Section 2-6 - Magnetic Induction & Friends

Faraday's Law of Induction & Lenz's Law

We did a demonstration using a coil of wire hooked up to a galvanometer, which detects any current induced in the wire.  We examined very qualitatively the effects of a magnet on the current in the wire (We can find the corresponding emfs by using Ohm's Relationship):

• Changing the strength of the B-field by bringing the magnet near to, and pulling the magnet away from, the loop, keeping the loop's area and orientation constant (e ~ DB).
• Changing the area of the loop, while keeping the B-field and the orientation of the loop constant (e ~ DA).
• Changing the orientation of the loop with respect to the field, while keeping the area of the loop and the loop's orientation constant (e ~ D(some function of q, which careful experimentation will disclose is the cosine of q)).

Now, there should be one question still bothering you.  What is it?

Try an example.  Suppose that a loop of wire is lying horizontally on the table before you, and that you are bringing a bar magnet down toward it, south pole first.  In which direction wil the induced current flow?

So, in general, as the magnetic field though a loop gets weaker,

the emf will be the same as it would be for a magnet being pulled away from the loop:

What would we do if the flux were changing due to a change in the area or the orientation of the loop?

We need to be able to twist the situation around a bit until it looks like a magnet moving towards or away from the loop. In that case, fewer B-field lines will be passing through the loop to the right, which we can once again simulate by pulling the magnet shown above away to the left. Then, we know that the current induced in the loop will try to form a south pole on the left side of the loop, and so the current must be going down the front side and up the back side of the loop.

Similarly, if we were to tilt the loop, so that fewer B-field lines pass through it, we could simulate that situation also with a north pole on the left being pulled away.

This last concept, that the induced emf will try to act to oppose any change in the system is called Lenz's Law; we'll remind ourselves of it by inserting a negative sign in the relationship developed above, which then appears as
e = ^(-) N df_M/dt.
This is known as Faraday's Law of Induction.

Faraday's Law does not come out of thin air; we can show that it is equivalent to an effect we've seen before.  Consider the very special case of a U-shaped wire situated with its plane perpendicular to a uniform magnetic field, B.  Consider further a light, conductive bar of length l, the ends of which lie on the wire and which is constrained to move along the wires without friction, as shown.

Let's assume that the bar is distance x from the left end and that it is moving to the right with speed v.  If we look at the loop formed by the bar and the three sides of the wire to the bar's left, we see that there is a magnetic flux given by
F_M = BAcosq_B,A = B*(lx)*1.
Faraday's Law then indicates that (since N = 1 here)
e = ^(-) N dF_M/dt = ^(-) d(Blx)/dt = ^(-) Bl d(x)/dt = ^(-) Blv.
The direction of the emf can be obtained this way:  Increasing the area of the loop increases the flux in that same way that increasing the B-field strength would, and we could do that by bringing the north pole of a bar magnet down toward the loop from above the page. Per the discussion above, this would make an induced north pole form just above the loop, and by the RHR, this pole would be formed at that location by a counter-clockwise current.

Now, instead consider the charges residing in the bar.  These are moving through a magnetic field with a speed v, and like all such charges, they will experience a deflecting force given by
F_M = qvB sinq_v,B (RHR)

In this case, the angle is 90^o, and the electrons wil be deflected toward the bottom of the page, while protons will be deflected toward the top of the page.  Since protons are almost 2000 times as heavy as electrons, and because they are connected together by atomic bonds, and because there may well be other outside forces acting on the bulk of this bar which will cancel the magnetic forces on the protons, we shall ignore said forces and concentrate on the forces on the electrons.  The force on charge q's worth of electrons will be
F_M = qvB.
As the electrons work their way from the top of the bar to the bottom, work will be done:
W = Fd cosq_F,d = F_Ml = qvBl.
Since the kinetic energy of the electrons doesn't change (drift velocity is constant), this work can be considered to be an increase in potential energy:
W = DPE (note that there's no negative sign, because the potential energy we're considering is not associated with the magnetic force).
What is the emf? It's the difference in potential between the two ends of the bar, and potential is the potential energy per unit charge:
e = DV = DPE/q = W/q = qvBl/q = Blv.
What's the direction of this emf?  We said that the electrons will experience a force toward the bottom of the page, which means that the current is actually flowing toward the top of the page in the bar (i.e., counter-clockwise around the loop), and so the emf is directed in the same manner.  Quelle surprise, eh?

The point of this is to demonstrate (for a very easy case, anyway) that the emf which seems to appear by magic in a coil is actually due to real forces acting on real charges moving though real (or at least, they are real in our present model) magnetic fields.

In fact, this emf will exist in the bar whether there is a loop connected to it or not.  In that case, the charges feel the magnetic force, same as before, but they can not escape around the loop, so instead, they will collect at the ends of the bar (one end positive and the other negative).  As a result, an internal electric field will be set up, runnng in this case from the top end to the bottom end.  Equilibrium wil be reached when the electric force on a charge is balanced by the magnetic force:
qE = qvB,
E = vB.
There is a potential difference associated with the electric field (here, d = l):
e = DV = ^(-) Ed = E l = [Bv] l = Bvl,
as before.

Just for fun, let's add a resistor to the circuit and investigate what happens to the energy in the system.  We've already decided that there is an emf in the loop, which will drive a current:
I = e/R = Blv/R.
The power dissipated in the resisor will be
P = I²R = B²l²v²/R.
Where does this energy originate?  Since there is a current in the bar, there is also a magnetic force acting (by the RHR) to the left of magnitude IlB.  There may well be magnetic forces on the other parts of the loop, but since the loop is presumably nailed down, those forces will be cancelled by mechanical forces.  This magnetic force should act to slow and eventually stop the bar's motion towards the right.  To keep the bar moving at a constant speed, as was stipulated at the top of this section, we need to apply our own force of magnitude  IlB to the right.

F_Applied =  IlB =  [Blv/R]lB = B²l²v/R.
Correspondingly, what power must be applied?
P = Fv = [B²l²v/R]v = B²l²v²/R.
So, we see that the energy must come form somewhere, either through an external agency doing work or perhaps from the kinetic energy of the bar as it slows.

Self Inductance

Let's examine a system for which it might be fairly easy to calculate L: the solenoid.  Consider a solenoid of N turns with length l and radius r.  If l >>r, we might think that the B-field generated inside the solenoid is approximately the same as that in an infinite solenoid:
B = m_onI =  m_o[N/l] I.
Each loop of the solenoid will experience a magnetic flux; since the magnitude of the B-field is constant all along the cross-sectional area and everywhere perpendicular to the cross-sectional area, the flux is easy to calculate:
F_M = BA = m_o[N/l] I pr².
Now, the emf is given by Faraday's Law:
e = ^(-) N dF_M/dt =  ^(-) N d[m_o[N/l] I pr²]/dt.
Since the current is the only quantity changing here, this can be rewritten as:
e = ^(-) [m_oN²pr²/ l ] dI/dt.
Comparison of this result to our expected form reveals that
L_solenoid = m_oN²pr²/l.

Applications of inductors include:
power spike and noise suppression (the rapidly changing currents associated with the spikes create a reverse emf in the inductor), high frequency filtres (same effect), and automobile ignition systems (rapidly disconnecting an inductor from a 12.6V driven current can produce a several thousand volt emf across the spark plug gap).

Mutual Inductance

Here is a useful example of mutual inductance.  Consider two coils with N₁ and N₂ turns, respectively, each wrapped around a frame made of soft iron, which is easily magnetized.

We'll drive a current through coil 1 with a varying voltage, V₁.  Since V₁ is varying in time, the current I₁ will do so also.  I₁ causes a magnetic field in coil 1, some of the lines of which pass through coil 2.  Now, the soft iron has the property of re-directing these magnetic field lines through itself, so that a very large fraction of them will pass through coil 2; in fact, we'll assume that 100% of them do so, although in real life, the figure is more like 96%.  Now, when I₁ changes, the flux through coil 2 will induce an emf, perhaps causing a current I₂ in coil 2, which will in turn cause its own magnetic field which also passes through both coils and is changing with time (although, the existence of I₂ is not necessary to this derivation).  At any rate, each loop in each of the two coil experiences a flux due to the fields generated by itself and by the other loops.  These fluxes are the same (regardless of their sources), since one interpretation of the flux is that it is proportional to the number of B-field lines passing through, and we stipulated that all lines pass through each coil.  So,
F_M1 = F_M2.
Now, if two quantities are always equal, then their time rates of change must also always be equal, so,
dF_M1/dt = dF_M2/dt.
Now, each coil will have an induced emf described by Faraday's Law:
e₁ = ^(-) N₁dF_M1/dt
e₂ = ^(-) N₂dF_M2/dt.
and so
e₁/N₁ = ^(-)dF_M1/dt = ^(-)dF_M2/dt = e₂/N₂
e₁/N₁ = e₂/N₂
By Kirchhoff's Loop Law, V₁ + e₁ =0 and V₂ + e, =0, so
V₁/V₂ = N₁/N₂.
So, we see that we can use such a device (called a transformer) to step-up or step-down the amplitude of a time varying voltage.  We did a demonstration with a transformer with a turn ratio of 10:1.  By applying a 10V sinusoidal voltage to the primary coil, we could obtain a 1 volt output across the secondary coil, or, by reversing the coils, a 100 volt output.

You may see transformers every day.  The power company produces electricity at about 1000V, then steps it up to 100,000V for transmission over long distances (thousands of miles).  As the lines get closer to the consumer, other transformers lower the voltage back down to the thousands, then to the hundreds.  Normally, each house has a transformer just outside (either on a pole or often just sitting on the front lawn) which lowers the voltage to 220V.  Why not lower it right away to the familiar 110V? Typically, electric ranges and air conditioners run on 220V.

RC Circuits

The quantity RC is often referred to as the capacitive time constant of the circuit,
t_C = RC,
which can be thought of as the time necessary for some quantity to decay to 1/e of its initial value.  Conceptually, it is similar to the notion of a half-life, except that is the time necessary to fall to 1/2 of the initial value.
So, we could write that
q(t) = e_BC (1 - e^{- t/t}C);     I(t) = [e_B/R]e^{- t/t}C     (Charging from empty case)
q(t) = e_BC e^{- t/t}C;     I(t) = - [e_B/R]e^{- t/t}C     (Discharging from full case).

LR Circuits

If we wait for a very long time, the current will achieve an ultimate value of
I_Final = e_B/R.
At that time, let's quickly switch the circuit to remove the battery.

Since the inductor will oppose the drop in the current, it will work to keep the current flowing.  Kirchhoff's Loop Law in that case is:

IR =  - L ddI/dt, 
the solution to which is
I(t) = [e_B/R]e^{- tR/L}. 

Here, we can define the inductive time constant t_L as L/R, so that these equations become:
I(t) = [e_B/R](1 - e^{- t/t}L),
and
I(t) = [e_B/R]e^{- t/t}L.

Consider this.  In each of the cases discussed above, the RC and the LR circuits, current flowed even after the battery was disconnected, even if it decreased pretty rapidly towards zero.  Since there was current flowing through a resistor, we know that  energy was being dissipated as heat.  From where did this energy come if not from the battery?  We'll, in the case of the RC circuit, we've already discussed how energy can be stored in a capacitor:
U_C = ¹/₂ [1/C] Q².
So, we might surmise that energy had to have been stored in the inductor as well.  We will assert now (and show later) that the energy stored in the magnetic field in an inductor is
U_L = ¹/₂ L I².