Chapter 12: Sources of Magnetic FieldsContent check — University Physics 2, Chapter 12: Sources of Magnetic Fields

Using Example 12.1, at what distance would P have to be to measure a magnetic field half of the given answer?

The wire loop forms a full circle of radius R and current I. What is the magnitude of the magnetic field at the center?

Using Example 12.3, keeping the currents the same in wires 1 and 3, what should the current be in wire 2 to counteract the magnetic fields from wires 1 and 3 so that there is no net magnetic field at point P?

Two wires, both carrying current out of the page, have a current of magnitude 2.0 mA and 3.0 mA, respectively. The first wire is located at (0.0 cm, 5.0 cm) while the other wire is located at (12.0 cm, 0.0 cm). What is the magnitude of the magnetic force per unit length of the first wire on the second and the second wire on the first?

Using Example 12.5, at what distance would you have to move the first coil to have zero measurable magnetic field at point P?

Consider using Ampère’s law to calculate the magnetic fields of a finite straight wire and of a circular loop of wire. Why is it not useful for these calculations?

What is the ratio of the magnetic field produced from using a finite formula over the infinite approximation for an angle $\theta$ of (a) $85\text{°}\text{?}$ (b) $89\text{°}\text{?}$ The solenoid has 1000 turns in 50 cm with a current of 1.0 A flowing through the coils

Repeat the calculations from the previous example for $I_0=0.040\text{A}.$

For calculating magnetic fields, what are the advantages and disadvantages of the Biot-Savart law?

How can you decide if a wire is infinite?

How would you orient two long, straight, current-carrying wires so that there is no net magnetic force between them? (Hint: What orientation would lead to one wire not experiencing a magnetic field from the other?)

Is $\overrightarrow{B}$ constant in magnitude for points that lie on a magnetic field line?

What happens to the length of a suspended spring when a current passes through it?

Is Ampère’s law valid for all closed paths? Why isn’t it normally useful for calculating a magnetic field?

Explain why $\overrightarrow{B}=0$ inside a long, hollow copper pipe that is carrying an electric current parallel to the axis. Is $\overrightarrow{B}=0$ outside the pipe?

If you cut a bar magnet into two pieces, will you end up with one magnet with an isolated north pole and another magnet with an isolated south pole? Explain your answer.

Ten amps flow through a square loop where each side is 20 cm in length. At each corner of the loop is a 0.01-cm segment that connects the longer wires as shown. Calculate the magnitude of the magnetic field at the center of the loop.

The accompanying figure shows a current loop consisting of two concentric circular arcs and two perpendicular radial lines. Determine the magnetic field at point P.

Two long wires, one of which has a semicircular bend of radius R, are positioned as shown in the accompanying figure. If both wires carry a current I, how far apart must their parallel sections be so that the net magnetic field at P is zero? Does the current in the straight wire flow up or down?

The magnitude of the magnetic field 50 cm from a long, thin, straight wire is $8.0\text{μT}.$ What is the current through the long wire?

A long, straight, horizontal wire carries a left-to-right current of 20 A. If the wire is placed in a uniform magnetic field of magnitude $4.0\times {10}^{\text{−5}}\text{T}$ that is directed vertically downward, what is the resultant magnitude of the magnetic field 20 cm above the wire? 20 cm below the wire?

The accompanying figure shows two long, straight, horizontal wires that are parallel and a distance 2a apart. If both wires carry current I in the same direction, (a) what is the magnetic field at $P_1?$ (b) $P_2?$

Consider the area between the wires of the preceding problem. At what distance from the top wire is the net magnetic field a minimum? Assume that the currents are equal and flow in opposite directions.

Two long, straight wires are parallel and 10 cm apart. One carries a current of 2.0 A, the other a current of 5.0 A. (a) If the two currents flow in opposite directions, what is the magnitude and direction of the force per unit length of one wire on the other? (b) What is the magnitude and direction of the force per unit length if the currents flow in the same direction?

A circuit with current I has two long parallel wire sections that carry current in opposite directions. Find magnetic field at a point P near these wires that is a distance a from one wire and b from the other wire as shown in the figure.

When the current through a circular loop is 6.0 A, the magnetic field at its center is $2.0\times {10}^{\text{−4}}\text{T}.$ What is the radius of the loop?

A flat, circular loop has 20 turns. The radius of the loop is 10.0 cm and the current through the wire is 0.50 A. Determine the magnitude of the magnetic field at the center of the loop.

Two flat, circular coils, each with a radius R and wound with N turns, are mounted along the same axis so that they are parallel a distance d apart. What is the magnetic field at the midpoint of the common axis if a current I flows in the same direction through each coil?

A current I flows around the rectangular loop shown in the accompanying figure. Evaluate ${\displaystyle \oint \overrightarrow{B}}·d\overrightarrow{l}$ for the paths A, B, C, and D.

The coil whose lengthwise cross section is shown in the accompanying figure carries a current I and has N evenly spaced turns distributed along the length l. Evaluate ${\displaystyle \oint \overrightarrow{B}}·d\overrightarrow{l}$ for the paths indicated.

A long, straight wire of radius R carries a current I that is distributed uniformly over the cross-section of the wire. At what distance from the axis of the wire is the magnitude of the magnetic field a maximum?

A long, solid, cylindrical conductor of radius 3.0 cm carries a current of 50 A distributed uniformly over its cross-section. Plot the magnetic field as a function of the radial distance r from the center of the conductor.

A solenoid is wound with 2000 turns per meter. When the current is 5.2 A, what is the magnetic field within the solenoid?

If a current is 2.0 A, how many turns per centimeter must be wound on a solenoid in order to produce a magnetic field of $2.0\times {10}^{\mathrm{-3}}\text{T}$ within it?

Determine the magnetic field on the central axis at the opening of a semi-infinite solenoid. (That is, take the opening to be at $x=0$ and the other end to be at
$x=\infty .$)

A solenoid with 25 turns per centimeter carries a current I. An electron moves within the solenoid in a circle that has a radius of 2.0 cm and is perpendicular to the axis of the solenoid. If the speed of the electron is $2.0\times {10}^5\text{m/s},$ what is I?

A toroid with a square cross section 3.0 cm $\times$ 3.0 cm has an inner radius of 25.0 cm. It is wound with 500 turns of wire, and it carries a current of 2.0 A. What is the strength of the magnetic field at the center of the square cross section?

A solenoid has a ferromagnetic core, n = 1000 turns per meter, and I = 5.0 A. If B inside the solenoid is 2.0 T, what is $\chi$ for the core material?

The magnetic dipole moment of the iron atom is about $2.1\times {10}^{\mathrm{-23}}\text{A}·{\text{m}}^2.$ (a) Calculate the maximum magnetic dipole moment of a domain consisting of ${10}^{19}$ iron atoms. (b) What current would have to flow through a single circular loop of wire of diameter 1.0 cm to produce this magnetic dipole moment?

A current of 1.5 A flows through the windings of a large, thin toroid with 200 turns per meter and a radius of 1 meter. If the toroid is filled with iron for which $\chi =3.0\times {10}^3,$ what is the magnetic field within it?

Three long, straight, parallel wires, all carrying 20 A, are positioned as shown in the accompanying figure. What is the magnitude of the magnetic field at the point P?

The accompanying figure shows a long, straight wire carrying a current of 10 A. What is the magnetic force on an electron at the instant it is 20 cm from the wire, traveling parallel to the wire with a speed of $2.0\times {10}^5\text{m/s?}$ Describe qualitatively the subsequent motion of the electron.

(a) Use the result of the previous problem to calculate the magnetic field between, above, and below the pair of infinite sheets shown in the accompanying figure. (b) Repeat your calculations if the direction of the current in the lower sheet is reversed.

How is the fractional change in the strength of the magnetic field across the face of the toroid related to the fractional change in the radial distance from the axis of the toroid?

A toroid with an inner radius of 20 cm and an outer radius of 22 cm is tightly wound with one layer of wire that has a diameter of 0.25 mm. (a) How many turns are there on the toroid? (b) If the current through the toroid windings is 2.0 A, what is the strength of the magnetic field at the center of the toroid?

A reasonably uniform magnetic field over a limited region of space can be produced with the Helmholtz coil, which consists of two parallel coils centered on the same axis. The coils are connected so that they carry the same current I. Each coil has N turns and radius R, which is also the distance between the coils. (a) Find the magnetic field at any point on the z-axis shown in the accompanying figure. (b) Show that dB/dz and $\frac{d^{2}B}{d{z}^2}$ are both zero at z = 0. (These vanishing derivatives demonstrate that the magnetic field varies only slightly near z = 0.)

A thin, nonconducting disk of radius R is free to rotate around the axis that passes through its center and is perpendicular to the face of the disk. The disk is charged uniformly with a total charge q. If the disk rotates at a constant angular velocity $\omega ,$ what is the magnetic field at its center?

Consider the axial magnetic field $B_y=$ ${\mu }_{0}I{R}^2\text{/}2(y^2+R^2{)}^{3\text{/}2}$ of the circular current loop shown below. (a) Evaluate ${\displaystyle {\int }_{\text{−}a}^a{B}_{y}dy.}$ Also show that $\underset{a\to \infty }{\text{lim}}{\displaystyle {\int }_{\text{−}a}^a{B}_y}dy=$ ${\mu }_{0}I.$ (b) Can you deduce this limit without evaluating the integral? (Hint: See the accompanying figure.)

A long, straight, cylindrical conductor contains a cylindrical cavity whose axis is displaced by a from the axis of the conductor, as shown in the accompanying figure. The current density in the conductor is given by $\overrightarrow{J}=J_0\widehat{k},$ where $J_0$ is a constant and $\widehat{k}$ is along the axis of the conductor. Calculate the magnetic field at an arbitrary point P in the cavity by superimposing the field of a solid cylindrical conductor with radius $R_1$ and current density $\overrightarrow{J}$ onto the field of a solid cylindrical conductor with radius $R_2$ and current density $\text{−}\overrightarrow{J}.$ Then use the fact that the appropriate azimuthal unit vectors can be expressed as ${\widehat{\theta }}_1=\widehat{k}\times {\widehat{r}}_1$ and ${\widehat{\theta }}_2=\widehat{k}\times {\widehat{r}}_2$ to show that everywhere inside the cavity the magnetic field is given by the constant $\overrightarrow{B}=$ $\frac{1}{2}{\mu }_0{J}_{0}k\times a,$ where $a=r_1-r_2$ and $r_1=r_1{\widehat{r}}_1$ is the position of P relative to the center of the conductor and $r_2=r_2{\widehat{r}}_2$ is the position of P relative to the center of the cavity.

Show that the magnetic field of a thin wire and that of a current loop are zero if you are infinitely far away.

A very long, thick cylindrical wire of radius R carries a current density J that varies across its cross-section. The magnitude of the current density at a point a distance r from the center of the wire is given by $J=J_0\frac{r}{R},$ where $J_0$ is a constant. Find the magnetic field (a) at a point outside the wire and (b) at a point inside the wire. Write your answer in terms of the net current I through the wire.

Magnetic field inside a torus. Consider a torus of rectangular cross-section with inner radius a and outer radius b. N turns of an insulated thin wire are wound evenly on the torus tightly all around the torus and connected to a battery producing a steady current I in the wire. Assume that the current on the top and bottom surfaces in the figure is radial, and the current on the inner and outer radii surfaces is vertical. Find the magnetic field inside the torus as a function of radial distance r from the axis.

The accompanying figure shows a flat, infinitely long sheet of width a that carries a current I uniformly distributed across it. Find the magnetic field at the point P, which is in the plane of the sheet and at a distance x from one edge. Test your result for the limit $a\to 0.$

A nonconducting hard rubber circular disk of radius R is painted with a uniform surface charge density $\sigma .$ It is rotated about its axis with angular speed $\omega .$ (a) Find the magnetic field produced at a point on the axis a distance h meters from the center of the disk. (b) Find the numerical value of magnitude of the magnetic field when $\sigma =1{\text{C/m}}^2,$ $R=$ $\text{20 cm},$ $h=$ $\text{2 cm},$ and $\omega =$ $400\text{rad/sec},$ and compare it with the magnitude of magnetic field of Earth, which is about 1/2 Gauss.