23.1 Induced Emf and Magnetic Flux

zero

What is the value of the magnetic flux at coil 2 in the figure due to coil 1? (Assume the center of coil 1 lies in the plane of coil 2 and vice versa.)
The planes of the two coils are perpendicular.
[coil 1 is propped up vertically with its loop center facing perpendicular to coil 2, with current flowing counterclockwise.
coil 2 is laying flat on the ground with its loop center facing up/down]
nonzero
zero

zero

What is the value of the magnetic flux through the coil in the figure due to the wire?
[the wire stands vertical, with current flowing straight up, and the coil lies flat nearby]
The wire is perpendicular to the plane of the coil.
zero
nonzero

thumb up with current, field circulating in direction of bent fingers, fingers/magnetic field into page at the point of hitting coil. Thumb into page with magnetic field, current circulating in opposite direction of bent fingers to oppose increase.
counterclockwise

Referring to the figure below, what is the direction of the current induced in the coil for the following cases? (Explicitly show on paper how you follow the steps in the Problem-Solving Strategy for Lenz's law. Your instructor may ask you to turn in this work.)
[wire standing vertical with current flowing up, coil standing vertical to the right of wire with center hole perpendicular to wire]
(a)
the current in the wire increases
--Select--
clockwise
counterclockwise
no current

thumb up with current, field circulating in direction of bent fingers, fingers/magnetic field into page at the point of hitting coil. Thumb into page with magnetic field, current circulating in direction of bent fingers to oppose decrease.
clockwise

Referring to the figure below, what is the direction of the current induced in the coil for the following cases? (Explicitly show on paper how you follow the steps in the Problem-Solving Strategy for Lenz's law. Your instructor may ask you to turn in this work.)
[wire standing vertical with current flowing up, coil standing vertical to the right of wire with center hole perpendicular to wire]
(b)
the current in the wire decreases
--Select--
no current
counterclockwise
clockwise

thumb down with current, field circulating in direction of bent fingers, fingers/magnetic field out of page at the point of hitting coil. Thumb out of page with magnetic field, current circulating in opposite direction of bent fingers to oppose decrease.
clockwise

Referring to the figure below, what is the direction of the current induced in the coil for the following cases? (Explicitly show on paper how you follow the steps in the Problem-Solving Strategy for Lenz's law. Your instructor may ask you to turn in this work.)
[wire standing vertical with current flowing up, coil standing vertical to the right of wire with center hole perpendicular to wire]
(c)
the current in the wire suddenly changes direction
--Select--
counterclockwise
no current
clockwise

N = 42
A = 0.150m^2
A final = 0
t = 0.100s
emf = ?V
B = 1.60T
θ = 0

ΔΦ = Δ(BAcosθ)
ΔΦ = 1.60T x 0.150m^2
ΔΦ = 0.24Tm^2

emf=NΔΦ/Δt
emf = 42 x 0.24Tm^2 /0.1s
emf = 100.8Tm^2/s
emf = 101V

Suppose a 42-turn coil lies in the plane of the page in a uniform magnetic field that is directed out of the page. The coil originally has an area of 0.150 m^2. It is stretched to have no area in 0.100 s. What is the magnitude (in V) and direction (as seen from above) of the average induced emf if the uniform magnetic field has a strength of 1.60 T?
magnitude
V

magnetic field/thumb out of page, fingers curl counterclockwise.
counterclockwise

Suppose a 42-turn coil lies in the plane of the page in a uniform magnetic field that is directed out of the page. The coil originally has an area of 0.150 m^2. It is stretched to have no area in 0.100 s. What is the magnitude (in V) and direction (as seen from above) of the average induced emf if the uniform magnetic field has a strength of 1.60 T?
direction
clockwise
counterclockwise
the magnitude is zero

B = 2.50T
θ = 0
diameter = 2.19cm = 0.0219m
t = 0.320s
R = 0.0100Ω
I = ?A

r = d/2 = 0.01095m
A = πr^2
A = π x (0.01095m)^2
A = 0.0003767m^2

ΔΦ = Δ(BAcosθ)
ΔΦ = 2.50T x 0.0003767m^2 x 1
ΔΦ = 0.0009418Tm^2

emf=NΔΦ/Δt
emf = 0.0009418Tm^2 /0.320s
emf = V = 0.002943V

V = IR
V/R = I
I = 0.002943V /0.0100Ω
I = 0.294A

An MRI technician moves his hand from a region of very low magnetic field strength into an MRI scanner's 2.50 T field with his fingers pointing in the direction of the field. His wedding ring has a diameter of 2.19 cm, and it takes 0.320 s to move it into the field.
(a)
What average current is induced in the ring if its resistance is 0.0100 Ω? (Enter the magnitude in amperes.)
A

P = ?W
emf = V = 0.002943V
I = 0.294A
P = IV
p = 0.294A x 0.002943V
P = 0.000865W

An MRI technician moves his hand from a region of very low magnetic field strength into an MRI scanner's 2.50 T field with his fingers pointing in the direction of the field. His wedding ring has a diameter of 2.19 cm, and it takes 0.320 s to move it into the field.
(a)
What average current is induced in the ring if its resistance is 0.0100 Ω? (Enter the magnitude in amperes.)
0.294A
(b)
What average power is dissipated (in W)?
W

B = ?T
I = 0.294A
r = 0.01095m
µ0 = 4πe-7Tm/A

B = µ0 x I /2r
B = 4πe-7Tm/A x 0.294A /(2 x 0.01095m)
B = 1.69e-5T

An MRI technician moves his hand from a region of very low magnetic field strength into an MRI scanner's 2.50 T field with his fingers pointing in the direction of the field. His wedding ring has a diameter of 2.19 cm, and it takes 0.320 s to move it into the field.
(a)
What average current is induced in the ring if its resistance is 0.0100 Ω? (Enter the magnitude in amperes.)
0.294A
(b)
What average power is dissipated (in W)?
0.000865W
(c)
What average magnetic field is induced at the center of the ring? (Enter the magnitude in teslas.)
T

ring hole is parallel, so ring, where current is flowing, is antiparallel.
antiparallel

An MRI technician moves his hand from a region of very low magnetic field strength into an MRI scanner's 2.50 T field with his fingers pointing in the direction of the field. His wedding ring has a diameter of 2.19 cm, and it takes 0.320 s to move it into the field.
(d)
What is the direction of this induced magnetic field relative to the MRI's field?
parallel
antiparallel
The magnitude is zero.

N = 1100 turns
d = 20.0cm = 0.200m
B = 5.25e-5T
emf = ?V
Δθ = 90degrees
Δcosθ = -1
t = 10.0ms = 10.0e-3s

r = d/2 = 0.100m
A = πr^2
A = π(0.1m)^2
A = 0.0314m^2

ΔΦ = Δ(BAcosθ)
ΔΦ = 5.25e-5T x 0.0314m^2 x 1
ΔΦ = 1.65e-6Tm^2

emf=NΔΦ/Δt
emf = 1100 x 1.65e-6Tm^2 /10.0e-3s
emf = 181.5e-3Tm^2/s
emf = 0.182V

An emf is induced by rotating a 1100 turn, 20.0 cm diameter coil in the Earth's 5.25 ✕ 10^−5 T magnetic field. What average emf (in V) is induced, given the plane of the coil is originally perpendicular to the Earth's field and is rotated to be parallel to the field in 10.0 ms?
V

B = µ0 x I /2r
B = 1/r
It is proportional to 1/r.

Approximately how does the emf induced in the loop in the figure depend on the distance r of the center of the loop from the wire? (Assume r is much greater than the radius of the coil.)
The wire is in the plane of the coil.
It is proportional to 1/r^2.
It is proportional to 1/r.
It is proportional to r.
It is proportional to r^2.
It does not depend on r.

thumb down, fingers curve around into back of coil and out of coil forward.
+z

A coil is placed next to a straight wire. The current in the wire is as shown in the diagram below. The coil and wire lie in the same plane with the +z axis perpendicular to the plane of the coil.
[the wire is left of the coil, current flowing down through wire]
(a) As the current in the wire increases, find the direction of the induced current in the coil by answering the following questions.
(i) What is the direction of the magnetic field due to the current carrying wire in the center of the coil?
---Select---
+x
−x
+y
−y
+z
−z

increase

A coil is placed next to a straight wire. The current in the wire is as shown in the diagram below. The coil and wire lie in the same plane with the +z axis perpendicular to the plane of the coil.
[the wire is left of the coil, current flowing down through wire]
(a) As the current in the wire increases, find the direction of the induced current in the coil by answering the following questions.
(ii) As the current in the wire increases, how will the magnetic flux in the coil change? ---Select---
decrease
remain the same
increase

induced magnetic flux opposes inducer, +z.
-z

A coil is placed next to a straight wire. The current in the wire is as shown in the diagram below. The coil and wire lie in the same plane with the +z axis perpendicular to the plane of the coil.
[the wire is left of the coil, current is flowing down through wire]
(a) As the current in the wire increases, find the direction of the induced current in the coil by answering the following questions.
(iii) What is the direction of the induced magnetic flux in the coil? ---Select---
+x
−x
+y
−y
+z
−z

induced magnetic field/thumb into page, fingers/induced current curl clockwise.
clockwise

A coil is placed next to a straight wire. The current in the wire is as shown in the diagram below. The coil and wire lie in the same plane with the +z axis perpendicular to the plane of the coil.
[the wire is left of the coil, current is flowing down through wire]
(a) As the current in the wire increases, find the direction of the induced current in the coil by answering the following questions.
(iv) What is the direction of the induced current in the coil? ---Select---
clockwise
counterclockwise
no current

induced current opposes decrease, thumb/induced magnetic field now flows with inducing magnetic field out of page, fingers curl/induced current flow counterclockwise.
counterclockwise

A coil is placed next to a straight wire. The current in the wire is as shown in the diagram below. The coil and wire lie in the same plane with the +z axis perpendicular to the plane of the coil.
[the wire is left of the coil, current is flowing down through wire]
(b) Now suppose the current in the wire decreases, what is the direction of the induced current in the coil?
---Select---
clockwise
counterclockwise
no current

wire's current/thumb now flows up, fingers/magnetic field curl and hit coil into page, thumb into page with magnetic field, fingers curl clockwise, induced current opposes this increase so current flows counterclockwise.
counterclockwise

A coil is placed next to a straight wire. The current in the wire is as shown in the diagram below. The coil and wire lie in the same plane with the +z axis perpendicular to the plane of the coil.
[the wire is left of the coil, current is flowing down through wire]
(c) Now suppose the current in the wire suddenly changes direction, what is the direction of the induced current in the coil?

ΔΦ = 2.30e-5Tm^2
θ = 43.0°
r = 8.50cm = 0.0850m
B = ?
A = πr^2
A = π(0.0850m)^2
A = 0.02270m^2

ΔΦ = Δ(BAcosθ)
ΔΦ/ΔAcosθ = ΔB
ΔB = 2.30e-5Tm^2 /(0.02270m^2 x cos43.0°)
ΔB = 138.5e-5T
ΔB = 1.385e-3T
ΔB = 1.39mT

The magnitude of the magnetic flux through the surface of a circular plate is 2.30 x 10^-5 T·m^2 when it is placed in a region of uniform magnetic field that is oriented at 43.0° to the vertical. The radius of the plate is 8.50 cm. Determine the strength of the magnetic field.
mT

N = 1
r = 4.50cm = 0.045m
A = πr^2
A = π(0.045m)^2
A = 0.006362m^2
ΔB = 0.325 T - 0.490 T = -0.165T
t = 0.120s
R = 3.50 Ω
I = ?mA
θ = 0

ΔΦ = Δ(BAcosθ)
ΔΦ = 0.165T x 0.006362m^2
ΔΦ = 0.001050Tm^2

emf=NΔΦ/Δt
emf = 1 x 0.001050Tm^2 /0.120s
emf = V = 0.008748Tm^2/s
emf = V = 0.008748V

V = IR
V/R = I
I = 0.008748V /3.50Ω
I = 0.002499A
I = 2.499mA

A single turn coil of radius 4.50 cm is held in a vertical plane and a magnet is rapidly moved relative to the coil as shown in the diagram below. The field inside the coil changes from 0.490 T to 0.325 T in 0.120 s. If the resistance of the coil is 3.50 Ω, what is the magnitude and direction of the induced current in the coil as viewed from the side of the magnet?
magnitude
mA

inducing magnetic field/thumb points right, induced field opposes inducing field, thumb/induced field points left, fingers/induced current curl clockwise.
clockwise

A single turn coil of radius 4.50 cm is held in a vertical plane and a magnet is rapidly moved relative to the coil as shown in the diagram below. The field inside the coil changes from 0.490 T to 0.325 T in 0.120 s. If the resistance of the coil is 3.50 Ω, what is the magnitude and direction of the induced current in the coil as viewed from the side of the magnet?
[magnet is going through the center of the loop towards the right]
direction
---Select---
clockwise
counterclockwise
no current

fingers/current clockwise up coil, thumb/magnetic field up.
The ring will move away from the coil and fly upward.

A wire of length L is wound around an iron cylinder mounted on a base. The two ends of the wire are connected to a battery via a switch that is initially open. A metal ring with a diameter larger than that of the cylinder sits on top of the coil. What happens to the metal ring when the switch is closed?
[the current flows clockwise up the coil]
The ring will remain stationary.
The ring will move toward the coil.
The ring will move away from the coil and fly upward.

I = ?A
ΔΦ = 1.90e-4Wb
r = 38.0cm = 0.38m
A = πr^2
A = π(0.38m)^2
A = 0.4536m^2
N/L = 235turns/m
B solenoid = µ0(N/L)I

ΔΦ = Δ(BAcosθ)
ΔΦ/A = B
B = 1.90e-4Wb /0.4536m^2
B = 4.189e-4Wb/m^2

B solenoid = µ0(N/L)I
B/(µ0 x N/L) = I
I = 4.189e-4Wb/m^2 /(4πe-7Tm/A x 235/m)
I = 0.001419e3WbA/m^2T
I = 1.42A

Determine the current I flowing through a solenoid, if the magnetic flux inside its core is found to be 1.90 x 10^-4 Wb. The radius of the solenoid is 38.0 cm and the number of turns per meter is 235.
A

v = ?m/s
emf = V = 1.00V
B = 1.30T
L = 32.5cm = 0.325m
ΔA = LΔx
v = Δx/Δt

ΔΦ = Δ(BAcosθ)
emf = NΔΦ/Δt
emf = BΔA/Δt
emf = BLΔx/Δt
emf = BL(Δx/Δt)
emf = BL(v)
emf/BL = v
1.00V /(1.30T x 0.325m) = v
v = 2.37m/s

At what speed (in meters per second) must the sliding rod in the figure below move to produce an emf of 1.00 V in a 1.30 T field directed perpendicular to the rods and into the page, given the rod's length is 32.5 cm?
m/s

L = 31.0cm = 0.31m
v = Δx/Δt = 4.00m/s
emf = 8.90V
B = ?T

ΔA = LΔx
ΔΦ = Δ(BAcosθ)
emf = NΔΦ/Δt
emf = NBΔA/Δt
emf = NBLΔx/Δt
emf = NBL(Δx/Δt)
emf = BL(v)
emf/Lv = B
B = 8.90V /(0.31m x 4m/s)
B = 7.18Vs/m^2
B = 7.18T

A conducting rod of length L = 31.0 cm slides over two horizontal metal bars with a constant speed v = 4.00 m/s to the right. The entire set up is in a region of uniform magnetic field that is directed perpendicular to the rods and into the page. What is the strength of the magnetic field if an emf of magnitude 8.90 V is induced?
T