Competitive Physics: Thermodynamics, Electromagnetism and Relativity

Wang Jinhui

Chapter 7 Magnetism - all with Video Answers

Educators

Chapter Questions

Problem 1

Determine the magnetic force per unit length between two parallel, thin and long current-carrying wires that are separated by a distance $r$ and held stationary. The currents in the wires are $I_1$ and $I_2$ respectively, possibly traveling in different directions. Now, wire 1 is still held fixed but wire 2 is gently released. The current in wire 1 is maintained at $I_1$ by an external battery. We know from the previous result that wire 2 will begin to move towards or away from wire 1 . Since the kinetic energy of wire 2 increases, have we violated the fact that a magnetic force cannot produce work? If not, suggest possible forms of energy that this kinetic energy originated from, in the case where wire 2 is not connected to any external entity and the case where the current in wire 2 is also maintained at $I_2$ by an external battery. Do not worry about how these energies are actually converted to the kinetic energy of wire 2 .

Supratim Pal

Numerade Educator

01:34

Problem 2

A charge $q$ moves along the positive $\mathrm{y}$-axis while another charge $-q$ moves along the positive $\mathrm{x}$-axis on fixed rails. Both start at the same time from the origin, and move with constant speed $0<v \ll c$. What is the magnetic force between the charges? Remember to indicate the direction too. Can you spot something wrong? Now, what if $v=0$ ? Argue, physically and mathematically, why your expressions are invalid.

Dominador Tan

Numerade Educator

02:45

Problem 3

Referring to the left figure, a current $I$ flowing along the edges of one face of a cube produces a magnetic field $B$ at the center of the cube. What is the magnetic field at the center of the same cube in the right figure? The cubes are isolated systems.
(GRAPH CAN'T COPY)

Khoobchandra Agrawal

Numerade Educator

01:12

Problem 4

An infinitely long and thin wire carrying current $I_1$ in the negative ydirection lies along the $\mathrm{y}$-axis $(x=0, z=0)$. Another short and thin wire carrying current $I_2$ in the positive x-direction lies parallel to the x-axis. Its two ends are at $(0,0, h)$ and $(l, 0, h)$. Find the force on the short wire due to the long wire.

Penny Riley

Numerade Educator

01:42

Problem 5

A uniform external magnetic field $\boldsymbol{B}$ permeates all space in the positive z-direction. Consider two points A and B in the xy-plane. An arbitrary connection of thin wires that lie entirely in the xy-plane is used to transfer $I$ amount of total current from A to B. No charge is accumulated anywhere, except for the terminals $\mathrm{A}$ and $\mathrm{B}$, possibly. If the linear distance between $\mathrm{A}$ and $\mathrm{B}$, is $l$, determine the total torque experienced by the wires connecting A to B, about terminal A. Note that the wires may merge or split freely.

Prem Bijarniya

Numerade Educator

06:05

Problem 6

A pair of two identical coils of radius $R$ are placed symmetrically along a common axis and are separated by a distance $d$. The common axis coincides with the z-axis, with the origin located at the center of the coils. The coils carry identical currents $I$ in the same direction and each has $N$ turns in total. Find the magnetic field strength along the axis of the coils $B(z)$ as a function of $z$. Determine the distance $d$ such that $\frac{\partial^2 B}{\partial z^2}=0$ at the center of the two coils. This set-up is known as the Helmholtz coil and is useful in generating a relatively uniform magnetic field between the coils (possibly to cancel Earth's local field).

Keshav Singh

Numerade Educator

00:46

Problem 7

An infinite wire is bent as shown in the figure below. Find the magnetic field at point $\mathrm{P}$.
(GRAPH CAN'T COPY)

- -

Numerade Educator

02:34

Problem 8

A thin, vertical solenoid with $\eta$ turns per unit length, length $l$ and radius $R$ carries a current $I$ anti-clockwise with respect to the positive z-axis which is along its symmetrical axis. Determine the magnetic field everywhere along the z-axis. To enforce the steady current condition, assume that thin wires whose contributions to the magnetic field can be neglected - are used to transfer current to and from infinity.
Now, consider a vertical solenoid with $\eta$ turns per unit length, inner radius $r_0$ and outer radius $r_1$ which carries a uniform current $I$ anti-clockwise. Determine the magnetic field everywhere along the $\mathrm{z}$-axis.

Manik Pulyani

Numerade Educator

06:42

Problem 9

An insulating sphere with a uniform volume charge density $\rho$ and radius $R$ rotates about an axis through its center with a constant angular frequency $\omega$. Find the magnetic field at a point along the axis of rotation.

Keshav Singh

Numerade Educator

06:42

Problem 10

A long cylinder, with its axis oriented in the z-direction, carries an axial current. The current density, although symmetric about the cylindrical axis, is not uniform but varies with radial distance $r$ from the axis according to
$$
J(r)= \begin{cases}\frac{2 I_0}{\pi a^2}\left(1-\frac{r^2}{a^2}\right) & \text { for } r<a \\ 0 & \text { for } r \geq a\end{cases}
$$
where $a$ is the radius of the cylinder and $I_0$ is a constant with units of amperes. Determine the magnetic field due to the cylinder everywhere.

Salamat Ali

Numerade Educator

01:35

Problem 11

A small long cylinder of radius $\frac{R}{2}$ is carved out of a long cylinder of radius $R$ as shown on the next page (a cross section is depicted). The "wire" then carries a uniform current $I$, coming out of the page. Find the magnetic field at point $\mathrm{P}$.
(GRAPH CAN'T COPY)

Donya Dobbin

Numerade Educator

03:31

Problem 12

The figure below shows the cross section of a current distribution that extends indefinitely in the z-direction. It is given by two overlapping circles of radius $b$ with a separation $2 a$ between their centers. The shaded regions of the left and right circles carry a uniform current density $J$ into and out of the page respectively. No current flows across the intersection of the two circles. Determine the magnetic field at a point in the overlapping region, in this cross section.
(GRAPH CAN'T COPY)

Khoobchandra Agrawal

Numerade Educator

01:40

Problem 13

A toroid is a donut with a uniform cross section. For example, a toroid with a circular cross section can be formed by bending a solenoid along a circular ring and connecting both ends together. Now, suppose that we coil a wire into a toroid with an arbitrary cross section and $N \gg 1$ densely wound turns. If the resultant toroid carries a current $I$, determine the magnetic field due to this set-up everywhere.

Keshav Singh

Numerade Educator

09:15

Problem 14

Consider a long solenoid with $\eta$ turns per unit length and radius $R$ that carries a steady current $I$. We categorize the solenoid into two halves about its center - one half contains the North pole while the other half contains the South. What is the net magnetic flux that leaves the solenoid through the lateral surface of the North half? Next, a field line propagates at a radial distance $r$ from the solenoid axis, in the direction of the solenoid axis and towards the North pole, at the central cross section of the solenoid. For what values of $r$ will the field line exit from the lateral surface of the solenoid? If it does not, what is its radial distance from the solenoid axis as it leaves the North end of the solenoid?

Khoobchandra Agrawal

Numerade Educator

03:43

Problem 15

A particle of mass $m$ and charge $q$ is currently located on the interior of a square room of side length $l$ as shown in the figure below (top view). A uniform magnetic field $B$ is directed into the page in the figure. Suppose that the particle is now propelled at an initial velocity $v_0=\frac{q B l}{8 m}$ perpendicular to a wall of the room, as depicted in the figure. Determine the time it requires to return to its initial position, assuming that its collisions with the walls are perfectly elastic.
(GRAPH CAN'T COPY)

Mahnoor Amin

Numerade Educator

01:52

Problem 16

Determine the instantaneous torque experienced by the loop below when it is placed in a uniform magnetic field $\boldsymbol{B}$, whose direction is depicted in the figure. The loop is composed of two semicircles of radius $r$ that subtend a right angle. $\boldsymbol{B}$ bisects the angle between the semicircles.
In light of the previous set-up, propose a definition for the magnetic dipole moment $\boldsymbol{\mu}$ of a non-planar loop C with $N$ turns, carrying a current $I$, such that the torque that it experiences in a uniform magnetic field $\boldsymbol{B}$ is
$$
\tau=\mu \times B
$$
(GRAPH CAN'T COPY)

Dominador Tan

Numerade Educator

View

Problem 17

A particle with a charge magnitude $q$ and mass $m$ is moving in the xy-plane. It is launched from $(-a, 0)$ with speed $v$. The region above $y=f(x)$ is filled with a uniform and constant magnetic $B$ pointing in the negative z-direction. It is observed that regardless of the direction of its initial velocity, as long as the charge enters the magnetic field region in the region $x \leq 0$, it will pass through $(a, 0)$ via a path symmetric about the $\mathrm{y}$-axis. (International Physics Olympiad)
(a) What is the sign of the charge?
(b) With what speed does the charge pass through point $(a, 0)$ ?
(c) Find the function $f(x)$.

Eduard Sanchez

Numerade Educator

05:31

Problem 18

A solenoid of $N$ turns carrying a constant current $I$ is wound around the equatorial plane of a uniform sphere of radius $R$ and mass $m$. The sphere is then placed on a rough, inclined plane with an angle of inclination $\theta$, as shown in the figure below. A uniform magnetic field $\boldsymbol{B}$ points upwards everywhere. The magnetic dipole moment of the coil is initially directed perpendicularly outwards from the plane and the sphere is initially stationary. If the sphere rolls without slipping subsequently, determine the clockwise angular velocity of the sphere $\dot{\phi}$, as a function of the clockwise angle $\phi$ that the sphere has rotated.
(GRAPH CAN'T COPY)

Jonathan Ibarra

Numerade Educator

05:01

Problem 19

A fixed infinite wire carries current $I$ along the positive z-axis. Supposing that a charged particle of mass $m$ and charge $q$ is initially launched at a radial velocity $v_0>0$ (positive outwards), at a perpendicular distance $r_0$ from the wire, determine the maximum and minimum radial distances that the charged particle can attain from the wire.

Nicholas Majtenyi

Numerade Educator

13:48

Problem 20

Two identical particles of mass $m$ and charge $q$ are placed along the $\mathrm{x}$ axis, in a region of uniform magnetic field $B$ in the positive z-direction, with arbitrary initial velocities. The charges still lie in the xy-plane in their resulting motion.
By denoting $r_1$ and $r_2$ as the position vectors of the two charges, write down the equations of motion of the charges. Now, express these in terms of the position vector of the center of mass $r_{C M}$ and the separation vector $r=r_1-r_2$. What is the motion of the center of mass of the two charges?
Now, supposing that we wish to ensure that the distance between the two charges is a constant $d$, show that there is a minimum $d$ for which this is possible. The angular velocity of the separation vector $r$ in the lab frame, that corresponds to the minimum $d$, undertakes a certain constant value $\omega$. After determining this $\omega$, show that the original position of the center of mass and the instantaneous positions of the two particles are always collinear if the initial velocity of the center of mass (which is non-zero) does not have a y-component.

Khoobchandra Agrawal

Numerade Educator

01:10

Problem 21

Two particles of the same mass $m$ and charges $q$ and $-q$ are placed along the $\mathrm{x}$-axis, in a region of uniform magnetic field $B$ in the positive z-direction. Let the position vectors of the charges $q$ and $-q$ be $\boldsymbol{r}_1$ and $\boldsymbol{r}_2$ respectively and define the separation vector as $r=r_1-r_2$. Only under certain special initial conditions can the two charges remain in the xy-plane while their separation vector remains at a constant magnitude $d$ and rotates at a constant angular velocity $\omega$. Determine the initial velocity of the center of mass of the two particles that results in such a motion. Given $\omega$, find $d$ and show that there exists a minimum magnitude of the angular velocity $\omega$ for such a motion to be possible.

Narayan Hari

Numerade Educator

01:01

Problem 22

A thin, vertical solenoid carrying anti-clockwise current $I$ with $\eta$ turns per unit length, length $l$ and radius $R$ can be used to focus off-axis charges. Define the origin at the bottom of the solenoid and the z-axis to be positive upwards (towards the top of the solenoid).
(a) Determine the longitudinal and radial magnetic field at a small perpendicular distance $r \ll R$ from the cylindrical axis as a function of z-coordinate $z$. Hint: we have calculated one of these fields in a previous problem.
(b) Now, suppose that particles with charge $q$ and mass $m$ are placed at radial distances $r \ll R$ at the bottom of the solenoid. They are given a large velocity $v_0$ in the positive $z$-direction such that their radial coordinates are approximately constant throughout the solenoid. If $l \gg R$, approximately how large should $v_0$ be as compared to the other parameters for this to occur? Now, determine the instantaneous velocities of these particles at the instance they exit from the top of the solenoid.
(c) Determine the time $t$ after this instance, at which they coincide with the z-axis, assuming that the magnetic field outside the solenoid is zero. Would this set-up function as a good lens to focus charges (of the same magnitude) with different initial radial distances?

Dominador Tan

Numerade Educator