Question 1

A closed flat loop conductor with radius $2.0 \mathrm {~m}$ is located in a changing uniform magnetic field. If the emf induced in the loop is $2.0 \mathrm {~V}$, what is the rate at which the magnetic field strength is changing if the magnetic field is oriented perpendicular to the plane in which the loop lies?
A) $0.080 \mathrm {~T} / \mathrm { s }$
B) $2.0 \mathrm {~T} / \mathrm { s }$
C) $1.0 \mathrm {~T} / \mathrm { s }$
D) $0.16 \mathrm {~T} / \mathrm { s }$

Accepted Answer

The rate of change of the magnetic field strength can be found using Faraday's law of electromagnetic induction, which states that the induced emf ($\epsilon$) in a loop is equal to the negative rate of change of the magnetic flux ($\Phi_B$) through the loop. The magnetic flux is given by $\Phi_B = BA$, where $B$ is the magnetic field strength and $A$ is the area of the loop. For a circle, $A = \pi r^2$. Given $r = 2.0 \, \mathrm{m}$ and $\epsilon = 2.0 \, \mathrm{V}$, we can solve for $\frac{dB}{dt}$ as follows:$$\epsilon = -\frac{d\Phi_B}{dt} = -\frac{d(BA)}{dt} = -A\frac{dB}{dt}$$$$2.0 = -\pi (2.0)^2 \frac{dB}{dt}$$$$\frac{dB}{dt} = -\frac{2.0}{\pi (2.0)^2} = -\frac{2.0}{4\pi} = -\frac{1}{2\pi} \, \mathrm{T/s}$$Since the question asks for the magnitude, we take the absolute value:$$\left| \frac{dB}{dt} \right| = \frac{1}{2\pi} \approx 0.16 \, \mathrm{T/s}$$Therefore, the rate at which the magnetic field strength is changing is $0.16 \, \mathrm{T/s}$.

Question 2

A flat circular loop of radius 0.10 m is rotating in a uniform magnetic field of 0.20 T. Find the magnetic flux through the loop when the plane of the loop and the magnetic field vector are
Perpendicular. 
A) $5.5 \times 10 ^ { - 3 } \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$
B) $0 \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$
C) $3.1 \times 10 ^ { - 3 } \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$
D) $6.3 \times 10 ^ { - 3 } \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$

Accepted Answer

The magnetic flux through a loop is given by $\Phi = B \cdot A \cdot \cos(\theta)$, where $B$ is the magnetic field strength, $A$ is the area of the loop, and $\theta$ is the angle between the magnetic field and the normal to the plane of the loop. When the plane of the loop and the magnetic field vector are perpendicular, $\theta = 0$ degrees, and $\cos(0) = 1$. The area $A$ of a circle is $\pi r^2$, so for a radius $r = 0.10$ m, $A = \pi \cdot (0.10)^2 = \pi \cdot 0.01$ m$^2$. Thus, the magnetic flux $\Phi = 0.20 \cdot \pi \cdot 0.01 = 6.28 \times 10^{-3}$ T$\cdot$m$^2$, which rounds to $6.3 \times 10^{-3}$ T$\cdot$m$^2$, matching choice D.

Question 3

A flat coil is wrapped with 200 turns of very thin wire on a square frame with sides $18 \mathrm {~cm}$ long. A uniform magnetic field is applied perpendicular to the plane of the coil. If the field changes uniformly from $0.50 \mathrm {~T}$ to $0.00 \mathrm {~T}$ in $8.0 \mathrm {~s}$, find the emf induced in the coil.
A) $2.1 \mathrm { mV }$
B) $4.1 \mathrm { mV }$
C) $0.41 \mathrm {~V}$
D) $0.21 \mathrm {~V}$

Accepted Answer

C

Question 4

A rectangular loop of wire that can rotate about an axis through its center is placed between the poles of a magnet in a magnetic field with a strength of $0.40 \mathrm {~T}$, as shown in the figure. The length of the loop $L$ is $0.16 \mathrm {~m}$ and its width $w$ is $0.040 \mathrm {~m}$. What is the magnetic flux through the loop when the plane of the loop is perpendicular to the magnetic field?

A) $13 \times 10 ^ { - 3 } \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$
B) $2.6 \times 10 ^ { - 3 } \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$
C) $2.6 \times 10 ^ { 3 } \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$
D) $0.80 \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$
E) $0 \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$

Accepted Answer

The answer of A rectangular loop of wire that can...

Question 5

As shown in the figure, a wire and a $10 - \Omega$ resistor are used to form a circuit in the shape of a square with dimensions $20 \mathrm {~cm}$ by $20 \mathrm {~cm}$. A uniform but non-steady magnetic field is directed into the plane of the circuit. The magnitude of the magnetic field is steadily decreased from $2.70 \mathrm {~T}$ to $0.90 \mathrm {~T}$ in a time interval of $96 \mathrm {~ms}$. What is the induced current in the circuit, and what is its direction through the resistor?

A) $75 \mathrm {~mA}$, from $b$ to $a$
B) $45 \mathrm {~mA}$, from $b$ to $a$
C) $45 \mathrm {~mA}$, from $a$ to $b$
D) $110 \mathrm {~mA}$, from $a$ to $b$
E) $75 \mathrm {~mA}$, from $a$ to $b$

Accepted Answer

The answer of As shown in the figure, a wire...

Question 6

A flat circular loop having one turn and radius 5.0 cm is positioned with its plane
perpendicular to a uniform 0.60-T magnetic field. The area of the loop is suddenly
reduced to essentially zero in 0.50 ms. What emf is induced in the loop?

Accepted Answer

The answer of A flat circular loop having one turn...

Question 7

The magnetic flux through a coil changes steadily from $4.0 \times 10 - 5 \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$ to $5.0 \times 10 - 5 \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$ in $0.10 \mathrm {~s}$. What emf is induced in this coil?
A) $1.0 \times 10 ^ { - 4 } \mathrm {~V}$
B) $5.0 \times 10 ^ { - 4 } \mathrm {~V}$
C) $4.0 \times 10^{ - 4} \mathrm {~V}$
D) None of the given answers are correct.

Accepted Answer

The emf induced in a coil is given by the rate of change of magnetic flux through the coil. In this case, the magnetic flux changes from $4.0 \times 10 - 5 \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$ to $5.0 \times 10 - 5 \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$ in $0.10 \mathrm {~s}$, so the emf induced is:$$\varepsilon = \frac{\Delta \Phi}{\Delta t} = \frac{(5.0 \times 10 - 5 \mathrm {~T} \cdot \mathrm { m } ^ { 2 }) - (4.0 \times 10 - 5 \mathrm {~T} \cdot \mathrm { m } ^ { 2 })}{0.10 \mathrm {~s}} = 1.0 \times 10 ^ { - 4 } \mathrm {~V}$$

Question 8

A flat circular loop of radius 0.10 m is rotating in a uniform magnetic field of 0.20 T. Find the magnetic flux through the loop when the plane of the loop and the magnetic field vector are
Parallel. 
A) $3.1 \times 10 ^ { - 3 } \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$
B) $5.5 \times 10 ^ { - 3 } \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$
C) $6.3 \times 10 - 3 \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$
D) $0 \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$

Accepted Answer

When the plane of the loop and the magnetic field vector are parallel, the magnetic flux through the loop is zero because the angle between the magnetic field and the normal to the plane of the loop is 90 degrees, leading to a cosine factor of zero in the flux calculation.

Question 9

A circular conducting loop with a radius of $0.50 \mathrm {~m}$ and a small gap filled with a $10.0 - \Omega$ resistor is oriented in the $x y$-plane. If a uniform magnetic field of $1.0 \mathrm {~T}$, making an angle of $30 ^ { \circ }$ with the $z$-axis, increases to $10.0 \mathrm {~T}$, in $4.0 \mathrm {~s}$, what is the magnitude of the current that will be caused to flow in the loop if it has negligible resistance?
A) $0 \mathrm {~A}$
B) $0.01530393 \mathrm {~A}$
C) $0.15303932 \mathrm {~A}$
D) $0.08835729 \mathrm {~A}$

Accepted Answer

The answer of A circular conducting loop with a radius...

Question 10

A flat circular loop of radius $0.10 \mathrm {~m}$ is rotating in a uniform magnetic field of $0.20 \mathrm {~T}$. Find the magnetic flux through the loop when the plane of the loop and the magnetic field vector are at an angle of $30 ^ { \circ }$.
A) $0 \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$
B) $6.3 \times 10 ^ { - 3 } \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$
C) $5.5 \times 10 ^ { - 3 } \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$
D) $3.1 \times 10 ^ { - 3 } \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$

Accepted Answer

The answer of A flat circular loop of radius \(0.10...

Question 11

A rectangular loop of wire that can rotate about an axis through its center is placed between the poles of a magnet in a magnetic field with a strength of $0.40 \mathrm {~T}$, as shown in the figure. The length of the loop $L$ is $0.16 \mathrm {~m}$ and its width $w$ is $0.040 \mathrm {~m}$. What is the magnetic flux through the loop when the plane of the loop is parallel to the magnetic field?

A) $2.6 \times 10 ^ { 3 } \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$
B) $0 \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$
C) $13 \times 10 ^ { - 3 } \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$
D) $2.6 \times 10 ^ { - 3 } \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$
E) $0.80 \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$

Accepted Answer

The answer of A rectangular loop of wire that can...

Question 12

A $2.00$-m long metal wire is formed into a square and placed in the horizontal $x y$-plane. A uniform magnetic field is oriented at $30 ^ { \circ }$ above the horizontal with a strength of $0.344 \mathrm {~T}$. What is the magnetic flux through the square due to this field?
A) $0.0745 \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$
B) $0.0430 \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$
C) $0.172 \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$
D) $0.298 \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$

Accepted Answer

The area of the square is $(2.00 \, \text{m} / 4)^2 = 0.25 \, \text{m}^2$. The magnetic flux, $\Phi = B A \cos(\theta)$, where $B = 0.344 \, \text{T}$, $A = 0.25 \, \text{m}^2$, and $\theta = 30^\circ$. Thus, $\Phi = 0.344 \times 0.25 \times \cos(30^\circ) = 0.0430 \, \text{T} \cdot \text{m}^2$.

Question 13

A bar magnet is pushed through a coil of wire of cross-sectional area $0.020 \mathrm {~m} ^ { 2 }$ as shown in the figure. The coil has seven turns, and the rate of change of the strength of the magnetic field in it due to the motion of the bar magnet is $0.040 \mathrm {~T} / \mathrm { s }$. What is the magnitude of the induced emf in that coil of wire?

A) $5.6 \times 10 ^ { - 4 } \mathrm {~V}$
B) $5.6 \times 10 ^ { - 2 } \mathrm {~V}$
C) $5.6 \times 10 ^ { - 3 } \mathrm {~V}$
D) $5.6 \times 10 ^ { - 5 } \mathrm {~V}$
E) $5.6 \times 10 ^ { - 1 } \mathrm {~V}$

Accepted Answer

The answer of A bar magnet is pushed through a...

Question 14

A flat square coil of wire with 15 turns and an area of $0.40 \mathrm {~m} ^ { 2 }$ is placed with the plane of its area parallel to a magnetic field of $0.75 \mathrm {~T}$. The coil is flipped so its plane is perpendicular to the magnetic field in a time of $0.050 \mathrm {~s}$. What is the magnitude of the average induced emf in the coil?
A) $90 \mathrm {~V}$
B) $6.0 \mathrm {~V}$
C) $45 \mathrm {~V}$
D) $36 \mathrm {~V}$

Accepted Answer

The average induced emf is given by:$$\varepsilon = -N\frac{\Delta \Phi}{\Delta t}$$where:* $N$ is the number of turns in the coil* $\Delta \Phi$ is the change in magnetic flux through the coil* $\Delta t$ is the time interval over which the change in flux occursIn this case, the initial magnetic flux through the coil is zero, since the plane of the coil is parallel to the magnetic field. When the coil is flipped, the magnetic flux through the coil changes by:$$\Delta \Phi = BA = (0.75 \mathrm {~T})(0.40 \mathrm {~m} ^ { 2 }) = 0.30 \mathrm {~Wb}$$The time interval over which the change in flux occurs is $0.050 \mathrm {~s}$. Therefore, the average induced emf is:$$\varepsilon = -N\frac{\Delta \Phi}{\Delta t} = -(15)(-0.30 \mathrm {~Wb}) / (0.050 \mathrm {~s}) = 90 \mathrm {~V}$$

Question 15

A conductor is formed into a flat loop that encloses an area of $1.0 \mathrm {~m} ^ { 2 }$. The plane of the loop is oriented at a $30.0 ^ { \circ }$ angle with the $x y$-plane. A uniform time-varying magnetic field is oriented parallel to the $z$-axis. If the emf induced in the loop is $25.0 \mathrm {~V}$, what is the rate at which the magnetic field strength is changing?
A) $29 \mathrm {~T} / \mathrm { s }$
B) $22 \mathrm {~T} / \mathrm { s }$
C) $13 \mathrm {~T} / \mathrm { s }$
D) $50 \mathrm {~T} / \mathrm { s }$

Accepted Answer

The emf induced in the loop is given by:$$\varepsilon = - \frac{d\Phi_B}{dt}$$where $\varepsilon$ is the emf, $\Phi_B$ is the magnetic flux through the loop, and $t$ is time. The magnetic flux is given by:$$\Phi_B = BA\cos\theta$$where $B$ is the magnetic field strength, $A$ is the area of the loop, and $\theta$ is the angle between the magnetic field and the normal to the loop.Substituting these equations into the first equation, we get:$$\varepsilon = - \frac{d}{dt}(BA\cos\theta)$$$$\varepsilon = -B\frac{dA}{dt}\cos\theta - BA\frac{d\theta}{dt}\sin\theta$$Since the area of the loop is constant, we have:$$\frac{dA}{dt} = 0$$Therefore, the emf induced in the loop is given by:$$\varepsilon = -BA\frac{d\theta}{dt}\sin\theta$$Solving for $\frac{d\theta}{dt}$, we get:$$\frac{d\theta}{dt} = -\frac{\varepsilon}{BA\sin\theta}$$Substituting the given values, we get:$$\frac{d\theta}{dt} = -\frac{25.0 \mathrm {~V}}{(1.0 \mathrm {~m} ^ { 2})(22 \mathrm {~T})\sin(30.0 ^ { \circ })}$$$$\frac{d\theta}{dt} = -22 \mathrm {~T} / \mathrm { s }$$Therefore, the rate at which the magnetic field strength is changing is $22 \mathrm {~T} / \mathrm { s }$.

Question 16

A flat circular coil having 16 turns, each of diameter 20 cm, is in a uniform and steady
0.13-T magnetic field.
(a) Find the total magnetic flux through the coil when the field is perpendicular to the
plane of the coil.
(b) If the coil is rotated in 10 ms so its plane is parallel to the field, find the average
induced emf in the coil.

Accepted Answer

The answer of A flat circular coil having 16 turns,...

Question 17

A flux of $4.0 \times 10 ^ { - 5 } \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$ is maintained through a coil of area $7.5 \mathrm {~cm} ^ { 2 }$ for $0.50 \mathrm {~s}$. What emf is induced in this coil during this time by this flux?
A) $3.0 \times 10 ^ { - 5 } \mathrm {~V}$
B) $4.0 \times 10 ^ { - 5 } \mathrm {~V}$
C) $2.0 \times 10 ^ { - 5 } \mathrm {~V}$
D) $8.0 \times 10 ^ { - 5 } \mathrm {~V}$
E) No emf is induced in this coil.

Accepted Answer

No emf is induced because the magnetic flux through the coil does not change over the given time period. According to Faraday's law of electromagnetic induction, an emf is induced in a coil only if there is a change in magnetic flux through the coil.

Question 18

As shown in the figure, a uniform magnetic field $B$ is confined to a cylindrical volume of radius $0.050 \mathrm {~m}$. This field is directed into the plane of the page and is increasing at a constant rate of $0.900 \mathrm {~T} / \mathrm { s }$. Calculate the magnitude and direction (clockwise or counterclockwise) of the current induced in a circular wire ring of radius $0.16 \mathrm {~m}$ and resistance $7.1 \Omega$ that encircles the magnetic field region.

Accepted Answer

The answer of As shown in the figure, a uniform...

Question 19

A rectangular loop of wire that can rotate about an axis through its center is placed between the poles of a magnet in a magnetic field with a strength of $0.40 \mathrm {~T}$, as shown in the figure. The length of the loop $L$ is $0.16 \mathrm {~m}$ and its width $w$ is $0.040 \mathrm {~m}$. What is the magnetic flux through the loop when the plane of the loop makes an angle of $60 ^ { \circ }$ with the magnetic field?

A) $0.80 \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$
B) $2.6 \times 10 ^ { - 3 } \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$
C) $1.3 \times 10 ^ { - 3 } \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$
D) $2.2 \times 10 ^ { - 3 } \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$
E) $0 \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$

Accepted Answer

The answer of A rectangular loop of wire that can...

Question 20

A flat coil having 40 turns, each one of cross-sectional area 12.0 cm2, is oriented with its
plane perpendicular to a uniform magnetic field. The field varies steadily from 0.00 T to
1.20 T in 20.0 ms. What emf is induced in the coil during this time?

Accepted Answer

The answer of A flat coil having 40 turns, each...

Quiz 19: Dc Circuits

A flat circular loop of radius 0.10 m is rotating in a uniform magnetic field of 0.20 T. Find the magnetic flux through the loop when the plane of the loop and the magnetic field vector are Perpendicular.

A flat circular loop having one turn and radius 5.0 cm is positioned with its plane perpendicular to a uniform 0.60-T magnetic field. The area of the loop is suddenly reduced to essentially zero in 0.50 ms. What emf is induced in the loop?

The magnetic flux through a coil changes steadily from $4.0 \times 10 - 5 \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$ to $5.0 \times 10 - 5 \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$ in $0.10 \mathrm {~s}$ . What emf is induced in this coil?

A flat circular loop of radius 0.10 m is rotating in a uniform magnetic field of 0.20 T. Find the magnetic flux through the loop when the plane of the loop and the magnetic field vector are Parallel.

A flat circular loop of radius $0.10 \mathrm {~m}$ is rotating in a uniform magnetic field of $0.20 \mathrm {~T}$ . Find the magnetic flux through the loop when the plane of the loop and the magnetic field vector are at an angle of $30 ^ { \circ }$ .

A $2.00$ -m long metal wire is formed into a square and placed in the horizontal $x y$ -plane. A uniform magnetic field is oriented at $30 ^ { \circ }$ above the horizontal with a strength of $0.344 \mathrm {~T}$ . What is the magnetic flux through the square due to this field?

A flux of $4.0 \times 10 ^ { - 5 } \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$ is maintained through a coil of area $7.5 \mathrm {~cm} ^ { 2 }$ for $0.50 \mathrm {~s}$ . What emf is induced in this coil during this time by this flux?

A flat coil having 40 turns, each one of cross-sectional area 12.0 cm2, is oriented with its plane perpendicular to a uniform magnetic field. The field varies steadily from 0.00 T to 1.20 T in 20.0 ms. What emf is induced in the coil during this time?

Quiz 19: Dc Circuits

A flat circular loop of radius 0.10 m is rotating in a uniform magnetic field of 0.20 T. Find the magnetic flux through the loop when the plane of the loop and the magnetic field vector are Perpendicular.

A flat circular loop having one turn and radius 5.0 cm is positioned with its plane perpendicular to a uniform 0.60-T magnetic field. The area of the loop is suddenly reduced to essentially zero in 0.50 ms. What emf is induced in the loop?

A flat circular loop of radius 0.10 m is rotating in a uniform magnetic field of 0.20 T. Find the magnetic flux through the loop when the plane of the loop and the magnetic field vector are Parallel.

A flat circular loop of radius 0.10 m0.10 \mathrm {~m}0.10 m is rotating in a uniform magnetic field of 0.20 T0.20 \mathrm {~T}0.20 T . Find the magnetic flux through the loop when the plane of the loop and the magnetic field vector are at an angle of 30∘30 ^ { \circ }30∘ .

A flux of 4.0×10−5 T⋅m24.0 \times 10 ^ { - 5 } \mathrm {~T} \cdot \mathrm { m } ^ { 2 }4.0×10−5 T⋅m2 is maintained through a coil of area 7.5 cm27.5 \mathrm {~cm} ^ { 2 }7.5 cm2 for 0.50 s0.50 \mathrm {~s}0.50 s . What emf is induced in this coil during this time by this flux?

A flat coil having 40 turns, each one of cross-sectional area 12.0 cm2, is oriented with its plane perpendicular to a uniform magnetic field. The field varies steadily from 0.00 T to 1.20 T in 20.0 ms. What emf is induced in the coil during this time?

A flat circular loop of radius $0.10 \mathrm {~m}$ is rotating in a uniform magnetic field of $0.20 \mathrm {~T}$ . Find the magnetic flux through the loop when the plane of the loop and the magnetic field vector are at an angle of $30 ^ { \circ }$ .

A flux of $4.0 \times 10 ^ { - 5 } \mathrm {~T} \cdot \mathrm { m } ^ { 2 }$ is maintained through a coil of area $7.5 \mathrm {~cm} ^ { 2 }$ for $0.50 \mathrm {~s}$ . What emf is induced in this coil during this time by this flux?