Chapter 28

A +6.00-\(\mu\)C point charge is moving at a constant 8.00 \(\times\) 10\(^6\) m/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 \(\overrightarrow{B}\) it produces at the following points: (a) \(x =\) 0.500 m, \(y =\) 0, \(z =\) 0; (b) \(x =\) 0, \(y = -\)0.500 m, \(z =\) 0; (c) \(x =\) 0, \(y =\) 0, \(z = +\)0.500 m; (d) \(x =\) 0, \(y4 = -\)0.500 m, \(z = +\)0.500 m?

A -4.80-\(\mu\)C charge is moving at a constant speed of 6.80 \(\times\) 10\(^5\) m/s in the +\(x\) direction relative to a reference frame. At the instant when the point charge is at the origin, what is the magnetic-field vector it produces at the following points: (a) \(x =\) 0.500 m, \(y =\) 0, \(z =\) 0; (b) \(x =\) 0, \(y =\) 0.500 m, \(z =\) 0; (c) \(x =\) 0.500 m, \(y =\) 0.500 m, \(z =\) 0; (d) \(x =\) 0, \(y =\) 0, \(z =\) 0.500 m?

A short current element \(\overrightarrow{dl}\) S = (0.500 mm)\(\hat{\imath}\) carries a current of 5.40 A in the same direction as \(\overrightarrow{dl}\). Point P is located at \(\overrightarrow{r} =\) (-0.730 m)\(\hat{\imath}\) + (0.390 m)\(\hat{k}\) . Use unit vectors to express the magnetic field at \(P\) produced by this current element.

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

A square wire loop 10.0 cm on each side carries a clockwise current of 8.00 A. Find the magnitude and direction of the magnetic field at its center due to the four 1.20-mm wire segments at the midpoint of each side.

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 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 cm from a long, straight household current of 10 A?

A very long, straight horizontal wire carries a current such that 8.20 \(\times\) 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 cm directly above it?

The body contains many small currents caused by the motion of ions in the organs and cells. Measurements of the magnetic field around the chest due to currents in the heart give values of about 10 \(\mu\)G. Although the actual currents are rather complicated, we can gain a rough understanding of their magnitude if we model them as a long, straight wire. If the surface of the chest is 5.0 cm from this current, how large is the current in the heart?

Certain bacteria (such as \(Aquaspirillum\) \(magnetotacticum\)) tend to swim toward the earth's geographic north pole because they contain tiny particles, called magnetosomes, that are sensitive to a magnetic field. If a transmission line carrying 100 A is laid underwater, at what range of distances would the magnetic field from this line be great enough to interfere with the migration of these bacteria? (Assume that a field less than 5\(\%\) of the earth's field would have little effect on the bacteria. Take the earth's field to be 5.0 \(\times\) 10\(^{-5}\) T, and ignore the effects of the seawater.)

(a) How large a current would a very long, straight wire have to carry so that the magnetic field 2.00 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 separated by a distance 2\(a\) and are parallel to the \(x\)-axis. Let 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?

Currents in dc transmission lines can be 100 A or higher. Some people are concerned that the electromagnetic fields from such lines near their homes could pose health dangers. For a line that has current 150 A and a height of 8.0 m above the ground, what magnetic field does the line produce at ground level? Express your answer in teslas and as a percentage of the earth's magnetic field, which is 0.50 G. Is this value cause for worry?

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

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 direct current to a 100-W light bulb connected across a 120-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 the lamp cord? (Model the lamp cord as a very long straight wire.)

The magnetic field around the head has been measured to be approximately 3.0 \(\times\) 10\(^{-8}\) G. Although the currents that cause this field are quite complicated, we can get a rough estimate of their size by modeling them as a single circular current loop 16 cm (the width of a typical head) in diameter. What is the current needed to produce such a field at the center of the loop?

A closely wound, circular coil with radius 2.40 cm has 800 turns. (a) What must the current in the coil be if the magnetic field at the center of the coil is 0.0770 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 single circular current loop 10.0 cm in diameter carries a 2.00-A current. (a) What is the magnetic field at the center of this loop? (b) Suppose that we now connect 1000 of these loops in series within a 500 cm length to make a solenoid 500 cm long. What is the magnetic field at the center of this solenoid? Is it 1000 times the field at the center of the loop in part (a)? Why or why not?

A closely wound coil has a radius of 6.00 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 cm from the center of the coil, the magnetic field is 6.39 \(\times\) 10\(^{-4}\) T?

Two concentric circular loops of wire lie on a tabletop, one inside the other. The inner wire has a diameter of 20.0 cm and carries a clockwise current of 12.0 A, as viewed from above, and the outer wire has a diameter of 30.0 cm. What must be the magnitude and direction (as viewed from above) of the current in the outer wire so that the net magnetic field due to this combination of wires is zero at the common center of the wires?

A closed curve encircles several conductors. The line integral \(\oint\overrightarrow{B}\) \(\cdot\) \(\overrightarrow{dl}\) around this curve is 3.83 \(\times\) 10\(^{-4}\) T \(\cdot\) 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.

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 55.0 cm long and 2.80 cm in diameter. What current will you need to produce the necessary field?

A solenoid that is 35 cm long and contains 450 circular coils 2.0 cm in diameter carries a 1.75-A current. (a) What is the magnetic field at the center of the solenoid, 1.0 cm from the coils? (b) Suppose we now stretch out the coils to make a very long wire carrying the same current as before. What is the magnetic field 1.0 cm from the wire's center? Is it the same as that in part (a)? Why or why not?

A 15.0-cm-long solenoid with radius 0.750 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 toroidal solenoid has an inner radius of 12.0 cm and an outer radius of 15.0 cm. It carries a current of 1.50 A. How many equally spaced turns must it have so that it will produce a magnetic field of 3.75 mT at points within the coils 14.0 cm from its center?

An ideal toroidal solenoid (see Example 28.10) has inner radius \(r_1 =\) 15.0 cm and outer radius \(r_2 =\) 18.0 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 cm; (b) 16.0 cm; (c) 20.0 cm?

A wooden ring whose mean diameter is 14.0 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 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 long solenoid with 60 turns of wire per centimeter carries a current of 0.15 A. The wire that makes up the solenoid is wrapped around a solid core of silicon steel (\(K_m =\) 5200). (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 \(\overrightarrow{B_0}\) due to the solenoid current; (ii) the magnetization \(\overrightarrow{M}\); (iii) the total magnetic field \(\overrightarrow{B}\). (b) In a sketch of the solenoid and core, show the directions of the vectors \(\overrightarrow{B}\), \(\overrightarrow{B_0}\), and \(\overrightarrow{M}\) inside the core.

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

At a particular instant, charge \(q_1 = +\)4.80 \(\times\) 10\(^{-6}\) C is at the point (0, 0.250 m, 0) and has velocity \(\vec{v_1}\) = (9.20 \(\times\) 10\(^5\) m/s)\(\hat{\imath}\). Charge \(q_2 = -\)2.90 \(\times\) 10\(^{-6}\) C is at the point (0.150 m, 0, 0) and has velocity \(\vec{v_2} =\) (-5.30 \(\times\) 10\(^5\) m/s)\(\hat{\jmath}\). At this instant, what are the magnitude and direction of the magnetic force that \(q_1\) exerts on \(q_2\)?

Two long, parallel transmission lines, 40.0 cm apart, carry 25.0-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.

A long, straight wire carries a current of 8.60 A. An electron is traveling in the vicinity of the wire. At the instant when the electron is 4.50 cm from the wire and traveling at a speed of 6.00 \(\times\) 10\(^4\) m/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?

An electron is moving in the vicinity of a long, straight wire that lies along the \(x\)-axis. The wire has a constant current of 9.00 A in the \(-x\)-direction. At an instant when the electron is at point (0, 0.200 m, 0) and the electron's velocity is \(\vec{v} =\) (5.00 \(\times\) 10\(^4\) m/s)\(\hat{\imath}\) - (3.00 \(\times\) 10\(^4\) m/s)\(\hat{\jmath}\), what is the force that the wire exerts on the electron? Express the force in terms of unit vectors, and calculate its magnitude.

An electric bus operates by drawing direct current from two parallel overhead cables, at a potential difference of 600 V, and spaced 55 cm apart. When the power input to the bus's motor is at its maximum power of 65 hp, (a) what current does it draw and (b) what is the attractive force per unit length between the cables?

A long, straight wire with a circular cross section of radius \(R\) carries a current \(I\). Assume that the current density is not constant across the cross section of the wire, but rather varies as \(J =\) \(ar\), where a 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 \(a\) 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 long, straight, solid cylinder, oriented with its axis in the \(z\)-direction, carries a current whose current density is \(\overrightarrow{J}\). The current density, although symmetric about the cylinder axis, is not constant but varies according to the relationship $$\overrightarrow{J} = \frac{2I_0}{\pi{a}^2} [1-(\frac{r}{a})^2]\hat{k} \space for \space r \leq a$$ $$=0 \space for \space r \geq a$$ where a is the radius of the cylinder, \(r\) is the radial distance from the cylinder axis, and \(I_0\) is a constant having units of amperes. (a) Show that \(I_0\) is the total current passing through the entire cross section of the wire. (b) Using Ampere's law, derive an expression for the magnitude of the magnetic field \(\overrightarrow{B}\) in the region r \(\geq a\). (c) Obtain an expression for the current I contained in a circular cross section of radius \(r \leq a\) and centered at the cylinder axis. (d) Using Ampere's law, derive an expression for the magnitude of the magnetic field \(\overrightarrow{B}\) in the region \(r \leq a\). How do your results in parts (b) and (d) compare for \(r = a\)?

Long, straight conductors with square cross sections and each carrying current \(I\) are laid side by side to form an infinite current sheet (Fig. P28.73). The conductors lie in the \(xy\)-plane, are parallel to the \(y\)-axis, and carry current in the +\(y\)-direction. There are \(n\) conductors per unit length measured along the \(x\)-axis. (a) What are the magnitude and direction of the magnetic field a distance \(a\) below the current sheet? (b) What are the magnitude and direction of the magnetic field a distance a above the current sheet?

A long, straight, solid cylinder, oriented with its axis in the \(z\)-direction, carries a current whose current density is \(\overrightarrow{J}\). The current density, although symmetric about the cylinder axis, is not constant and varies according to the relationship $$\overrightarrow{J} = (\frac{b}{r})e^{(r a)/\delta}\hat{k} \space for \space r \leq a$$ $$=0 \space for \space r \geq a$$ where the radius of the cylinder is a = 5.00 cm, \(r\) is the radial distance from the cylinder axis, \(b\) is a constant equal to 600 A/m, and \(\delta\) is a constant equal to 2.50 cm. (a) Let \(I_0\) be the total current passing through the entire cross section of the wire. Obtain an expression for \(I_0\) in terms of \(b\), \(\delta\), and a. Evaluate your expression to obtain a numerical value for I0. (b) Using Ampere's law, derive an expression for the magnetic field \(\overrightarrow{B}\) in the region \(r \leq a\). Express your answer in terms of \(I_0\) rather than b. (c) Obtain an expression for the current \(I\) contained in a circular cross section of radius \(r \leq a\) and centered at the cylinder axis. Express your answer in terms of \(I_0\) rather than b. (d) Using Ampere's law, derive an expression for the magnetic field \(\overrightarrow{B}\) in the region \(r \leq a\). (e) Evaluate the magnitude of the magnetic field at \(r = \delta\), \(r = a\), and \(r = 2a\).

A wide, long, insulating belt has a uniform positive charge per unit area \(\sigma\) on its upper surface. Rollers at each end move the belt to the right at a constant speed \(v\). Calculate the magnitude and direction of the magnetic field produced by the moving belt at a point just above its surface. (\(Hint:\) At points near the surface and far from its edges or ends, the moving belt can be considered to be an infinite current sheet like that in Problem 28.73.)

To use a larger sample, the experimenters construct a solenoid that has the same length, type of wire, and loop spacing but twice the diameter of the original. How does the maximum possible magnetic torque on a bacterium in this new solenoid compare with the torque the bacterium would have experienced in the original solenoid? Assume that the currents in the solenoids are the same. The maximum torque in the new solenoid is (a) twice that in the original one; (b) half that in the original one; (c) the same as that in the original one; (d) one-quarter that in the original one.

The solenoid is removed from the enclosure and then used in a location where the earth's magnetic field is 50 \(\mu\)T and points horizontally. A sample of bacteria is placed in the center of the solenoid, and the same current is applied that produced a magnetic field of 150 \(\mu\)T in the lab. Describe the field experienced by the bacteria: The field (a) is still 150 \(\mu\)T; (b) is now 200 \(\mu\)T; (c) is between 100 and 200 \(\mu\)T, depending on how the solenoid is oriented; (d) is between 50 and 150 \(\mu\)T, depending on how the solenoid is oriented.