Chapter 28

A \(+6.00-\mu C\) point charge is moving at a constant \(8.00 \times 10^{6} \mathrm{m} / \mathrm{s}\) in the \(+y\) -direction, relative to a reference frame. At the instant when the point charge is at the origin of this reference frame, what is the magnetic-field vector \(\overrightarrow{\boldsymbol{B}}\) it produces at the following points: \((a) x=0.500 \mathrm{m}, y=0, z=0 ;\) (b) \(x=0\) \(y=-0.500 \mathrm{m}, \quad z=0 ; \quad(\mathrm{c}) \quad x=0, \quad y=0, \quad z=+0.500 \mathrm{m} ;\) (d) \(x=0, y=-0.500 \mathrm{m}, z=+0.500 \mathrm{m} ?\)

Fields within the Atom. In the Bohr model of the hydrogen atom, the electron moves in a circular orbit of radius \(5.3 \times 10^{-11} \mathrm{m}\) with a speed of \(2.2 \times 10^{6} \mathrm{m} / \mathrm{s}\) . If we are viewing the atom in such a way that the electron's orbit is in the plane of the paper with the electron moving clockwise, find the magnitude and direction of the electric and magnetic fields that the electron produces at the location of the nucleus (treated as a point).

An alpha particle (charge + 2e) and an electron move in opposite directions from the same point, each with the speed of \(2.50 \times 10^{5} \mathrm{m} / \mathrm{s}\) (Fig. \(28.32 ) .\) Find the magnitude and direction of the total magnetic field these charges produce at point \(P,\) which is 1.75 \(\mathrm{nm}\) from each of them.

Along, straight wire lies along the \(z\) -axis and carries a \(4.00-\mathrm{A}\) current in the \(+z\) -direction. Find the magnetic field (magnitude and direction) produced at the following points by a \(0.500-\mathrm{mm}\) segment of the wire centered at the origin: (a) \(x=2.00 \mathrm{m}, y=0\) , \(z=0 ;(b) x=0, y=2.00 \mathrm{m}, z=0 ;(\mathrm{c}) x=2.00 \mathrm{m}, y=2.00 \mathrm{m}\) \(z=0 ;(\mathrm{d}) x=0, y=0, z=2.00 \mathrm{m}\)

The Magnetic Field from a Lightning Bolt. Lightning bolts can carry currents up to approximately 20 kA. We can model such a current as the equivalent of a very long, straight wire. (a) If you were unfortunate enough to be 5.0 \(\mathrm{m}\) away from such a lightning bolt, how large a magnetic field would you experience? (b) How does this field compare to one you would experience by being 5.0 \(\mathrm{cm}\) from a long, straight household current of 10 \(\mathrm{A} ?\)

A very long, straight horizontal wire carries a current such that \(3.50 \times 10^{18}\) electrons per second pass any given point going from west to east. What are the magnitude and direction of the magnetic field this wire produces at a point 4.00 \(\mathrm{cm}\) directly above it?

(a) How large a current would a very long, straight wire have to carry so that the magnetic field 2.00 \(\mathrm{cm}\) from the wire is equal to 1.00 G (comparable to the earth's northward-pointing magnetic field)? (b) If the wire is horizontal with the current running from east to west, at what locations would the magnetic field of the wire point in the same direction as the horizontal component of the earth's magnetic field? (c) Repeat part (b) except the wire is vertical with the current going upward.

Two long, straight wires, one above the other, are seperated by a distance 2\(a\) and are parallel to the \(x\) -axis. I et the \(+y\) -axis be in the plane of the wires in the direction from the lower wire to the upper wire. Each wire carries current \(I\) in the \(+x\) -direction. What are the magnitude and direction of the net magnetic field of the two wires at a point in the plane of the wires (a) midway between them; (b) at a distance \(a\) above the upper wire; \((c)\) at a distance \(a\) below the lower wire?

A long, straight wire lies along the \(y\) -axis and carries a current \(I=8.00\) A in the \(-y\) -direction (Fig. 28.39\()\) . In addition to the magnetic field due to the current in the wire, a uniform magnetic field \(\overrightarrow{\boldsymbol{B}}_{0}\) with magnitude \(1.50 \times 10^{-6} \mathrm{T}\) is in the \(+x\) -direction \(\mathrm{What}\) is the total field (magnitude and direction) at the following points in the \(x z\) -plane: \((a) x=0, z=1.00 \mathrm{m}\) (b) \(x=1.00 \mathrm{m}, z=0 ;(\mathrm{c}) x=0\) \(z=-0.25 \mathrm{m} ?\)

Effect of Transmission Lines, Two hikers are reading a compass under an overhead transmission line that is 5.50 \(\mathrm{m}\) above the ground and carries a current of 800 \(\mathrm{A}\) in a horizontal direction from north to south. (a) Find the magnitude and direction of the magnetic field at a point on the ground directiy under the conductor. (b) One hiker suggests they walk on another 50 \(\mathrm{m}\) to avoid inaccurate compass readings caused by the current. Considering that the magnitude of the earth's field is of the order of \(0.5 \times 10^{-4} \mathrm{T},\) is the current really a problem?

Two long. parallel transmission lines, 40.0 \(\mathrm{cm}\) apart, carry \(25.0-\mathrm{A}\) and 75.0 - A currents. Find all locations where the net magnetic field of the two wires is zero if these currents are in (a) the same direction and (b) the opposite direction.

Two long, parallel wires are separated by a distance of \(2.50 \mathrm{cm} .\) The force per unit length that each wire exerts on the other is 4.00 \(\times 10^{-5} \mathrm{N} / \mathrm{m}\) , and the wires repel each other. The current in one wire is 0.600 \(\mathrm{A}\) . (a) What is the current in the second wire? (b) Are the two currents in the same direction or in opposite directions?

Lamp Cord Wires. The wires in a household lamp cord are typically 3.0 mm apart center to center and carry equal currents in opposite directions. If the cord carries current to a 100 -W light bulb connected across a \(120-\mathrm{V}\) potential difference, what force per meter does each wire of the cord exert on the other? Is the force attractive or repulsive? Is this force large enough so it should be considered in the design of lamp cord? (Model the lamp cord as a very long straight wire.)

A long, horizontal wire \(A B\) rests on the surface of a table and carries a current \(I\) . Horizontal wire \(C D\) is vertically above wire \(A B\) and is free to slide up and down on the two vertical metal guides \(C\) and \(D\) (Fig. 28.45\()\) . Wire \(C D\) is connected through the sliding contacts to another wire that also carries a current \(I,\) opposite in direction to the current in wire \(A B .\) The mass per unit length of the wire \(C D\) is \(\lambda\) . To what equilibrium height \(h\) will the wire \(C D\) rise, assuming that the magnetic force on it is due entirely to the current in the wire \(A B ?\)

A closely wound, circular coil with radius 2.40 \(\mathrm{cm}\) has 800 turns. a) What must the current in the coil be if the magnetic field at the center of the coil is 0.0580 \(\mathrm{T}\) ? b) At what distance \(x\) from the center of the coil, on the axis of the coil, is the magnetic field half its value at the center?

A closely wound, circular coil with a diameter of 4.00 cm has 600 turns and carries a current of 0.500 A. What is the magnitude of the magnetic field (a) at the center of the coil and (b) at a point on the axis of the coil 8.00 \(\mathrm{cm}\) from its center?

A closely wound coil has a radius of 6.00 \(\mathrm{cm}\) and carries a current of 2.50 A. How many turns must it have if, at a point on the coil axis 6.00 \(\mathrm{cm}\) from the center of the coil, the magnetic field is \(6.39 \times 10^{-4} \mathrm{T} ?\)

A closed curve encircles several conductors. The line integral \(\phi \overrightarrow{\boldsymbol{B}} \cdot d \vec{l}\) around this curve is \(3.83 \times 10^{-4} \mathrm{T} \cdot \mathrm{m} .\) (a) What is the net current in the conductors? (b) If you were to integrate around the curve in the opposite direction, what would be the value of the line integral? Explain.

A long, straight, cylindrical wire of radius \(R\) carries a current uniformly distributed over its cross section. At what location is the magnetic field produced by this current equal to half of its largest value? Consider points inside and outside the wire.

A 15.0 -cm-long solenoid with radius 2.50 \(\mathrm{cm}\) is closely wound with 600 turns of wire. The current in the windings is 8.00 A. Compute the magnetic field at a point near the center of the solenoid.

A solenoid is designed to produce a magnetic field of 0.0270 \(\mathrm{T}\) at its center. It has radius 1.40 \(\mathrm{cm}\) and length \(40.0 \mathrm{cm},\) and the wire can carry a maximum current of 12.0 \(\mathrm{A}\) . (a) What minimum number of turns per unit length must the solenoid have? (b) What total length of wire is required?

As a new electrical technician, you are designing a large solenoid to produce a uniform 0.150 T magnetic field near the center of the solenoid. You have enough wire for 4000 circular turns. This solenoid must be 1.40 \(\mathrm{m}\) long and 20.0 \(\mathrm{cm}\) in diameter. What current will you need to produce the necessary field?

A magnetic field of 37.2 T has been achieved at the MTT Francis Bitter National Magnetic Laboratory. Find the current needed to achieve such a field (a) 2.00 \(\mathrm{cm}\) from a long, straight wire; \((b)\) at the center of a circular coil of radius 42.0 \(\mathrm{cm}\) that has 100 turns; \((\mathrm{c})\) near the center of a solenoid with radius \(2.40 \mathrm{cm},\) length \(32.0 \mathrm{cm},\) and \(40,000\) turns.

A toroidal solenoid (see Example 28.10 ) has inner radius \(r_{1}=15.0 \mathrm{cm}\) and ourer radius \(r_{2}=18.0 \mathrm{cm} .\) The solenoid has 250 turns and carries a current of 8.50 A. What is the magnitude of the magnetic field at the following distances from the center of the torus: (a) \(12.0 \mathrm{cm} ;(\mathrm{b}) 16.0 \mathrm{cm} ;(\mathrm{c}) 20.0 \mathrm{cm} ?\)

A wooden ring whose mean diameter is 14.0 \(\mathrm{cm}\) is wound with a closely spaced toroidal winding of 600 turns. Compute the magnitude of the magnetic field at the center of the cross section of the windings when the current in the windings is 0.650 \(\mathrm{A}\) .

A toroidal solenoid with 400 turns of wire and a mean radius of 6.0 cm carries a current of 0.25 A. The relative permeability of the core is \(80 .\) (a) What is the magnetic field in the core? (b) What part of the magnetic field is due to atomic currents?

A toroidal solenoid with 500 turns is wound on a ring with a mean radins of 290 \(\mathrm{cm}\) . Find the current in the winding that is required to set up a magnetic field of 0.350 \(\mathrm{T}\) in the ring (a) if the ring is made of annealed iron \(\left(K_{m}=1400\right)\) and \((b)\) if the ring is made of sillicon steel \(\left(K_{m}=5200\right)\) .

The current in the windings of a toroidal solenoid is 2.400 A. There are 500 turns, and the mean radius is 25.00 \(\mathrm{cm}\) . The toroidal solenoid is filled with a magnetic material. The magnetic field inside the windings is found to be 1.940 \(\mathrm{T}\) . Calculate (a) the relative permeability and \((b)\) the magnetic susceptibility of the material that fills the toroid.

A long solenoid with 60 turns of wire per centimeter carries a current of 0.15 \(\mathrm{A}\) . The wire that makes up the solenoid is wrapped around a solid core of silicon steel \(\left(K_{\mathrm{m}}=5200\right) .\) (The wire of the solenoid is jacketed with an insulator so that none of the current flows into the core.) (a) For a point inside the core, find the magnitudes of (i) the magnetic field \(\overrightarrow{\boldsymbol{B}}_{0}\) due to the solenoid current; (ii) the magnetization \(\vec{M} ;\) (iii) the total magnetic field \(\overrightarrow{\boldsymbol{B}}\) . (b) In a sketch of the solenoid and core, show the directions of the vectors \(\overrightarrow{\boldsymbol{B}}, \overrightarrow{\boldsymbol{B}}_{0}\) , and \(\overrightarrow{\boldsymbol{M}}\) inside the core.

A long, straight wire carries a current of 2.50 A. An electron is traveling in the vicinity of the wire. At the instant when the electron is 4.50 \(\mathrm{cm}\) from the wire and traveling with a speed of \(6.00 \times 10^{4} \mathrm{m} / \mathrm{s}\) directly toward the wire, what are the magnitude and direction (relative to the direction of the current) of the force that the magnetic field of the current exerts on the electron?

Two identical circular, wire loops 40.0 \(\mathrm{cm}\) in diameter each carry a current of 1.50 \(\mathrm{A}\) in the same direction. These loops are parallel to each other and are 25.0 \(\mathrm{cm}\) apart. Line \(a b\) is normal to the plane of the loops and passes through their centers. A proton is fired at 2400 \(\mathrm{km} / \mathrm{sperpendicular}\) to line \(a b\) from a point midway between the centers of the loops. Find the magnitude and direction of the magnetic force these loops exert on the proton just after it is fired.

A negative point charge \(q=-7.20 \mathrm{mC}\) is moving in a reference frame. When the point charge is at the origin, the magnetic field it produces at the point \(x=25.0 \mathrm{cm}, y=0, z=0\) is \(\vec{B}=(6.00 \mu \mathrm{T}) \hat{j},\) and its speed is 800 \(\mathrm{km} / \mathrm{s}\) . (a) What are the \(x\) - \(y\) .and \(z\) -components of the velocity \(\vec{v}_{0}\) of the charge? (b) At this same instant, what is the magnitude of the magnetic field that the charge produces at the point \(x=0, y=25.0 \mathrm{cm}, z=0 ?\)

A neophyte magnet designer tells you that he can produce a magnetic field \(\vec{B}\) in vacuum that points everywhere in the \(x\) -direction and that increases in magnitude with increasing \(x\) . That is, \(\vec{B}=B_{0}(x / a) \hat{\imath},\) where \(B_{0}\) and \(a\) are constants with units of teslas and meters, respectively. Use Gauss's law for magnetic fields to show that this claim is impossible. (Hint: Use a Gaussian surface in the shape of a rectangular box, with edges parallel to the \(x\) ; \(y\) . and \(z\) -axes.)

A pair of long, rigid metal rods, each of length \(L,\) lie parallel to each other on a perfectly smooth table. Their ends are connected by identical, very light conducting springs of force constant \(k\) (Fig. 28.55 ) and negligible unstretched length. If a current \(I\) runs through this circuit, the springs will stretch. At what separation will the rods remain at rest? Assume that \(k\) is large enough so that the separation of the mods will be much less than \(L\) .

A circular wire loop of radius \(a\) has \(N\) turns and carries a current \(I .\) A second loop with \(N^{\prime}\) turns of radius \(a^{\prime}\) carries current \(I^{\prime}\) and is located on the axis of the first loop, a distance \(x\) from the center of the first loop. The second loop is tipped so that its axis is at an angle \(\theta\) from the axis of the first loop. The distance \(x\) is large compared to both \(a\) and \(a^{\prime}\) (a) Find the magnitude of the torque exerted on the second loop by the first loop. (b) Find the potential energy for the second loop due to this interaction. (c) What simplifications result from having \(x\) much larger than \(a ?\) From having \(x\) much larger than \(a^{\prime} ?\)

Helmholtz Coils. Fig. 28.59 is a sectional view of two circular coils with radius \(a\) , each wound with \(N\) turns of wire carrying a current \(I,\) circulating in the same direction in both coils. The coils are separated by a distance \(a\) equal to their radii. In this configuration the coils are called Helmholtz coils; they produce a very uniform magnetic field in the region between them. (a) Derive the expression for the magnitude \(B\) of the magnetic field at a point on the axis a distance \(x\) to the right of point \(P,\) which is midway between the coils. (b) Graph \(B\) versus \(x\) for \(x=0\) to \(x=a / 2\) . Compare this graph to one for the magnetic field due to the right-hand coil alone. (c) From part (a), obtain an expression for the magnitude of the magnetic field at point \(P .\) (d) Calculate the magnitude of the magnetic field at \(P\) if \(N=300\) turns, \(I=6.00 \mathrm{A},\) and \(a=8.00 \mathrm{cm} .\) ( e) Calculate \(d B / d x\) and \(d^{2} B / d x^{2}\) at \(P(x=0)\) . Discuss how your results show that the field is very uniform in the vicinity of \(P .\)

A long, straight wire with a circular cross section of radius \(R\) carries a current \(L\) . Assume that the current density is not constant across the cross section of the wire, but rather varies as \(J=\alpha r,\) where \(\alpha\) is a constant. (a) By the requirement that \(J\) integrated over the cross section of the wire gives the total current \(I,\) calculate the constant \(\alpha\) in terms of \(I\) and \(R\) (b) Use Ampere's law to calculate the magnetic field \(B(r)\) for (i) \(r \leq R\) and (ii) \(r \geq R\) . Express your answers in terms of \(I\) .

A conductor is made in the form of a hollow cylinder with inner and outer radii a and \(b\) , respectively. It carries a current I uniformly distributed over its cross section. Derive expressions for the magnitude of the magnetic field in the regions (a) \(r < a\) ; (b) \(a< r b\).