ELECTROMAGNETIC FIELDS

Maxwell's Third Equation. In 1831 Michael Faraday discovered that a changing magnetic field produces an electric current in a conductor. During his experiments with a coil connected to a galvanometer and a magnet he noticed that when the magnet was trust into the coil, the galvanometer showed a momentary current. When the magnet was at rest inside the coil nothing happened, but when the magnet was withdrawn the galvanometer again showed a momentary current in the opposite direction. Faraday concluded from this experiment that when the magnetic field inside the coil is changing because of the motion of the magnet, then a current is produced in the circuit consisting of the coil and galvanometer. When the magnetic field is no longer changing because the magnet is at rest, the current ceases. This phenomenon is known as electromagnetic induction and currents produced in this way are called induced currents.

Faraday's Law of Electromagnetic Induction. Because electric current was not clearly understood as the electrical charges in motion at the time that Faraday discovered electromagnetic induction in 1831, Faraday expressed the phenomenon of electromagnetic induction, not as an induced current, but as an induced EMF (ElectroMagnetic Force). He said that this induced EMF produced by changing magnetic field is directly proportional to the time rate of change of the lines of flux linking the coil and to the number of turns in the coil. That is,
EMF ∝ N(Δφ/Δt), (1)
where EMF is the average electromagnetic force measured in volts, N is the number of turns in the coil, and is the change of lines of flux occurring in the time interval Δt. The changing lines of flux is measured in webers and the time interval in seconds, so that ΔφΔt is measured in webers per second. The inducted EMF produces a voltage difference between the ends of the coiled wire, so that Faraday's Law can be written as
V_i = -N(Δφ/Δt), (2)
where V_i is the induced voltage difference across the coil. Faraday's Law can also be expressed as a differential equation in terms of instantaneous time rate of change of magnetic flux. That is,
V_i = -N(dφ/dt). (3)
The minus sign is placed in these equations because Lenz's Law.

Lenz's Law. In 1833 Heinrich Friedrich Emil Lenz, a Russian physicist, as a result of many original experiments on electromagnetic induction, formulated a very important law concerning induced EMF's. He found that in all cases of electromagnetic induction, the current which flows as the result of an induced EMF is in such a direction that the magnetic field produced by the current opposes the cause of the induced EMF. This is known as Lenz's Law and it may be illustrated in the following manner. Consider again the coil connected to a galvanometer into which a magnet is trusted. As the north pole of the magnet moves into the coil, an EMF is induced in the winding and a electric current is induced in the circuit that detected by deflection of the needle of the galvanometer. Now this current induced in the coil is in such a direction that its electromagnetic field produces a north pole at the end of the coil into which the magnet is entering. The north pole of the coil repels the north pole of the magnet and vice versa, tending to stop the inward motion of the magnet, the cause of the induced EMF. If now the north pole of magnet is withdrawn from the coil, the induced current is reversed in direction and produces a south pole at the end of coil out of which the north pole of magnet is withdrawn. The south pole of the coil attracts the north pole of the magnet and vice versa, tending to stop the outward motion of the magnet, the cause of the induced EMF. This is reason that the minus sign is put into equation (2) that expresses Faraday's Law of Electromagnetic Induction. The minus sign indicates that the direction of the EMF that is induced into the electrical circuit by the changing flux linking the circuit is in such a direction as to oppose the changing flux and its cause.

Lenz's Law is one of the many forms of the law of the conservation of energy. A brief consideration of electromagnetic induction from the point of view of energy will reveal its basis in the law of the conservation of energy. Work must be done and thus energy expended to generate an EMF. If this EMF was not in such a direction as to oppose the cause of the induced EMF, no work would have to be done to generate this induced EMF. Electrical energy could be gotten without the expenditure of any energy. Lenz's Law is therefore another way of saying that work must be done against a magnetic field and thus energy expended in order to get energy in the form of electrical energy.

Induced Electromotive Force. Faraday also investigated all the possible ways of producing induced currents that he summarized in his law. There are several ways of producing an induced current in an electric circuit. An electromagnetically induced current may be produced by a coil moving near a stationary magnet as well as by a magnet moving into and out of a stationary coil. It makes no difference whether the coil alone moves, or the magnet alone moves, or they both move. It is the relative motion of the coil and magnet that produces the current. If the relative motion ceases, the current ceases.

Consider the motion of a conductor in a magnetic field. A current is induced electromagnetically in a conductor because the magnetic field exerts a force on every charge within the conductor as it moves perpendicular to the field. When an electric charge q moves with a velocity v perpendicular to a magnetic field of uniform flux density B, a force F is exerted upon the charge directed perpendicular to the plane determined by v and B. The magnitude of this force may be found using Lorentz Force Law.
F = q(v × B).
As a conductor moves perpendicular to a magnetic field, the charges in the conductor are also moved perpendicular to the field. There is therefore exerted on them by the magnetic field a force whose direction is along the length of the conductor. Since the charges are free to move in a metallic conductor, the force exerted by the magnetic field moves the charges along the conductor from one end to the other, thus producing an electric current, if there is a complete circuit. If there is not a complete circuit, a potential difference exists between the ends of the conductor by a piling up an excess of the charge at one end and a deficiency at the other. Thus a moving conductor in a magnetic field is a source of electromotive force.

The electromotive force (EMF) developed within such a moving conductor in a magnetic field is defined as the work per unit charge done in moving any charge q along the length L of the conductor which is in the magnetic field with a flux density B. That is,
EMF = W/q.
Now the work W done by a force F is equal to the product of the force times the displacement in the direction of the force. Since by Lorentz Force Law the force F exerted on the charge q is equal to the product of magnitude of the charge q times the velocity v of the charge times the flux density B of the magnetic field, then the work done on the charge q by the force F in moving it a distance L is given the following formula.
W = qvBL.
Now if each side of this equation is divided by the charge q, then we get
W/q = vBL.
Since by definition the work per unit charge is equal to the electromotive force developed within the conductor, then this equation gives us an equation for electromotive force.
EMF = vBL, (4)
where in the RMKS system of units EMF is measured in volts, v is measured in meters per second, B is in webers per square meter, and L is the measured in meters. This is a form of the Generator Principle.

The conventional direction of the induced current and thus the polarity of the induced EMF (The conventional direction of current through a source of EMF is from negative to positive.) may be determined by the right hand rule. If the thumb and the first two fingers of the right hand are pointed in mutually perpendicular directions, the first finger points in the direction of the magnetic field, the thumb point in the direction of the motion of the conductor, then the second finger will point in the conventional direction of the current through the source. Thus if the direction of the magnetic field or of the motion of the conductor is reversed, then the conventional direction of the current will be reversed also.

At ordinary temperatures the free charge in a conductor encounters resistance when they move through the conductor. This means that, when a steady current is maintained in the conductor, electrical energy is continually being dissipated as heat energy. This requires a device in the circuit to take energy from some external source and continually convert it into electrical energy. Such a device must produce an electric field E whose line integral around the closed circuit is not zero. This electric field is required to do work to move the charges against the resistance forces. The source of such a field is said to be the seat of electromotive force. This electromotive force (EMF) around a closed circuit is defined as
EMF = ∫E·dλ. (5)

The Law of Magnetic Induction. Since the total magnetic flux φ passing through the cross-sectional area A is φ = B · A, where B = magnetic flux density. The time derivative of the flux is
dφ/dt = B · (dA/dt) + A · (dB/dt). (6)
Substituting into equation (3), we get
V_i = -N(dφ/dt) = -N[B · (dA/dt) + A · (dB/dt)] = -NB · (dA/dt) - NA · (dB/dt).
If B is constant, then dB/dt = 0 and NB · (dA/dt). (7)
If A is constant, then dA/dt = 0 and NA · (dB/dt). (8)

The Generator Principle. Consider the first case where B is constant. In a short interval of time or differential time, the conductor of length L moves a differential distance ds, sweeping over a differential area
dA = ds × L.
The time rate of change of area is
dA/dt = (ds/dt) × L = v × L,
since L is constant independent of time and ds/dt is the conductors velocity v.
Substituting this into equation (6), we get
V_i = -NB · (dA/dt) = -NB·(v × L) = N(v × B) · L, (9)
where according to identities of Vector Algebra, B · (v × L) = (B × v) · L = -(v × B) · L.
If the direction of v is at an angle β to respect to the direction of the magnetic flux density B and the conductor of length L cutting the magnetic flux, then
V_i = NBLv sin β. (10)
Equation (9) or (10) is known as the Generator Principle. The essential elements in the operation of an electrical generator consists of a rectangular coil having N turns rotating in a magnetic field of a uniform flux density B. The coil rotates about an axis that is perpendicular to the magnetic field. The two sides of the coil parallel with axis of the coil has a length l and a width through the axis is w. The two ends of the coil are connected to two slip rings that are concentric with the axis of the coil and rotate with the coil. The slip rings are insulated from each other and are connected to an external circuit by stationary brushes that make continuous contact with the slip rings. We will use the Generator Principle to derive an equation for the generator voltage appearing across the external circuit. At any instant of time the plane of the coil is oriented as some arbitrary angle θ with respect to the magnetic field. The two sides of the coil parallel to the axis of coil move in a direction that makes an angle β with respect to the lines of magnetic flux. Thus the angle between v and B is β and magnitude of the cross product of v and B is vB sin β. If we assume that the coil is driven counterclockwise with a constant angular velocity ω, then the velocity v = ωr, where r is radial distance to the axis of the coil and is equal to one half of the width of the coil, that is, 2r = w. The total length L of coil cutting the lines of magnetic flux is two times the length of the coil, that is, L = 2l. The other two sides of the coil through the axis of the coil do not cut the magnetic lines of flux. Using the Generator Principle, equation (10), we get
V_i = NBLv sin β = NB(2l)[ω(½w)]sinωt = NBωA sin ωt = ωANB sin ωt,
where A = lw is the cross-sectional area of the coil and β = ωt, since ω = dβ/dt. Now the coefficient of the sine function in this equation is ωANB and is equal to the maximum voltage induced in the coil. Hence,
V = V_maxsin ωt. (11)
The voltage induced in the coil varies with time according to a sine function and is alternately positive and negative. This is called an alternating voltage. When an alternating voltage is impress across an external electrical circuit it produces an alternating current. The alternating current generator is called an alternator; the rotating coil is the simplest form used in practical alternators.

Mutual Induction. In a earlier section two ways of producing an induced EMF and thus an induced current were described:
(a) by moving a magnet in the vicinity of a stationary coil, and
(b) by moving a coil in the vicinity of a stationary magnet.
An EMF may also be generated with both the coil and the magnet stationary. Consider two coils that are wound on the same iron core. One coil, called the primary, is connected across a D.C. source with a rheostat placed in series with coil to vary the current. The other coil, called the secondary, has a voltmeter connected across its terminals. As the current is varied in the primary, an EMF is induced in the secondary by the flux linking the secondary varying with the change of current in the primary. In this case neither the coil nor the iron core moves; the magnetic field about the coil is changed by varying the current in the primary. This changing magnetic field induces the EMF in the secondary. This phenomenon is known as mutual induction, and the EMF generated in the secondary is known as the electromotive force of mutual induction. It may be calculated by Faraday's Law. Both the induction coil and the transformer operate on this principle of mutual induction. Since by Faraday's Law,
V_i in the primary coil or , and
in the secondary coil or , then
(V₁/N₁) = (V₂/N₂)
or
(V₁/V₂) = (N₁/N₂). (12)
This is known as the Transformer Principle.

Self-Induction. When the current is varied in a coil, not only is there an EMF induced in a neighboring coil or conductor, but there is an EMF induced into the coil itself; that is, in the coil itself in which there is varying current. The flux of a magnetic field about a current carrying coil is not only linked with a neighboring coil but with each turn of the coil producing the magnetic field. As the current is varied in the coil, the flux linking each turn varies with the change of the current in the coil. Thus the changing magnetic field about a current-carrying coil induces an EMF into each turn of the coil itself. This EMF is called the electromotive force of self-induction and the phenomenon is known as self-induction. This EMF may be calculated by Faraday's Law. Using equation (8) and assuming that the area A is constant, we get
V_i = NA·(dB/dt) = -NA(μndI/dt) = -(nI)Aμn(dI/dt) = -(μn²lA)(dI/dt) = -l(dI/dt),
where B = μnI and dB = μndI,
N = number of turns in the coil = nl, with n = turns per unit length and l = length of coil, and
L = μn²lA = self-inductance measured in webers per ampere or henry. This unit is named in honor of the American physicist, Joseph Henry, who, while a schoolteacher at Albany Academy, discovered mutual inductance independently of Faraday.

Hence, the voltage across the coil is directly proportional to the time rate of change the current in the coil.
V_i = -L(dI/dt). (13)
If the current is increasing so that dI/dt is positive, the coil voltage is positive. If the current is decreasing, dI/dt is negative and the coil is coil voltage is negative.
Since V_i = -L(dI/dt) = -N(dφ/dt), then
L(dI/dt) = N(dφ/dt).
Therefore, LdI = Ndφ.
Integrating this differential equation between the limits the current from 0 to I and the flux from 0 to φ,
LI = Nφ = λ, (14)
where Nφ = λ is called the Magnetic Flux Linkages.
Solving this equation for L, we get
L (Nφ/I) = λ/I. (15)
In this equation (15) inductance is defined as the flux linkages per ampere of electrical current. The symbol L used for inductance is associated with the L that is the first letter of the word "Linkage"

Series Self-Inductances. Consider a series connection of two inductors. The problem is to determine how these two inductances combine to produce the total equivalent inductance of the circuit. The total voltage across the two inductors is the sum of the two coil voltages. That is,
V = V₁ + V₂.
The same current is present in both coils so that the two voltages are
V₁ = -L₁(dI/dt)
and
V₂ = -L₂(dI/dt).
Hence, since I = I₁ = I₂, then
V = V₁ + V₂ = -L₁(dI/dt) -L₂(dI/dt) = -(L₁ + L₂)(dI/dt) = -L(dI/dt) .
Therefore, the total inductance is
L = L₁ + L₂. (16)
Thus, the total inductance of a series combination of inductors is the sum of the separate inductances.

Parallel Self-Inductances. Consider two parallel connected inductors. The problem is to determine how these two inductances combine to produce the total equivalent inductance of the circuit. The total voltage across each coil are equal, that is, V = V₁ = V₂, so that
V = -L₁(dI₁/dt) = -L₂(dI₂/dt). (17)
Solve each of these equations for the time derivative of the current in each inductor. This gives
(dI₁/dt) = -(V/L₁)
and
(dI₂/dt) = -(V/L₂). (18)
Now the sum of the current into a junction is equal to the sum of currents out of the junction. That is,
I = I₁ + I₂. (19)
Taking the derivatives of both sides of this equation with respect to time, we get
(dI/dt) = (dI₁/dt) = (dI₂/dt). (20)
Substituting the equations (18) into this equation (20), we get
(dI/dt) = [-(V/L₁)] + [-(V/L₂)] = -V[(1/L₁) + (1/L₂)]. (21)
Now the total equivalent inductance of the parallel circuit is defined by
V = -L(dI/dt)
and solving this for time derivative, we get
(dI/dt) = -(V/L).
Equating this with equation (21) and canceling the V, we get
(1/L) = (1/L₁) + (1/L₂). (22)
Thus, the total inductance of parallel connected inductors is equal to the reciprocal of the sum of the reciprocals of the individual coil inductances.

D.C. Series RL Circuit. Consider the Direct Current circuit that consists of a coil with a self-induction L and a resistance R in series and connected to a battery V_e and with a switch when closed the circuit has a constant current i = V_e/R flowing in it and when open disconnects the battery and closes a short circuit path across the coil and resistance. When the switch is quickly closed, the current does not rise immediately to its maximum value I = V_e/R, but increases gradually, steadily approaching the maximum value when the battery EMF voltage V_e is applied. Let us find the equation that describes how the current i varies in time t from the moment the switch is quickly closed until the current reaches it maximum value I. As the current flows through the coil, the magnetic flux through the coil changes and an EMF is induced in the coil that, by Lenz's Law, opposes the applied battery's voltage. That is,
V_e - L(di/dt) = iR.
The term L(di/dt) is on the left side of this equation as an EMF and may be moved to the right side of this equation, regarding it as voltage drop. Separating the variables i and t, we get
di/[i - (V_e/R)] = -(R/L)dt.
Integrating this differential equation between the limits of i from 0 to i as t is from 0 to t, we get
ln[i - (V_e/R)] - ln(V_e/R)] = -(Rt/L), or
ln[i - (V_e/R)] = ln(-V_e/R)] + lne^-Rt/L,
since lne^-Rt/L = -(Rt/L).
Hence, solving for i as a function of t, we get
i - (V_e/R) = -(V_e/R)e^-Rt/L or,
since I = V_e/R,
i = I[1 - e^-Rt/L]. (23)
The current grows exponentially in time with a characteristic time constant τ = L/R. That is, after the time L/R has elapsed, the current i differs from its final value I = V_e/R by 1/e or 37 per cent.

If the switch is opened in the RL circuit, the current does not drop instantaneously to zero. Although the battery's EMF no longer drives charges around the circuit, the changing current through the coil creates an EMF which attempts to keep the current from dropping, acting like the battery's EMF. That is, the inductor's EMF is in the same direction as was the battery's EMF before the switch was opened. That is,
-L(di/dt) = iR.
To find how the current i varies with time t, let us rearrange this equation, separating the variables and then integrate that differential equation from t = 0 to t as the current i goes from I to zero.
di/i = -(R/L)dt, or
∫di/i = -(R/L)∫dt, or
ln i - ln I = -(Rt/L), or
ln(i/I) = -(Rt/L), or
i = I e^-Rt/L.
Thus the current decays exponentially in time with a characteristic time constant = L/R. That is, after the time L/R has elapsed, the current i drops from its initial value I = V_e/R to 1/e or 37 per cent of initial value.

The Energy Stored in Magnetic Field. The energy stored in a magnetic field of an inductor can be calculated from the definition of electric power as the time rate change that the energy is stored in the inductor. That is, since
P = dW/dt, then dW = Pdt = VIdt,
where the electrical power is equal to the product of the voltage across and current in the inductor.

When a current builds up in an inductor, an external source of EMF must do work against the induced back voltage. Hence, the differential amount of work dW done against the back voltage in the time interval dt is
dW = VIdt, (24)
where the positive sense is the direction of the current.
By equation (12) the back voltage is the self-induced voltage. That is,
V = -L(dI/dt).
Hence, substituting this into equation (24), we get
dW = -[-L(dI/dt)](LIdt) = LI(dI/dt)dt = LIdI.
Integrating this equation between the limits from 0 to W and from 0 to I, we get
W = ½LI².
Let U_m be the energy stored in the magnetic field of an inductor and is equal to the work done against the induced back voltage. Hence,
U_m = ½LI². (25)

Density of Energy Stored in Magnetic Field. Let us now compute the density of the energy stored in the magnetic field. Since L = μn²lA, then
U_m = ½LI² = ½(μn²lA)I² = ½(B²/μ)lA,
since B = μnI and is the uniform magnetic flux density of a long solenoid and lA is the volume of the magnetic field within the solenoid. Since energy density is the ratio of energy in a field to the volume of the field, then
u_m = U_m/lA = U_m(1/lA) = [½(B²/μ)lA](1/lA) = ½(B²/μ), (26)
where u_m is the magnetic energy density and in RMKS is measured in joules per meter³.

Faraday's Law and the Electric Field. According to Faraday's Law the changing magnetic flux creates an electric field which drives electric charges around the conducting circuit. Does Faraday's Law imply that a changing magnetic flux create electric field in space when there are no electric charges present to be set in motion? The answer is Yes. Faraday's Law states this fundamental effect of electromagnetic induction: a changing magnetic flux generates an electric field. This electric field generated by a changing magnetic flux is quite different from the electric field originating from an electric charge. Of course, the electric field E is still by definition the electric force per unit positive point charge, but the electric field generated by the changing magnetic flux is nonconservative, and the line integral of the electric field intensity E about the closed circuit is not zero. We will take the following as a general definition of an induced EMF.
EMF = ∫E · dλ ≠ 0. (27)
Now this line integral depends upon the rate at which the magnetic flux changes through the closed path λ. According to Faraday's Law, the EMF induced in any path is equal to the negative of the time rate of change of magnetic flux through the area enclosed by the chosen path, that is, by equation (3) with N = 1,
EMF = -(dφ/dt). (28)
Equating the right sides of equations (27) and (28) we can write the law of electromagnetic induction in the following general form.
EMF = ∫E · dλ = -(dφ/dt). (29)

Now since the total magnetic flux φ crossing a surface S is
φ = ∫∫_SB · dS, (30)
where B is the magnetic flux density that is defined as the number of lines of flux per unit area through a surface S whose normal is in the direction of B and dS is an element of the surface S. Substituting this into equation (29), we get
EMF = ∫E · dλ = -(d/dt)∫∫_SB · dS.
If the magnetic flux density B varies with time but the area S enclosed does not, then
EMF = ∫E · dλ = -(d/dt)∫∫_SB · dS = -∫∫_S(∂B/∂t) · dS.
Note that here we write ∂B/∂t rather than dB/dt because B is a function of the coordinates as well as of time, and we want here only its variation with time at a fixed point, not its variation due to the moving the circuit to B that has a different value.
EMF = ∫E · dλ = -∫∫(∂B/∂t) · dS, (31)
where S is the surface whose periphery is λ. This is the integral statement of Faraday's Law. It applies to any closed path, whether or not a conducting wire coincides with the path.

Curl of E. Using the integral statement of Faraday's Law, equation (31), and applying Stokes' Theorem from Vector analysis, we get
∫∫_S(∇ × E) · dS = -∫∫_S(∂B/∂t) · dS. (32)
Since both integrals refer to the same arbitrary surface S, then the integrands may be equated, and we get
∇ × E = -(∂B/∂t). (33)
This is the differential, or point statement, of Faraday's Law. It is also Maxwell's Third Equation of his theory of electromagnetic fields. It reduces to ∇ × E = 0 for the special case of a steady or static electric field.