Important points

• Changing magnetic flux induces acurrent in a closed circuit.

• The direction of the induced currentflow is such that it opposes the changein magnetic flux (Lenz’s law).

Important equations

• Magnetic Flux (uniform field)

ΦB = B⊥A = BA cos φ

• Faraday’s law

E = N |∆ΦB/∆t|

• Motional emf

E = vBL

Figure 1: Magnetic flux: ΦB = BA cos φ.1 Magnetic flux has it’s own special SIunit the Weber, or Wb. 1Wb ≡ 1T·m2.

PHYS 202 Notes, Week 6Greg Christian

February 23 & 25, 2016Last updated: 02/25/2016 at 12:36:40

This week we learn about electromagnetic induction.

Magnetic Induction

This section deals with magnetic induction, the phenomenon by whichchanging magnetic fields can induce currents in circuits. This meansthat changing magnetic fields are an additional source of electromotiveforce, or emf.

First off, let’s introduce the concept of magnetic flux, which is en-tirely analogous to the electric flux we already saw.

Magnetic Flux

As illustrated in Figure 1, magnetic flux, ΦB is the amount of magneticfield passing through the perpendicular component of a surface witharea A.1 Mathematically, this means

∆ΦB = B cos φ∆A. (1)

For a uniform field, this becomes

ΦB = BA cos φ, (2)

or defining B⊥ ≡ B cos φ,

ΦB = B⊥A. (3)

Note that for a surface parallel to the field, φ = 90◦ and ΦB = 0.

Faraday’s Law

As mentioned, changing magnetic fields can induce a current, or anemf, in a circuit. It turns out that the induced emf E is directly pro-portional to the change in magnetic flux per unit time. This is calledFaraday’s Law. Mathematically, Faraday’s law reads:

E =

∣∣∣∣∆ΦB∆t

∣∣∣∣ , (4)

where ΦB is the flux through the area of the circuit.

phys 202 notes, week 6 2

Equation (4) is the expression for the emf induced in a single closedloop. For a coil with N identical terms, it becomes

E = N∣∣∣∣∆ΦB

∆t

∣∣∣∣ . (5)

The change in magnetic flux per unit time can occur for differentreasons. For example, you could have a stationary current loop andsome magnetic field whose strength varies with time. Alternatively,you could have a constant magnetic field in some region of space anda loop being constantly moved into the field region. In this case, thearea of the loop is changing, leading to an overall change in flux.

Lenz’s Law

Faraday’s law only deals with the magnitude of the emf (and in turn,current) generated by a changing ΦB. To figure out the direction of theinduced current, we turn to Lenz’s Law, which states:

The current or emf induced by a changing magnetic flux is always in the direc-tion that opposes that change in flux.

What does this mean exactly? Recall that moving charges (currents)are the source of magnetic fields. So the induced current in some loopwill itself create a magnetic field. The direction of the current willalways be such that the newly created field opposes, or counteracts,the changing flux creating it. Figure 2 illustrates this for a few differentexample cases.

Figure 2: A few examples of Lenz’s law.

phys 202 notes, week 6 3

Figure 3: Motional emf.

Important points

• Mutual inductance M characterizesthe induced-emf interaction of twocoils.

• Self inductance L characterizes theself-induced emf in a single coil.

Important equations

• Mutual Inductance

M = |N2ΦB2/i1|= |N1ΦB1/i2|

E1 = M |∆i2/∆t|E2 = M |∆i1/∆t|

• Self Inductance

E = L |∆i/∆t|

• Transformers

V2/V1 = N2/N1

• Energy stored in an Inductor

U = LI2/2

Motional emf

A related concept to Faraday/Lenz’s law is motional emf. This is illus-trated in Figure 3, explained as follows:

• Consider a conducting metal rod moving in a uniform magneticfield, with a velocity #»v that’s perpendicular to the field ( #»v to theright,

#»B into the page).

• The magnetic field exerts an upwards force on the free positive chargesin the conductor.

• The magnetic force causes positive charge to collect at the top of therod and negative charge to collect at the bottom.

• The charge differences create an electric field which exerts a down-ward force

#»Fe on the charges.

• The magnetic and electric forces balance, meaning that

Fe = FB (6)

qE = qvB (7)

E = vB. (8)

• The potential difference between point a at the top of the rod andpoint b at the bottom of the rod is equal to Vab = EL. Thus

Vab = EL (9)

= vBL. (10)

Now what would happen if the rod was attached to a circuit loopas in Example 21.4? The potential difference Vab would act as an emffor the circuit with magnitude

E = vBL, (11)

just as was figured out from Faraday’s law in the example. The direc-tion of the emf will cause current to flow from point a to point b. Thiscorresponds to counter-clockwise in the loop of Example 21.4, a resultthat’s entirely consistent with Lenz’s law.

Mutual/Self Inductance and Transformers

Mutual Inductance

When two coils are placed together, a changing current in one of thecoils will induce an emf in the other one. This coupling phenomenonbetween coils is referred to as mutual inductance.

An illustration of mutual inductance is shown in Figure 4. We havetwo coils, coil 1 and coil 2. Coil 1 has N1 turns and coil 2 has N2.

phys 202 notes, week 6 4

2 The SI units of inductance are theWb/A, or henry, H.

3 Or 220V in much of the world.

Coil 1 carries some current i1 that is varying with time. This currentproduces a magnetic field which passes through the area of coil 2,creating a flux ΦB2. As i1 changes (with time), so does B1, and in turnΦB2. This generates an emf in coil 2 given by

E2 = N2∆ΦB2

∆t. (12)

Similarly, we could have a situation where coil 2 carries some cur-rent i2 and creates a flux ΦB1 in coil 1. By symmetry with the first case,we get

E1 = N1∆ΦB1

∆t. (13)

It turns out that we can define a quantity called the mutual induc-tance which relates the two cases (either a current in coil 1 inducing anemf in coil 2 or vice-versa). The mutual inductance2 is given by

M =

∣∣∣∣N2ΦB2

i1

∣∣∣∣ = ∣∣∣∣N1ΦB1

i2

∣∣∣∣ . (14)

From this, we can re-write Eqs. (12) & (13) as

E2 = M∣∣∣∣∆i1

∆t

∣∣∣∣ (15)

and

E1 = M∣∣∣∣∆i2

∆t

∣∣∣∣ . (16)

Figure 4: Mutual inductance.

Transformers

Mutual inductance can be taken advantage of to build a device called atransformer, which can change the voltage levels in alternating-current(AC) circuits (i.e. circuits where the current is constantly changing).They are incredibly useful in real life, for example taking the∼ 100, 000volts used in power transmission lines and turning them into the 120

volts3 delivered in your household plugs.

phys 202 notes, week 6 5

Figure 5: Self inductance.4 Like mutual inductance, L has units ofhenry, or Wb/A.

A transformer is made up of two coils as discussed in the last sectionon mutual inductance, with the same flux, ΦB, in each coil. The inducedemfs are then defined by

E1 = N1

∣∣∣∣∆ΦB∆t

∣∣∣∣ (17)

E2 = N2

∣∣∣∣∆ΦB∆t

∣∣∣∣ . (18)

Thus E1 and E2 are related by

E2

E1=

N2

N1.

More generally, we can talk about the voltage levels “transformed” bya transformer:

V2

V1=

N2

N1. (19)

Self Inductance

When only a single coil is involved, we get a phenomenon called selfinductance. This occurs because the field produced by a single coil willpass through its own area, creating a magnetic flux, and, in turn a selfinduced emf. This is illustrated in Figure 5.

In this case, we can define a term called the self inductance4, or L,according to the following equation:

N |ΦB| = L |i| . (20)

If ΦB and i change with time, then

N∣∣∣∣∆ΦB

∆t

∣∣∣∣ = L∣∣∣∣∆i∆t

∣∣∣∣ . (21)

And combining Eqs. (20) & (21), we get the equation for the self-induced emf:

E = L∣∣∣∣∆i∆t

∣∣∣∣ . (22)

The direction of the induced emf is found from Lenz’s law. Remem-ber that the cause of the induced emf is changing current in the circuit.Hence the direction of the induced emf always serves to oppose thatchange.

The inductance of a coil is a constant that depends on its geometricalproperties: size, shape, and number of turns. In circuits, we can designa specific element to have a particular inductance. This circuit elementis called an inductor. We’ll learn more about these in a bit.

phys 202 notes, week 6 6

Important points

• Inductors are frequently used in cir-cuits.

• Current in RL circuits has an expo-nential dependence with time.

• Current (and charge) in RC circuitsoscillates with a characteristic fre-quency.

Important equations

• R-L Circuit

i = E/R[1− e−(R/L)t

]• L-C Circuit

ω =√

1/(LC)

Figure 6: An RL circuit with constant(battery) emf.

Magnetic Field Energy

Similar to capacitors, inductors can store energy. Except now the en-ergy is stored in the form of magnetic fields rather than electric fields.The potential energy stored in an inductor with inductance L carryingcurrent I is given by

U =12

LI2. (23)

Doing some math, we can work out that the energy density, u, (po-tential energy per unit volume) depends on the magnetic field by

u =12

B2

µ0. (24)

This is entirely analogous to the energy density in a capacitor:

ucapacitor =12

ε0E2. (25)

Inductors in Circuits

RL Circuits

Inductors are typically used as circuit devices. Figure 6 shows an ex-ample of an RL circuit, that is, a circuit containing a resistor, an induc-tor, and a constant emf source (like a battery).

When we close the switch in the circuit in Figure 6, the currentvaries with time according to:

i =ER

[1− e−(R/L)t

]. (26)

Why is this? The reason comes from Kirchhoff’s loop rule, which statesthat the sum of potential differences along the loop must equal zero:

E − iR− vL = 0, (27)

where vL is the potential drop across the inductor (from b to c in thedrawing). From Eq. (22), this is equal to

vL = L∆i∆t

. (28)

Thus,

E − iR− L∆i∆t

= 0. (29)

Using the techniques of differential equations, Eq. (26) can be derivedfrom Eq. (29). However, this is outside the scope of this course.

Let’s take note of the “immediate” (t = 0) and “long time” (t→ ∞)

behavior of LC circuits. At t = 0, Eq. (26) becomes

i = 0. (30)

phys 202 notes, week 6 7

Figure 7: Time dependence of an LC cir-cuit.

Initially, there is no current flowing through the circuit; the inductoreffectively acts like an open circuit element. At long times, Eq. (26)becomes

i =ER

. (31)

The circuit looks like we only have a resistor connected to a battery.The inductor is behaving just like a “short,” or a piece of ideal con-ducting wire.

As with the RC circuits we saw a couple of chapters ago, we candefine a time constant by

τ =LR

. (32)

A plot of current vs. time for an LC circuit is shown in Figure 7.

LC Circuits

Another type of circuit involving inductors is called an LC circuit, con-sisting of a capacitor and an inductor. In this arrangement, the currentand the charge on the capacitor oscillate. Figure 8 illustrates what’sgoing on:

Figure 8: Diagram of an LC circuit.

1. Begin with the capacitor fully charged to some value Qmax and zerocurrent flow. All of the energy in the circuit is stored in the electric field

phys 202 notes, week 6 8

between the capacitor plates.2. The capacitor begins to discharge: current flows from the positive

to the negative terminals of the capacitor. Due to the induced emf inthe inductor coils, the discharge can’t happen instantly. Instead thecurrent slowly builds up to some maximum value Imax. The circuit’senergy is shared between the capacitor electric field and the magnetic fieldin the inductor.

3. When the current reaches Imax, the charge on the plates is zero. Allof the energy is stored in the magnetic field of the inductor.

4. The capacitor now begins to charge in the opposite sense, with pos-itive charge on the formerly-negative plate and vice-versa. Again,this charge doesn’t happen instantly but rather over time. The cir-cuit’s energy is shared between the capacitor electric field and the magneticfield in the inductor.

5. Eventually the capacitor becomes fully charged again in the oppo-site sense, and I = 0. All of the energy in the circuit is stored in theelectric field between the capacitor plates.

6. Now we repeat steps (1–5), except in the opposite direction, ad in-finitum.

So basically what’s happening is that we’re oscillating between nocurrent and maximum charge on the capacitor; and maximum cur-rent and no charge on the capacitor. In between, we have some non-maximal charge and some non-maximal current. Similarly, the energyin the circuit oscillates, between being 100% electric field in the capacitor and100% magnetic field in the inductor.

As with anything periodic, we can define a frequency for the oscil-lation. This is related to L and C by

ω =

√1

LC. (33)

phys 202 notes, week 6 9

Example Problems