Challenge Problemsthe magnetic field at point A. (b) The wire is now unwrapped so it is straight, centered on point C, and perpendicular to the line AC, but the same current is maintained in it. Now find the magnetic field at point A. (c) Which field is greater: the one in part (a) or in part (b)? By what factor? Why is this result physically reasonable? 28.77 . CALC 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 B1r2 for (i) r … R and (ii) r Ú R. Express your answers in terms of I. 28.78 . CALC The wire shown Figure P28.78 in Fig. P28.78 is infinitely long and carries a current I. Calculate I the magnitude and direction of a P the magnetic field that this curI rent produces at point P. 28.79 . 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 6 a; (b) a 6 r 6 b; (c) r 7 b. 28.80 . A circular loop has Figure P28.80 radius R and carries current I2 I2 in a clockwise direction (Fig. P28.80). The center of the loop is a distance D above a R long, straight wire. What are D the magnitude and direction of the current I1 in the wire if I1 the magnetic field at the center of the loop is zero? 28.81 . CALC A long, straight, solid cylinder, oriented with its S axis in the z-direction, carries a current whose current density is J . The current density, although symmetric about the cylinder axis, is not constant but varies according to the relationship S r 2 c 1 - a b d kN for r … a a pa ⴝ0 for r Ú a Jⴝ 2I0 2 where a is the radius of the cylinder, r is the radial distance from the cylinder axis, and I0 is a constant having units of amperes. (a) Show that I0 is the total current passing through the entire cross section of the wire. (b) Using Ampere’s law, derive an expression S for the magnitude of the magnetic field B in the region r Ú a. (c) Obtain an expression for the current I contained in a circular cross section of radius r … a and centered at the cylinder axis. (d) Using Ampere’sSlaw, derive an expression for the magnitude of the magnetic field B in the region r … a. How do your results in parts (b) and (d) compare for r = a? 28.82 . A long, straight, solid cylinder, oriented with its axis in S the z-direction, carries a current whose current density is J . The current density, although symmetric about the cylinder axis, is not constant and varies according to the relationship b J ⴝ a be1r - a2>dkN for r … a r S ⴝ0 for r Ú 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 d is a constant equal to 2.50 cm. (a) Let I0 be the total current passing 955 through the entire cross section of the wire. Obtain an expression for I0 in terms of b, d, and a. Evaluate your expression to obtain a numerical value for I0 . (b) Using Ampere’s law, derive an expresS sion for the magnetic field B in the region r Ú a. Express your answer in terms of I0 rather than b. (c) Obtain an expression for the current I contained in a circular cross section of radius r … a and centered at the cylinder axis. Express your answer in terms of I0 rather than b. (d) Using Ampere’s law, derive an expression for the S magnetic field B in the region r … a. (e) Evaluate the magnitude of the magnetic field at r = d, r = a, and r = 2a. 28.83 . An Infinite Current Sheet. Long, straight conduc- Figure P28.83 tors with square cross sections and each carrying current I are laid side by side to form an infinite current sheet (Fig. P28.83). x The conductors lie in the I y z 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? 28.84 . Long, straight conductors Figure P28.84 with square cross section, each carP rying current I, are laid side by side to form an infinite current sheet with current directed out of the d R plane of the page (Fig. P28.84). A second infinite current sheet is a distance d below the first and is S parallel to it. The second sheet carries current into the plane of the page. Each sheet has n conductors per unit length. (Refer to Problem 28.83.) Calculate the magnitude and direction of the net magnetic field at (a) point P (above the upper sheet); (b) point R (midway between the two sheets); (c) point S (below the lower sheet). 28.85 . CP A piece of iron has magnetization M = 6.50 * 10 4 A>m. Find the average magnetic dipole moment per atom in this piece of iron. Express your answer both in A # m2 and in Bohr magnetons. The density of iron is given in Table 14.1, and the atomic mass of iron (in grams per mole) is given in Appendix D. The chemical symbol for iron is Fe. CHALLENGE PROBLEMS 28.86 ... A wide, long, insulating belt has a uniform positive charge per unit area s 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 28.83.) 28.87 ... CP Two long, straight conducting wires with linear mass density l are suspended from cords so that they are each horizontal, parallel to each other, and a distance d apart. The back ends of the wires are connected to each other by a slack, lowresistance connecting wire. A charged capacitor (capacitance C) is now added to the system; the positive plate of the capacitor (initial charge +Q 0) is connected to the front end of one of the wires, and the negative plate of the capacitor (initial charge -Q 0) is connected to the front end of the other wire (Fig. P28.87). Both of