Competitive Physics: Thermodynamics, Electromagnetism and Relativity

Wang Jinhui

Chapter 8 Currents and EMI - all with Video Answers

Educators

Chapter Questions

Problem 1

Consider a thin, long cylindrical shell branching along its axis. Half of the shell is filled with a material of resistivity $\rho_1$ while the other half is filled with one of resistivity $\rho_2$ (the cross section of the shell is uniform). If a steady and uniform current density $J$ flows from the former material to the latter, parallel to the axis of the cylindrical shell, determine the surface charge density of the charges trapped at the interface.

Bruce Edelman

Numerade Educator

06:40

Problem 2

Consider two thin, concentric cylindrical shells, made of conducting material, that have length $l$ and radii $a$ and $b$, with $a<b$. The gap between the two shells is filled with a material with conductivity $\sigma(r)=\frac{k}{r}$ where $r$ is the perpendicular distance from the axis. The inner and outer shells are maintained at a constant potential difference $V_0$ with the inner shell having the higher potential. Determine the resistance of this set-up when a steady current flows between the shells. Determine the electric field between the two shells and hence the charge density $\rho$ everywhere within the mediating material.

Khoobchandra Agrawal

Numerade Educator

02:41

Problem 3

A homogeneous matter of unknown resistivity $\rho$ fills the region $x \geq 0$. Physicists then work in the other half-space to determine $\rho$. They draw a pale square with corners A, B, C, D at coordinates $\left(0,-\frac{a}{2}, \frac{a}{2}\right),\left(0, \frac{a}{2}, \frac{a}{2}\right),\left(0, \frac{a}{2},-\frac{a}{2}\right)$, $\left(0,-\frac{a}{2},-\frac{a}{2}\right)$. Subsequently, they inject a steady current $I$ into point A and withdraw the same amount from point B via tiny electrodes. If the (positive) potential difference between points $\mathrm{C}$ and $\mathrm{D}$ is measured as $V$, determine $\rho$.

Dading Chen

Numerade Educator

03:59

Problem 4

If a volume $\mathrm{V}$ is filled with a conducting material of uniform conductivity $\sigma$, determine the total electric charge $q(t)$ enclosed by the volume $\mathrm{V}$, given that the total initial charge inside $\mathrm{V}$ is $q_0$.

Keshav Singh

Numerade Educator

01:32

Problem 5

Two bulbs are connected to opposite sides of a circular loop of wire, as shown in the figure below. A changing magnetic field (confined within the loop) induces an emf in the loop that causes the two bulbs to light at first. All wires lie in the same plane. When the switch is closed, what happens to the bulbs? Subsequently, the wires containing the closed switch remain connected at points A and B and are lifted up (out of the page) and moved gradually towards the other side, until the configuration in the right figure is finally attained (where all wires again lie in the same plane). Describe the responses of the bulbs throughout this process.
(GRAPH CAN'T COPY)

Pawan Yadav

Numerade Educator

01:02

Problem 6

There are two long solenoids that are encircled by loops with two resistors $R_1$ and $R_2$, as shown in the figure below. The right solenoid produces a magnetic flux $\Phi_B^r=\alpha t$ out of the page through itself where $t$ is the time elapsed from a reference point while the left solenoid produces a magnetic flux $\Phi_B^l=\beta t$ out of the page through itself. The voltmeters are ideal and possess infinite resistance. If each voltmeter measures the voltage across its ends via a path through itself, determine the readings of the voltmeters.
(GRAPH CAN'T COPY)

Dominador Tan

Numerade Educator

04:57

Problem 7

Imagine a square loop of edge length $3.00 \mathrm{~m}$ and resistance $10.0 \Omega$ in this page. It is placed in a uniform external $0.100 \mathrm{~T}$ magnetic field that is directed perpendicularly into this page. The loop is then compressed (the lengths of the edges remain unchanged but the angles between them may vary) into a rhombus in the same plane, with a separation $3.00 \mathrm{~m}$ between two opposite vertices. Note that the edges can possibly protrude out of this page en route but they must still remain in this page in the final configuration. If this process takes $0.100 \mathrm{~s}$, what is the average current generated in the loop? What is its direction? Remember to account for the self-inductance of the loop.

Zachary Warner

Numerade Educator

03:56

Problem 8

A long solenoid of radius $R$ produces a magnetic field $B(t)=B_0 \sin \omega t$ out of the page (defined as the positive $\mathrm{Z}$-axis). Now, consider two points $A$ and $B$ in the plane of the cross section of the solenoid. If points A and B are outside the solenoid and their position vectors from the center of the solenoid in the current plane subtend an angle $\theta$, determine the voltage from points $\mathrm{A}$ to B along any line that does not cut through the solenoid. B is located anticlockwise of A, where the anti-clockwise direction is determined by applying the right-hand-grip rule to the positive z-direction.

Salamat Ali

Numerade Educator

06:18

Problem 9

A rectangular loop of dimensions $l$ and $h$ moves with a constant velocity $u$ away from a long wire that carries a steady current $I_1$ in the plane of the loop. The total resistance of the loop is $R$. Derive an expression for the current $I_2$ in the loop at the instant the closer side of the loop is a distance $r$ from the wire. We have done a similar problem before but use the flux rule this time. Notice that the resistor heats up. What is providing this energy or rather, doing work on the system? The magnetic force seems to be doing work! Resolve this apparent paradox.
(GRAPH CAN'T COPY)

Vishal Gupta

Numerade Educator

05:13

Problem 10

A square loop is exiting a constant and uniform magnetic field $\boldsymbol{B}$ with a constant velocity $v$ perpendicular to the magnetic field. Find the emf induced in the loop when the left end of the loop is at a distance $x$ from the right boundary of the magnetic field. Find the force required to maintain the velocity of the loop. The loop has resistance $R$ and negligible self-inductance.
(GRAPH CAN'T COPY)

Vishal Gupta

Numerade Educator

02:17

Problem 11

A wheel with six spokes is placed in a perpendicular magnetic field $B=0.5 \mathrm{~T}$ as shown in the figure. The field is directed into the plane of the paper and permeates the entire wheel. The external wires are connected to the wheel's center and a point on its rim. When the switch $\mathrm{S}$ is closed, there is an initial current of $6 \mathrm{~A}$ through the battery and the wheel begins to rotate. The resistance of the spokes and the rim may be neglected. You may find the result of Problem 5 (Chapter 7 ) to be helpful in the following questions.
(a) What is the direction of rotation of the wheel? Explain.
(b) The radius of the wheel is $r=0.2 \mathrm{~m}$. Calculate the initial torque on the wheel about its center.
(c) Describe qualitatively the angular velocity of the wheel as a function of time. Let the emf of the battery be $\varepsilon=1 \mathrm{~V}$.
(GRAPH CAN'T COPY)

Khoobchandra Agrawal

Numerade Educator

08:47

Problem 12

A loop of width $w$ and self-inductance $L$ is exiting a constant and uniform magnetic field $B$, that is perpendicular to the plane of the loop. An external force is exerted on the loop to sustain its velocity at a constant $v$. Determine the current in the loop as a function of time $t$ if there was no initial current. Determine the external force required to maintain the loop at a constant velocity as a function of time. Lastly, verify that the power delivered by the external force to the coil is consistent with the rate of increase in energy in all other entities, except that which exerts the external force.
(GRAPH CAN'T COPY)

Saikat Chatterjee

Numerade Educator

07:37

Problem 13

A ring with mass $m$, radius $r$ and resistance $R$ is rotating about a diameter in a region of constant and uniform external magnetic field $B$. Initially, the magnetic field passes through the ring perpendicularly, resulting in maximum flux through the ring, and the ring rotates with initial angular speed $\omega_0$. Neglect the self-inductance of the ring.
(a) Find the relation between the total angle $\phi$ that the ring rotates before it stops, and the other given variables.
(b) Find the number of complete rounds that the ring manages to rotate when it initially spins at 8 rotations per second, $m=1 \mathrm{~kg}, R=1 \Omega$, $r=30 \mathrm{~cm}$, and $B=1 \mathrm{~T}$.

Sheh Lit Chang

University of Washington

01:19

Problem 14

A bar of mass $m$ and resistance $R$ slides without friction in a horizontal plane that is perpendicular to a constant and uniform magnetic field $\boldsymbol{B}$, moving on parallel rails as shown. The perfectly conductive rails are separated by a distance $d$ and a battery maintains a constant $\varepsilon$ between these two rails. Assuming that the bar starts from rest at $t=0$, find the speed of the bar as a function of $t$. Neglect the self-inductance of this set-up.
(GRAPH CAN'T COPY)

Penny Riley

Numerade Educator

07:20

Problem 15

A uniform external magnetic field $B$ is directed vertically downwards, passing through the slope of an inclined plane which supports a wire loop as shown in the figure. A rod, of mass $m$, length $l$ and resistance $R_3$ is sliding down without friction, starting from rest. The wire loop behaves like a "rail" for the rod. Neglect all self- and mutual inductances in this set-up.
(a) When the rod has an instantaneous velocity $v$ (positive down the slope), find the magnitude and direction of the induced current flowing in each wire segment.
(b) Find the magnitude and direction of the magnetic force acting on the rod.
(c) Find the velocity and acceleration of the rod as functions of time.
(d) Find the instantaneous "power" of the magnetic force ${ }^{13}$ on the rod. You can leave the expression in terms of $v$.
(e) Show that the rate of change of the total mechanical energy of the rod equals to negative of the dissipated power.
(GRAPH CAN'T COPY)

Vishal Gupta

Numerade Educator

15:13

Problem 16

A rectangular loop of width $h$ and length $l$ is currently traveling in the xyplane (the length is aligned with the $\mathrm{x}$-direction). The resistance of the entire loop is $R$. A magnetic field $B=B_0 \cos (k x-\omega t)$, where $k$ and $\omega$ are constants, in the positive z-direction (out of the page) permeates all space. Given that the only possible force on the loop is the magnetic force, determine the condition for the loop to be able to travel at a constant velocity in the positive $\mathrm{x}$-direction at all possible velocities. When the previous condition is
${ }^{13}$ We really mean the power delivered by the component of magnetic force caused by the current in the rod.
not met, there is a terminal velocity of the loop when the loop is given an initial velocity in the $\mathrm{x}$-direction. Determine this terminal velocity. Neglect the self-inductance of the loop.

Linda Winkler

Numerade Educator

04:16

Problem 17

An outer solenoid of cross sectional area $A_1, \eta_1$ turns per unit length and anti-clockwise current $I_1$ (relative to the positive z-axis defined along its solenoid axis) encloses an inner, coaxial solenoid of cross sectional area $A_2$, $\eta_2$ turns per unit length and anti-clockwise current $I_2$. The currents in both perfectly conducting solenoids are maintained at their respective values by current sources. If the inner solenoid is now pulled out of the outer solenoid at a constant velocity $v$ in the positive $z$-direction, determine the emfs produced by the current sources. You may assume the individual magnetic fields of the solenoids to be uniform inside them and zero outside of them. Determine the ratio of these emfs and explain why it makes sense.

Ghazala Khan

Numerade Educator

07:51

Problem 18

On a smooth, insulating and neutral large ring of radius $R$, there is a small ring of mass $m$ which carries charge $q$. The large ring is placed in a uniform magnetic field of strength $B(t)$ and perpendicular to the plane of the ring (xy-plane), $B(t)=B_0+\alpha t$ in the positive z-direction. Find the force of the small ring acting on the big ring thereafter and describe the motion of the small ring. (Singapore Physics Olympiad)

Mahnoor Amin

Numerade Educator

01:10

Problem 19

A coil carrying a constant current $I$ with $N$ densely wound rounds of area $A$ is placed in a region of uniform magnetic field $\boldsymbol{B}$. Define $\theta$ to be the angle between the magnetic dipole moment and the magnetic field. Determine the external mechanical work required to rotate the magnetic dipole from $\theta=\frac{\pi}{2}$ to a general angle $\theta$ along a single axis of rotation, without any increase in kinetic energy. Now, there is an apparent paradox as the dipole does not gain any energy, and yet we are supplying (possibly negative) external work to it. Furthermore, we know that the magnetic force cannot perform any work on the coil (to counteract the mechanical work). Where does this external mechanical work then go? Note that we have analyzed a similar set-up before.

Ajay Singhal

Numerade Educator

01:33

Problem 20

Find the self inductance of a $N$-turn toroid which has a rectangular cross section with inner radius $r_1$, outer radius $r_2$ and height $h$.

Shahab Ullah

Numerade Educator

02:36

Problem 21

A finite solenoid of length $l$, radius $r$ and $N$ turns is placed in a long solenoid with turns per unit length $\eta$ and radius $R, r \ll R$. The axes of the solenoids coincide. Find the mutual inductance of this system.

Keshav Singh

Numerade Educator

01:14

Problem 22

If the self-inductance of an equilateral triangle loop of side length $l$ is $L$, determine the self-inductance of the loop shown in the figure below. It is akin to two sides of a tetrahedron formed by four equilateral triangles of length $l$.
(GRAPH CAN'T COPY)

Dominador Tan

Numerade Educator

06:50

Problem 23

If the self-inductance of an equilateral triangle loop of side length $l$ is $L$, determine the mutual inductance of the two loops that are placed side by side as shown in the figure below. One loop is an equilateral triangle of length $l$ while the other loop is a rhombus formed by joining two of such equilateral triangles together (and removing the common side).
(GRAPH CAN'T COPY)

Vishal Gupta

Numerade Educator

04:02

Problem 24

A primitive transformer is made by overlapping two solenoids of the same length and radius (that are not connected) and inserting an iron core within them to ensure that the magnetic fluxes through the turns of the solenoids are identical. The primary circuit is connected to one solenoid with $N_1$ turns while the secondary circuit is connected to the other solenoid with $N_2$ turns. To specify the direction of coupling of the mutual inductors, define $I_1$ to be the clockwise current in the primary circuit and $I_2$ to be the anti-clockwise current in the secondary circuit. The magnetic flux through the secondary solenoid due to a positive current $I_1$ in the primary solenoid reinforces the magnetic flux through the secondary solenoid due to its own positive current $I_2$ - the converse statement holds as well. Show that the ratio of the emfs produced by the two solenoids in the primary and secondary circuits is $\frac{\varepsilon_2}{\varepsilon_1}=\frac{N_2}{N_1}$.
Now, you may think that this set-up violates the conservation of energy as we can ramp up the emf in the secondary circuit by increasing $N_2$. To ease your worries, consider the situation where the primary solenoid is connected to an $\mathrm{AC}$ source with a clockwise $\operatorname{emf} \varepsilon=\varepsilon_0 \cos \omega t$ (i.e. no resistance in the primary circuit) while the secondary solenoid is connected to a resistor $R$. Find the steady state currents in the two circuits. Show that the conservation of energy holds in the steady state situation by computing the various rates of changes of energy. You may have to define some quantities of your own.

Khoobchandra Agrawal

Numerade Educator

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Problem 25

By applying Ampere-Maxwell law, show that the magnetic field (in the non-relativistic approximation $v \ll c$ ) due to a point charge $q$ traveling at a constant velocity $v$ is
$$
\boldsymbol{B}=\frac{\mu_0 q \boldsymbol{v} \times \hat{\boldsymbol{r}}}{r^2}
$$
where $r$ is the instantaneous vector joining the point charge to the location at which the magnetic field is of concern. You may assume that the electric field due to the moving charge is still given by Coulomb's law in the nonrelativistic regime.

Lainey Roebuck

Numerade Educator

01:23

Problem 26

Two identical square loops that are made of perfectly conducting material initially carry currents $I_1$ and $I_2$ when they are infinitely far apart. They are then brought closer such that the second loop now carries current $I_2^{\prime}$. Determine the current $I_1^{\prime}$ in the first loop at this juncture.

Supratim Pal

Numerade Educator

08:47

Problem 27

A conducting rectangular loop of self-inductance $L$ and mass $m$ is initially stationary at the edge of a region of uniform magnetic field $B$ (pointing out of the page) as shown in the figure below. The length of the loop along the vertical is $l$. The loop initially carries no current. If you now displace the loop towards the right, describe the motion that the loop will undergo. You do not need to solve the equation of motion of the loop.
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Saikat Chatterjee

Numerade Educator

06:15

Problem 28

Defining the z-axis to be positive upwards, the region $z \leq 0$ is covered with an infinite superconducting material. A small magnet, which can be modeled as a small magnetic dipole with a magnetic dipole moment $\boldsymbol{\mu}$ that is pointing in the positive z-direction, is currently levitating at a z-coordinate $h$. If the mass of the magnet is $M$, determine $h$.

Eduard Sanchez

Numerade Educator

11:03

Problem 29

Consider a finite solenoid, with $\eta$ turns per unit length, that is carrying a current $I$. The length of its cylindrical axis squared is much larger than its cross sectional area $A$. Argue qualitatively why this finite solenoid should experience a compressive force in the axial direction, in addition to the magnetic pressure discussed in Section 8.10.1. Determine the magnitude of this compressive force via the principle of virtual work. Finally, determine the force between two of such solenoids with their ends placed near each other and their axes aligned. The two solenoids carry currents $I_1$ and $I_2$, which are not necessarily in the same direction.

Brandy Heflin

Numerade Educator