PHYSICS 3 (ELECTRICITY AND MAGNETISM) - CHAPTER 5 pps

When the bar magnet is moved closer to, or farther from, the loop, an electromotive force emf is induced in the loop.. 5.2 Faraday’s Law of Induction : The induced electromotive force e

Chapter 5 ELECTROMAGNETIC INDUCTION

5.1 Faraday’s Law of Induction

1) In Fig 5.1 an ammeter is connected in the circuit of a conducting loop When the bar magnet is moved closer to, or farther from, the loop, an electromotive force (emf) is induced in the loop The ammeter indicates currents in different directions depending on the relative motion of magnet and loop Notice that, when the magnet stops moving, the current returns to zero as indicated by the ammeter

Fig 5.1 : Electromagnetic Induction The magnetic flux ΦB through an area A in a magnetic field B

r

is defined as (Fig 5.2)

ΦB = ∫B.dA

r r

where the integration is taken over the area

Fig 5.2 Faraday’s Law of Induction : The induced electromotive force

e =

dt

dΦB

2) Lenz’s Law : An induced current has a direction such that the magnetic field due to the current opposes the changes in the magnetic flux that induced the current The induced emf has the same direction as the induced current

3) Electromotive force and the induced electric field : An emf is induced by a changing magnetic flux even if the loop through which the flux is changing is not a physical conductor The induced emf is related

to the electric field E by

e = ∫Edsr

r

(5.3)

r = dt

dΦB

a changing magnetic field B

r induces an electric field E

r

5.2 Inductors

1) Inductance

L =

i

NΦB

The inductance per unit length near the middle of a long solenoid of cross sectional area A and n turns per unit length

2) Self-induction : If the current in a coil changes with time, an emf is induced in the coil

e =

-dt

di

The direction of e is found from the Lenz’s law : the direction of e acts to oppose the changes that produces it

5.3 Series RL circuit

1) Rise of current (Fig 5.3)

For t < 0, the switch K is at 2, i = 0A

For t > 0, the switch K is at 1 Apply the loop rule

E =

dt

di

R(1-e

-t/τ

where τ = L

R : inductive time constant

Fig 5.3 Fig 5.4

2) Decay of current (Fig 5.4)

For t < 0, the switch K is at 1, i = E

R For t > 0, the switch K is at 2 Apply the loop rule

0 =

dt

di

Re

-t/τ

(5.9)

where τ = L

R is the inductive time constant

5.4 Magnetic Energy

Magnetic energy stored in an inductance

UB =

2

1

Density of magnetic energy = magnetic energy / volume

uB =

o

2

2

B

5.5 Mutual Induction

If coil 1 and 2 are near each other, a changing current in either coil can induce an emf in the other This mutual induction is described by

e1 =

-dt

di

e2 =

-dt

di

where M (measured in henries) is the mutual inductance for the coil arrangement

Problems

5.1) A small loop of area 10cm2 is placed inside a long solenoid that has 800 turns/cm and carries a sinusoidally varying current I of amplitude 1A and angular frequency 300rad/s The central axes of the loop and the solenoid coincide What is the amplitude of the electromotive force induced in the loop ?

5.2) In Fig P5.1, the magnetic flux through the loop increases according to the relation ΦB = 6t2 + 7t where ΦB

is in miliwebers and t is in seconds What is the magnitude of the electromotive force induced in the loop when t = 2s ? Is the direction of the current through R to the right or left ?

5.3) In Fig P5.2, the triangle ABC is moving into a magnetic field B with velocity v Find the electromotive force e(t) induced in the loop If the triangle has resistance R, find the magnitude and direction of the current i in the triangle AB = a, BC = b

Fig P5.1 Fig P5.2 5.4) A rectangular coil of N turns and of length a, width b is rotated at frequency f in a uniform magnetic field

B indicated in Fig P5.2 The coil is connected to co-rotating cylinders, against which metal brushes slide

to make contact Find the electromotive force induced in the coil

5.5) In Fig P5.3, a rectangular loop of wire with length a = 2cm, width b = 0.8cm and resistance R = 0.4mΩ is placed near an infinitely long wire carrying current i = 4A At t = 0, r = ro = 0.1cm The loop is then moved away from the wire at constant speed v = 3mm/s Find the magnitude of the magnetic flux through the loop and the current induced in the loop

5.6) Find the mutual inductance between the long wire and the rectangular loop (of N turns) in Fig P5.3 5.7) In Fig P5.4, a long rectangular conducting loop, of width L, resistance R, and mass m, is hung in a horizontal, uniform magnetic field B that exists only above line a-a The loop is then dropped During its fall, it accelerates until it reaches a certain terminal speed v Find an expression for v

5.8) In Fig P5.5, the current in the infinitely long wire is i = αt, the rectangle has resistance R Find the value and the direction of the induced current in the rectangle

5.9) A rectangular loop of n closely packed turns is positioned near a long straight wire as shown in Fig P5.6 What is the mutual inductance M for the loop-wire combination ?

Fig P5.6

Homeworks 5

H5.1 In Fig H5.1, the magnetic flux through the loop increases according to the relation ΦB = at2 + bt where ΦB

is in miliwebers and t is in seconds What is the magnitude of the electromotive force induced in the loop when t = 1s ? What is the magnitude and the direction of the current through R [Ω] ?

near an infinitely long wire carrying current i = 10A At t = 0, r = ro = 0.1cm The loop is then moved away from the wire at constant speed v = 10cm/s Find the magnitude and the direction of the current induced in the loop