March 11, 2014 Chapter 29 1

Electromagnetic Induction!

Notes !  Exam 4 next Tuesday

•  Covers Chapters 27, 28, 29 in the book •  Magnetism, Magnetic Fields, Electromagnetic Induction •  Material from the week before spring break and this week •  Homework sets 7 and 8

!  I will be out of town next week •  Prof. Joey Huston will teach lectures

March 11, 2014 Chapter 29 2

Physics for Scientists & Engineers 2 3

Magnetic field !  The force that a magnetic field exerts on a charge moving with velocity

v is given by

!  The direction is sideways (RH rule). !  The magnitude of the force

!  If the charge moves perpendicular to the magnetic field then

FB = qvBsinθ

F = qvB

FB = q

v ×B

Physics for Scientists & Engineers 2 4

Review: Magnetic field !  The unit of magnetic field strength the tesla (T)

!  Another unit of magnetic field strength that is often used but is not an SI unit is the gauss (G)

!  Typically the Earth’s magnetic field is about 0.5 G at the surface. !  A charged particle with charge q and mass m traveling

with speed v perpendicular to a constant magnetic field B will travel in a circle with radius r given by , so that and angular frequency

Ns N1 T = 1 1Cm Am

=

-41 G = 10 T 10 kG = 1 T

r = mvqB

Br = pq

ω = qBm

Physics for Scientists & Engineers 2 5

!  If we run a current i through a conductor of width h in a constant magnetic field B, we induce a voltage VH across the conductor that is given by

!  where n is the number of electrons per unit volume and e is the charge of an electron

Hall effect

H H HV V dhne V hneBdv di i

= = =

d

h

Physics for Scientists & Engineers 2 6

magnetic dipole moment !  We define the magnitude of the magnetic dipole moment of a coil to

be

!  We can express the torque on a coil in a magnetic field as

!  The magnetic potential energy of a magnetic dipole in a magnetic field is given by

µ = NiA

τ =µ ×B

U = −µ ⋅B = −µBcosθ

Physics for Scientists & Engineers 2 7

Magnetic field of a wire !  Magnetic field of a long, straight wire

with current i:

!  Magnetic field in the center of a current-carrying loop of radius R

!  The force between two current- carrying wires is given by

F12 =µ0i1i2L2πd

B(r⊥ ) =µ0i2πr⊥

Physics for Scientists & Engineers 2 8

Solenoid and Toroid !  The magnetic field inside an ideal solenoid is

given by •  The density of the coil loops is given by n

!  The magnetic field inside an ideal toroidal magnet is given by •  The number of loops

is given by N

0B inµ=

0

2NiBr

µπ

=

March 11, 2014 Chapter 29 9

Induction - Faraday’s Experiments !  A magnetic field is generated when electrical charges are in

motion !  An electric field is generated when a magnetic field is in

motion (relative to a conductor) or otherwise changes as a function of time

!  Consider a wire loop connected to an ammeter and a bar magnet

!  While the magnet is stationary, no current flows in the loop

!  The experiments relevant to this chapter were performed in the 1830s by the British chemist and physicist Michael Faraday and independently by the American physicist Joseph Henry

March 11, 2014 Chapter 29 10

Faraday’s Experiments

!  If the magnet is moved toward the loop a counterclockwise current flows in the loop as indicated by the positive current in the ammeter

!  If the magnet is reversed so the south pole points toward the loop and moved toward the loop, current flows in the loop in the opposite direction as indicated by the negative current in the ammeter

March 11, 2014 Chapter 29 11

Faraday’s Experiments

!  If the north pole of the magnet points toward the loop, and the magnet is then moved away from the loop, a negative clockwise current, as indicated on the ammeter, is induced in the loop

!  If the south pole of the magnet points toward the loop, and the magnet is moved away from the loop a positive current is induced as indicated by the ammeter

Faraday’s Experiments

March 11, 2014 Chapter 29 12

If a constant current is flowing through loop 1, no current is induced in loop 2. If the current in loop 1 is increased, a current is induced in loop 2 in the opposite direction

If current is flowing in loop 1 in the same direction as before and is then decreased, the current induced in loop 2 flows in the same direction as the current in loop 1

March 11, 2014 Chapter 29 13

Law of Induction !  From these observations we see that a changing magnetic field induces a current in a loop

!  We can visualize the change in magnetic field as a change in the number of magnetic field lines passing through the loop

!  Faraday’s Law of Induction states that: A potential difference is induced in a loop when the number of magnetic field lines passing through the loop changes with time

!  The rate of change of magnetic field lines determines the induced potential difference

March 11, 2014 Chapter 29 14

Magnetic Flux !  When we introduced Gauss’s Law for the electric field, we

defined the electric flux as

!  For the magnetic field, we can define magnetic flux in analogy as

!  Integration of the magnetic flux over a closed surface yields zero:

!  This result is often termed Gauss’s Law for Magnetic Fields !  There are no free magnetic charges, no magnetic

monopoles, no separate north poles or separate south poles

B•dA∫∫ = 0

ΦE =E•dA∫∫

ΦB =B•dA∫∫

March 11, 2014 Chapter 29 15

!  Consider the special case of a flat loop of area A in a constant magnetic field B

!  We can re-write the magnetic flux as

!  If the magnetic field is perpendicular to the plane of the loop

!  If the magnetic field is parallel to the plane of the loop

!  The unit of magnetic flux is the weber (Wb) given by

Magnetic Flux

ΦB = BAcosθ

ΦB = BAcos0° = BA

ΦB = BAcos90° = 0

1 Wb=1 T m2

March 11, 2014 Chapter 29 16

Faraday’s Law of Induction !  We can then recast Faraday’s Law of Induction in terms

of the magnetic flux as The magnitude of the potential difference, ΔVind, induced in a conducting loop is equal to the time rate of change of the magnetic flux through the loop.

!  Faraday’s Law of Induction is thus contained in

!  We can change the magnetic flux in several ways including changing the magnitude of the magnetic field, changing the area of the loop, or by changing the angle the loop with respect to the magnetic field

ΔVind = −

dΦB

dt

March 11, 2014 Chapter 29 17

Induction in a Flat Loop !  Let’s explore Faraday’s Law for the case of a flat loop and

a magnetic field that is constant in space, but varies in time !  The magnetic flux for this case is

!  The induced emf is then

!  Carrying out the derivative we get

!  Taking dθ/dt = ω we get

ΦB = BAcosθ

ΔVind = −

dΦB

dt= − d

dt(BAcosθ)

ΔVind = −Acosθ dB

dt− Bcosθ dA

dt+ ABsinθ dθ

dt

ΔVind = −Acosθ dB

dt− Bcosθ dA

dt+ωABsinθ

March 11, 2014 Chapter 29 18

Induction in a Flat Loop - Special Cases

!  If we leave two of the three variables (A,B,θ) constant, then we have the following three special cases •  We leave the area of the loop and its orientation relative to the

magnetic field constant, but vary the magnetic field in time

•  We leave the magnetic field as well as the orientation of the loop

relative to the magnetic field constant, but change the area of the loop that is exposed to the magnetic field

•  We leave the magnetic field constant and keep the area of the loop fixed as well, but allow the angle between the two to change as a function of time

A,θ constant: ΔVind = −Acosθ dB

dt

B,θ constant: ΔVind = −Bcosθ dA

dt

A,B constant: ΔVind =ωABsinθ

March 11, 2014 Chapter 29 21

Changing Magnetic Field !  A direct current of 600 mA is delivered to an ideal solenoid,

resulting in a magnetic field of 0.025 T !  Then the current is increased according to

PROBLEM !  If a circular loop of radius 3.4 cm with 200 windings is

located inside the solenoid and perpendicular to the magnetic field, what is the induced voltage at t = 2.0 s in this loop? SOLUTION

!  First, we compute the area of the coil

i(t)= i0 1+ 2.4s−2t 2( )

March 11, 2014 Chapter 29 22

Changing Magnetic Field !  Since it is circular, its area is πR2

!  However, there are N windings in the coil !  The effective area of the coil is then

!  The magnetic field inside an ideal solenoid is

!  Because the magnetic field is linearly proportional to the current, we obtain the time dependence of the magnetic field in this case

!  The area of the loop and the angle are kept constant so

A = NπR2 = 200π(0.034 m)2 = 0.73 m2

B = µ0in

B(t)= B0 1+ 2.4s−2( )t 2⎡⎣ ⎤⎦ , B0 = 0.025 T

A,θ constant: ΔVind = −Acosθ dB

dt

March 11, 2014 Chapter 29 23

Changing Magnetic Field !  For the induced voltage we then find

A,θ constant: ΔVind = −Acosθ dB

dt

ΔVind = −Acosθ dB

dt

= −Acosθ d

dtB0 1+ 2.4s−2( )t 2⎡⎣ ⎤⎦⎡⎣ ⎤⎦

= −(0.73 m2 )(0.025 T)(cos0°)(4.8s−2 )tat t = 2.0 s, ΔVind = −0.17 V

= −AB0 cosθ 2 2.4s−2( )t⎡⎣ ⎤⎦

= −Acosθ B0 2.4s−2( )2t⎡⎣ ⎤⎦

March 11, 2014 Chapter 29 26

Potential Difference Induced by a Moving Loop

!  A rectangular loop of width 3.1 cm and depth 4.8 cm is pulled out of the gap between two permanent magnets, with a field of 0.073 T throughout the gap PROBLEM

!  If the loop is removed with a constant velocity of 1.6 cm/s, what is the induced voltage in the loop as a function of time? SOLUTION

!  The magnetic field as well as the orientation of the loop relative to the field remains constant

!  What changes is the area of the loop that is exposed to the magnetic field

March 11, 2014 Chapter 29 27

Potential Difference Induced by a Moving Loop

!  With the narrow gap, there will be very little field outside the gap

!  The area of the loop exposed to the field is !  d(t) is the depth of the part of the loop in

the magnetic field at time t

!  While the entire loop is in the magnetic field no potential difference is produced

!  Letting the time of arrival of the right edge of the loop at the right end of the gap be t = 0, we have

!  This holds until the left edge of the loop reaches the right end of the gap, after which the exposed area is zero

A t( ) = w( ) d t( )( )

d t( ) = d0 − vt

A t( ) = w( ) d t( )( ) =w d0 − vt( )

March 11, 2014 Chapter 29 28

Potential Difference Induced by a Moving Loop

!  The left edge arrives at time

!  The induced potential difference is

!  During the time interval between 0 and 3 s, a constant potential difference of 36 μV is induced, and no potential difference is induced outside this time interval

ΔVind = −Bcosθ dA

dt

= (0.031 m)(0.016 m/s)(0.073 T)cos 0°( )=3.6 ⋅10-5 V = 36 µV

= −Bcosθ d

dtw d0 − vt( )⎡⎣ ⎤⎦

=wvBcosθ

tf =

dv= 4.8 cm

1.6 cm/s= 3.0 s