 0.35 T. If the coil has 1500 turns,determine the maximum voltage induced in the coil.

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A motor has coils turning inside magnetic fields, and a coil turning inside a magnetic field induces an emf. This emf, known as the back emf, acts against the applied voltage that's causing the motor to spin in the first place, and reduces the current flowing through the coils of the motor.

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Induced emf & electric field induction (Generators)

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Induced emf & electric field induction (Generators)

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The generator coil shown in Figure 1 is rotated through one-fourth of a revolution (from θ = 0º to θ = 90º ) in 15.0 ms. The 200-turn circular coil has a 5.00 cm radius and is in a uniform 1.25 T magnetic field. What is the average emf induced?
Hence, calculate the maximum emf, emf₀, of the generator .

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2. Calculate the e.m.f. generated between the fixed and the free ends of a helicopter blade 9.45 m long that is rotating at 3.5 revs per second.
The vertical component of the Earth's field has a flux density of 4.0 x 10⁻⁵ T.

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1. Calculate the emf generated between the wing tips of an aircraft that is flying horizontally at 200 ms-1 in a region where the vertical component of the Earth's magnetic field is 4.0 x 10⁻⁵ T , if the aircraft has a wingspan of 25 m.

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When a straight conductor is moved through a magnetic field an emf is induced between its ends. This movement must be in such a direction that the conductor cuts through the lines of magnetic flux, and will be a maximum when it moves at right angles to the field .
Let the length of the conductor be L and the flux density of the field be B.
If the conductor moves with velocity v at right angles to the field then the flux cut per second will be BvL (since the conductor will sweep out an area vL every second).

But the rate of cutting flux is equal to the e.m.f. induced in the conductor. Therefore

E = BL v

If the conductor cuts through the flux at an angle θ ((b) in Figure 1), where θ is the angle between the magnetic field and the direction of motion, the equation becomes

E = BLvsinθ

You can see that the maximum e.m.f is generated when the conductor moves at right angles to the field. (θ = 90o and so sinθ = sin90 = 1).

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Lenz’s law of electromagnetic induction states that the direction of the current induced in a conductor by a changing magnetic field (as per Faraday’s law of electromagnetic induction) is such that the magnetic field created by the induced current opposes the initial changing magnetic field which produced it. The direction of this current flow is given by Fleming’s right hand rule.

This can be hard to understand at first – so let’s look at an example problem. Remember that when a current is induced by a magnetic field, the magnetic field that this induced current produces will create its own magnetic field. This magnetic field will always be such that it opposes the magnetic field that originally created it. In the example below, if the magnetic field “B” is increasing – as shown in (1) – the induced magnetic field will act in opposition to it.
When the magnetic field “B” is decreasing – as shown in (2) – the induced magnetic field will again act in opposition to it. But this time ‘in opposition’ means that it is acting to increase the field – since it is opposing the decreasing rate of change.

Lenz’s law is based on Faraday’s law of induction. Faraday’s law tells us that a changing magnetic field will induce a current in a conductor. Lenzs law tells us the direction of this induced current, which opposes the initial changing magnetic field which produced it. This is signified in the formula for Faraday’s law by the negative sign (‘–’).

This change in the magnetic field may be caused by changing the magnetic field strength by moving a magnet towards or away from the coil, or moving the coil into or out of the magnetic field. In other words, we can say that the magnitude of the EMF induced in the circuit is proportional to the rate of change of flux.

Lenz’s Law Formula

Lenz’s law states that when an EMF is generated by a change in magnetic flux according to Faraday’s Law, the polarity of the induced EMF is such, that it produces an induced current whose magnetic field opposes the initial changing magnetic field which produced it

The negative sign used in Faraday’s law of electromagnetic induction, indicates that the induced EMF (ε) and the change in magnetic flux (δΦB) have opposite signs. The formula for Lenz’s law is shown below:

Where:

ε = Induced emf

δΦB = change in magnetic flux

N = No of turns in coil

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A square frame of 20 cm x 20 cm is wrapped with 250 turns of wire. The total resistance of the coil is 2.50 Ohm. The coil is situated in a uniform magnetic field where the plane is perpendicular to the field. Determine the magnitude of the induced e.m.f in the coil when the magnetic field is changed from 0 to 0.600 T in 0.75 s.Calculate the induced current generated when the field is changing.

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Faraday's law is a fundamental relationship which comes from Maxwell's equations. It serves as a succinct summary of the ways a voltage (or emf) may be generated by a changing magnetic environment. The induced emf in a coil is equal to the negative of the rate of change of magnetic flux times the number of turns in the coil. It involves the interaction of charge with magnetic field.

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Any change in the magnetic environment of a coil of wire will cause a voltage (emf) to be "induced" in the coil. No matter how the change is produced, the voltage will be generated. The change could be produced by changing the magnetic field strength, moving a magnet toward or away from the coil, moving the coil into or out of the magnetic field, rotating the coil relative to the magnet, etc.

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Electromagnetic Induction or Induction is a process in which a conductor is put in a particular position and magnetic field keeps varying or magnetic field is stationary and a conductor is moving. This produces a Voltage or  EMF (Electromotive Force) across the electrical conductor. Michael Faraday discovered Law of Induction in 1830.
Suppose while shopping you go cashless and  your parents use cards. The shopkeeper always scans or swipes the card. Shopkeeper does not take a photo of the card or tap it.  Why does he swipe/scan it? And how does this swiping deduct money from the card? This happens because of the ‘Electromagnetic Induction’.

Can moving objects produce electric currents? How to determine a relationship between electricity and magnetism? Can you imagine the scenario if there were no computers, no telephones, no electric lights. The experiments of Faraday has led to the generation of generators and transformers.
The induction of an electromotive force by the motion of a conductor across a magnetic field or by a change in magnetic flux in a magnetic field is called ‘Electromagnetic Induction’.

This either happens when a conductor is set in a moving magnetic field (when utilizing AC power source) or when a conductor is always moving in a stationary magnetic field.
This law of electromagnetic induction was found by Michael Faraday. He organized a leading wire according to the setup given underneath, connected to a gadget to gauge the voltage over the circuit. So when a bar magnet passes through the snaking, the voltage is measured in the circuit. The importance of this is a way of producing electrical energy in a circuit by using magnetic fields and not just batteries anymore. The machines like generators,  transformers also the motors work on the principle of electromagnetic induction.

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A strip of metal has dimension of 1.2 cm wide & 1.5 x 10^-3 cm thick. It carries current of 0.50 A along its length.If the metal contains 5 x 10^22 electrons per cm^3,determine the mean drift velocity of these electrons.

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After applying a voltage across it current, “I” starts flowing from left to right that is along the x-direction. We place the metal block in a magnetic field of density “B”. The direction of the magnetic field along the y-axis. Now by Fleming’s Left-Hand Rule, the charge carriers will experience a force depending on both – direction of current and direction of magnetic field. As per Fleming’s rule, the conduction electrons will deflect towards the bottom of the block. As a result, there will be a change in concentration of electrons at upper and lower portion of the block. Consequently, a tiny potential difference appears across the block along the z-axis. The electric field created due to this shifting of electrons also opposes the shifting of electrons towards the surface of the metal block. Hence there may be two forces acting on the charge carriers.

1. The force due to Hall effect

2. The force due to created electric field

These forces are opposite to each other. After the establishment of the certain electric field due to Hall effect the system becomes in equilibrium. At that condition the force acting on the charge carriers (conduction electrons) due to the established electric field and due to Hall effect become same, and opposite. Hence there would not be any further shifting of electrons towards the surface of the block and the system become in an equilibrium condition.
Let us concentrate on a single charge carrier moving through a conductor placed inside a magnetic field.
The charge of the charge carrier is e (say)
The magnetic flux density of the field is B (say).
Now we can write the magnetic force acting on the current carrying conductor of active length L and current I as

When the direction of B and I are perpendicular to each other.

We can rewrite the above equation as

Where V is the velocity of the charge passing through the conductor.

From the above equation, we can write the force acting on a single charge carrier as,

Where v is the drift velocity of the charge carriers.
Now we can write the field created due to Hall effect as, EH.
Hence, we can write the force acting on the charge carrier due to the field as

Now at equilibrium

Now we consider N is charge carrier concentration.

From equation 1 and 2, we can write

We call the term  as the Coefficient of Hall Effect or simply Hall Coefficient. We define Hall Coefficient as the Hall field per unit magnetic field density per unit current density.

Hall effect is a very useful phenomenon and helps to
Determine the Type of Semiconductor
By knowing the direction of the Hall Voltage, one can determine that the given sample is whether n-type semiconductor or p-type semiconductor. This is because Hall coefficient is negative for n-type semiconductor while the same is positive in the case of p-type semiconductor.

Calculate the Carrier Concentration
The expressions for the carrier concentrations of electrons (n) and holes (p) in terms of Hall coefficient are given by

Determine the Mobility (Hall Mobility)
Mobility expression for the electrons (μn) and the holes (μp), expressed in terms of Hall coefficient is given by,

Where, σn and σp represent the conductivity due to the electrons and the holes, respectively.

Measure Magnetic Flux Density
This equation can be readily deduced from the equation of Hall voltage and is given by

Further, there are many commercially available types of equipment based on the principle of Hall effect including Hall-effect sensors and Hall-effect probes.