F = -----------------
         4π ε₀ r²

Other units: μF, nF, pF

In SI base units: F = A² s⁴ kg⁻¹ m⁻²

Moving coil galvanometer is an electromagnetic device that can measure small values of current. It consists of permanent horseshoe magnets, coil, soft iron core, pivoted spring, non-metallic frame, scale, and pointer.

Principle of Moving Coil Galvanometer

Torque acts on a current-carrying coil suspended in the uniform magnetic field. Due to this, the coil rotates. Hence, the deflection in the coil of a moving coil galvanometer is directly proportional to the current flowing in the coil.

Construction of Moving Coil Galvanometer

It consists of a rectangular coil of a large number of turns of thinly insulated copper wire wound over a light metallic frame. The coil is suspended between the pole pieces of a horseshoe magnet by a fine phosphor – bronze strip from a movable torsion head. The lower end of the coil is connected to a hairspring of phosphor bronze having only a few turns.

The other end of the spring is connected to a binding screw. A soft iron cylinder is placed symmetrically inside the coil. The hemispherical magnetic poles produce a radial magnetic field in which the plane of the coil is parallel to the magnetic field in all its positions. A small plane mirror attached to the suspension wire is used along with a lamp and scale arrangement to measure the deflection of the coil.
Working of Moving Coil Galvanometer

Let PQRS be a single turn of the coil. A current I flows through the coil. In a radial magnetic field, the plane of the coil is always parallel to the magnetic field. Hence the sides QR and SP are always parallel to the field. So, they do not experience any force. The sides PQ and RS are always perpendicular to the field.

PQ = RS = l, length of the coil and PS = QR = b, breadth of the coil. Force on PQ, F = BI (PQ) = BIl. According to Fleming’s left-hand rule, this force is normal to the plane of the coil and acts outwards.
Force on RS, F = BI (RS) = BIl. This force is normal to the plane of the coil and acts inwards. These two equal, oppositely directed parallel forces having different lines of action constitute a couple and deflect the coil. If there are n turns in the coil, the moment of the deflecting couple = n BIl – b

Hence the moment of the deflecting couple = nBIA

The suspension wire twists when the coil deflects. On account of elasticity, a restoring couple is set up in the wire. This couple is proportional to the twist. If θ is the angular twist, then, the moment of the restoring couple = Cθ, where C is the restoring couple per unit twist. At equilibrium, deflecting couple = restoring couple nBIA = Cθ

Hence we can write, nBIA = Cθ

I = (C / nBA) × θ where C is the torsional constant of the spring; i.e. the restoring torque per unit twist. A pointer attached to the spring indicates the deflection θ on the scale.
The Sensitivity of Moving Coil Galvanometer

The sensitivity of a Moving Coil Galvanometer is the ratio of the change in deflection of the galvanometer to the change in current. Therefore we write, Sensitivity = dθ/di. If a galvanometer gives a larger deflection for a small current it is a sensitive galvanometer. The current in Moving Coil galvanometer is: I = (C/nBA) × θ

Therefore, θ = (nBA/C) × I. Differentiating on both sides wrt I, we have: dθ/di = (nBA/C).

To sum up, the sensitivity of Moving Coil Galvanometer increases by:

Advantages and Disadvantages of Moving Coil Galvanometer

In Figure 1(a), a conductor is carrying current away from us, and is placed between the poles of a permanent magnet. The right-hand screw rule gives the direction of the flux lines around this conductor as clockwise, thus they reinforce the flux from the magnet above the conductor and oppose the flux from the magnet below the conductor.

The result of the interaction between these fluxes is shown in Figure 1(b). The flux field above the conductor is strong and the lines are compressed. The flux field below the conductor is weak and flux lines are far apart. The tendency for the compressed lines to want to straighten and move further apart will exert a force on the conductor, attempting to push it downwards and out of the field.

If the conductor current is reversed, or the magnet poles are reversed, this force will happen in the opposite direction and will push the conductor upwards.

Where: F = force (newtons, N); B = field flux density (teslas, T )
i = current in conductor (amperes, A); l = length of conductor (metres, m)

The three directions can be related to each other by using Fleming’s left-hand rule, as depicted in Figure 2(b). Stated in words, this rule is:

If the thumb and first two fingers of the left hand are held mutually at right angles to each other, and the forefinger points in the direction of the magnetic field or flux, and the middle finger points in the direction of the current i , then the thumb will point in the direction of the motion.

The strength of a magnetic field is usually measured in terms of a quantity called the magnetic flux density of the field, B. A definition of B requires a consideration of the forces produced by electromagnetic fields.

 When a wire carrying a current is placed in a magnetic field the wire experiences a force due to the interaction between the field and the moving charges in the wire. A very good demonstration is the so-called catapult field experiment in which a wire carrying a d.c. current can be made to move in the field of two flat magnets.

τ=w2Fsinθ+w2Fsinθ=wFsinθ

 Now, each vertical segment has a length l that is perpendicular to B, so that the force on each is F=IlB

. Entering F into the expression for torque yields

τ=wIlBsinθ

.

If we have a multiple loop of N turns, we get N times the torque of one loop. Finally, note that the area of the loop is A = wl; the expression for the torque becomes

τ=NIABsinθ

.

This is the torque on a current-carrying loop in a uniform magnetic field. This equation can be shown to be valid for a loop of any shape. The loop carries a current I, has N turns, each of area A, and the perpendicular to the loop makes an angle θ with the field B. The net force on the loop is zero.

Mainly Lorentz force is responsible for Hall effect. All of we know that when we place a current carrying conductor inside a magnetic field, the conductor experiences a mechanical force to a direction depending upon the direction of magnetic field and the direction of current in the conductor. The electric current means a flow of charge. In metal it is entirely due to the flow of electrons, in semiconductor, it is due to flow of free electrons as well as holes. In semiconductor, holes move in the direction of conventional current and free electrons move in the opposite of the direction of conventional current. As the electrons have charge, they experience a force while flowing through a conductor placed inside a magnetic field. Due to this force, the electrons get diverted towards one side of the conducted during flowing. As the following charges get shifted to one side of the conductor, there may be a tiny potential difference appeared across the cross-section of the conductor. We call this entire phenomenon as hall effect.

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, E_H.
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 (μ_n) and the holes (μ_p), expressed in terms of Hall coefficient is given by,

Where, σ_n and σ_p 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.

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. 

 

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 (δΦ_B) have opposite signs. The formula for Lenz’s law is shown below:

Where:

 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 = 90^o and so sinθ = sin90 = 1). 

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. 

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 ( N_P ) compared to the number of coil turns on the secondary winding ( N_S ).

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.

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

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.

(Refer to part 3)

Where E_b 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 E_b = V – I_aR_a.

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

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.

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

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.

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.

Likewise, when the tip of the vector is vertical it represents the positive peak value, ( +Am ) at 90^o or π/2 and the negative peak value, ( -Am ) at 270^o 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 30^o.

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.

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.

 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. 

 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. 

 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.

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.

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 90^o out-of-phase with each other. But we can use this 90^o 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.

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. 

 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. 

 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: 

 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. 

 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). 

 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.