Chapter 32

Inductance L and the stored magnetic energy

RL and LC circuits RLC circuit

Resistance, Capacitance and InductanceOhm’s Law defines resistance:

VR

I

Capacitance, the ability to hold charge:Q

CV

Capacitors store electric energy once charged:

Resistors do not store energy, instead they transform electrical energy into thermo energy at a rate of:

22V

P V I I RR

2

21 1

2 2E

QU C V

C

Inductance, the ability to “hold” current (moving charge).Inductors store magnetic energy once “charged” with current, i.e., current flows through it.

Inductance, the definitionIWhen a current flows through a

coil, there is magnetic field established. If we take the solenoid assumption for the coil:

E EL

+

–

0B nI

When this magnetic field flux changes, it induces an emf, EL, called self-induction:

0 20

BL

d NAB d NA nId dI dIn V L

dt dt dt dt dt

E

or: L

dILdt

E

This defines the inductance L, which is constant related only to the coil. The self-induced emf is generated by current flowing though a coil. According to Lenz Law, the emf generated inside this coil is always opposing the change of the current which is delivered by the original emf.

For a solenoid: 20L n V

Wheren: # of turns per unit length.N: # of turns in length l.A: cross section areaV: Volume for length l.

Inductor

We used a coil and the solenoid assumption to introduce the inductance. But the definition

holds for all types of inductance, including a straight wire. Any conductor has capacitance and inductance. But as in the capacitor case, an inductor is a device made to have a sizable inductance.

LLdIdt

E

An inductor is made of a coil. The symbol is Once the coil is made, its inductance L is defined. The self-induced emf over this inductor under a changing current I is given by:

L

dILdt

E

Unit for Inductance

The SI unit for inductance is the henry (H)

Named for Joseph Henry:

1797 – 1878 American physicist First director of the Smithsonian Improved design of electromagnet Constructed one of the first motors Discovered self-inductance

AsV

1H1

Discussion about Some Terminology

Use emf and current when they are caused by batteries or other sources

Use induced emf and induced current when they are caused by changing magnetic fields

When dealing with problems in electromagnetism, it is important to distinguish between the two situations

Example: Inductance of a coaxial cable

2

ln2

bo

B a

o

μ IB dA dr

πrμ I b

π a

ln2

oB μ bL

I π a

BL

d dIL

dt dt

EStart from the definition

We have , or B Bd LdI LI

So the inductance is

Put inductor L to use: the RL Circuit

An RL circuit contains a resistor R and an inductor L.

There are two cases as in the RC circuit: charging and discharging. The difference is that here one charges with current, not charge.

Charging: When S2 is connected to

position a and when switch S1 is closed (at time t = 0), the current begins to increase

Discharging: When S2 is connected to

position b.

PLAYACTIVE FIGURE

RL Circuit, charging Applying Kirchhoff’s loop rule to the

circuit in the clockwise direction gives

0d I

ε I R Ldt

τ 1 1Rt L tε εI e e

R R

Here because the current is increasing, the induced emf has a direction that should oppose this increase.

τ L

R

Solve for the current I, with initial condition that I(t=0) = 0, we find

Where the time constant is defined as:

RL Circuit, discharging When switch S2 is moved to

position b, the original current disappears. The self-induced emf will try to prevent that change, and this determines the emf direction (Lenz Law).

0d I

I R Ldt

τ Rt L tε εI e e

R R

E0 RI t

Applying Kirchhoff’s loop rule to the previous circuit in the clockwise direction gives

Solve for the current I, with initial condition that we find

Energy stored in an inductor

In the charging case, the current I from the battery supplies power not only to the resistor, but also to the inductor.

From Kirchhoff’s loop rule, we have

d I

ε I R Ldt

Multiply both sides with I:

2 d IεI I R LI

dtThis equation reads: powerbattery=powerR+powerL

So we have the energy increase in the inductor as:

LdU d ILI

dt dt

Solve for UL: 2

0

1

2

I

LU LId I LI

Stored energy type and the Energy Density of a Magnetic Field

Given UL = ½ L I2 and assume (for simplicity) a solenoid with L = o n2 V

Since V is the volume of the solenoid, the magnetic energy density, uB is

This applies to any region in which a magnetic field exists (not just the solenoid)

2 221

2 2L oo o

B BU μ n V V

μ n μ

2

2L

Bo

U Bu

V μ

So the energy stored in the solenoid volume V is magnetic (B) energy.

And the energy density is proportional to B2.

RL and RC circuits comparison

RL RC

Charging

Discharging

Energy

Rt LεI e

R

1 Rt LεI e

R

21

2LU LI2

21( )

2 2C

QU C V

C

t

RCεI t e

R

t

RCQI t e

RC

Magnetic field Electric field

Energy density

2

2Bo

Bu

μ 21

2E ou ε E

Energy Storage Summary

Inductor and capacitor store energy through different mechanisms Charged capacitor

Stores energy as electric potential energy When current flows through an inductor

Stores energy as magnetic potential energy

A resistor does not store energy Energy delivered is transformed into thermo energy

LC CircuitsLC: circuit with an inductor and a capacitor.

Initial condition: either the C or the L has energy stored in it.

The “show” starts: when the switch S closes, t = 0 and the time starts.

Your physics intuition: neither C nor L consumes energy, the initially stored energy will oscillate between the C and the L.

LC Circuits, the calculationInitial condition: Assume that the capacitor was initially charged to Qmax. when the switch S closes, t = 0 and the time starts.

0C L

q dIV L

C dt +E

Here q is the charge in the capacitor at time t. Because charges flow out of the capacitor to form the current I, we have:

dqI

dt

Apply Kirchhoff’s loop rule:

Combine these two equations:2

2

10

d II

dt LC

Solve for the current I: maxI I sin t 2 1, and max maxI Q

LC with

Here we also have maxq Q cos t

LC Circuits, the oscillation of charge and current

Oscillations: simply plot the results, we find out that the charge stored in the capacitor and the current “stored” in the inductor oscillate. The phase difference is T/2.

This means that when the capacitor is fully charged, the current is zero. When the capacitor has no charges in it, the current reaches its maximum in magnitude through the inductor.

maxI I sin t maxq Q cos t

q

21

2LU LI2

2C

QU

C

From the formulas for the energies stored in a capacitor and an inductor, we know that this oscillation happens between electric energy and magnetic energy.

LC Circuits, the oscillation of energy

21

2LU LI2

2C

qU

C

maxI I sin t maxq Q cos t

From the following four formulas

We have the oscillation of the energies in the capacitor and the inductor:

2

2 2

2Cmax

C max

QU cos t E cos t

C

2 2 21

2L

L max maxU LI sin ωt E sin ωt

From energy conservation:

, or C Lmax max max maxE E I ωQ

Move from the ideal LC circuit to the real-life RLC circuit

In actual circuits, there is always some resistance, therefore, there is some energy transformed to thermo energy by the resistance in the system and dissipates to the environment.

Radiation is also inevitable in this type of circuit, and energy will be radiated out of the LC system as electromagnetic wave through space.

The total energy in the circuit continuously decreases as a result of these processes

Here we will only discuss about the energy dissipated through the resistance.

The RLC Circuit and the analysis Concentrating the resistance in the

system into a resistor, with the inductor and the capacitor, we model the circuit with an RLC Circuit.

PLAYACTIVE FIGURE

The capacitor is charged with the switch at position a. At time t = 0, the switch is thrown to position b to form the RLC circuit.

Apply Kirchhoff’s loop rule:

0 and q dI dq

L IR , IC dt dt

We have:2

2 22

10 and

d q R dqq ,

dt L dt LC

Solve for q:

221

with 2

RtL

max d d

Rq Q e cos t ,

LC L

And: 2

2

RtL

max d d d

RI Q e sin t cos t

L

Mutual Inductance

The magnetic flux through the area enclosed by a circuit often varies with time because of time-varying currents in nearby circuits

This process is known as mutual induction because it depends on the interaction of two circuits

Mutual Inductance and transformers The current in coil 1 sets up a magnetic

field that varies as I1. When magnetic field lines pass through

coil 2, cause the magnetic flux in coil 2 to change and induce current I2 in coil 2.

This process is called mutual inductance. If coil 1 has a current I1 and N1 turns, and

coil 2 has N2 turns. When the field lines that go through coil 1 completely go through coil 2, we have a transformer. Coil 1 and 2 are called prime and second coils. The terminal voltages at these two coils are 1 1

2 2

V N

V N

If coil 2 connects to a resistor R2, the resistance coil 1 “sees” is

From energy conservation: 1 1 2 2V I V I

We have: 2 1

1 2

I N

I N

2

11 2

2

NR R

N