Chapter 29

A single loop of wire with an area of 0.0900 m\(^2\) is in a uniform magnetic field that has an initial value of 3.80 T, is perpendicular to the plane of the loop, and is decreasing at a constant rate of 0.190 T/s. (a) What emf is induced in this loop? (b) If the loop has a resistance of 0.600 \(\Omega\), find the current induced in the loop.

In a physics laboratory experiment, a coil with 200 turns enclosing an area of 12 cm\(^2\) is rotated in 0.040 s from a position where its plane is perpendicular to the earth's magnetic field to a position where its plane is parallel to the field. The earth's magnetic field at the lab location is 6.0 \(\times\) 10\(^{-5}\) T. (a) What is the total magnetic flux through the coil before it is rotated? After it is rotated? (b) What is the average emf induced in the coil?

A closely wound search coil (see Exercise 29.3) has an area of 3.20 cm\(^2\), 120 turns, and a resistance of 60.0 \(\Omega\). It is connected to a charge- measuring instrument whose resistance is 45.0 \(\Omega\). When the coil is rotated quickly from a position parallel to a uniform magnetic field to a position perpendicular to the field, the instrument indicates a charge of 3.56 \(\times\) 10\(^{-5}\) C. What is the magnitude of the field?

A circular loop of wire with a radius of 12.0 cm and oriented in the horizontal \(xy\)-plane is located in a region of uniform magnetic field. A field of 1.5 T is directed along the positive z-direction, which is upward. (a) If the loop is removed from the field region in a time interval of 2.0 ms, find the average emf that will be induced in the wire loop during the extraction process. (b) If the coil is viewed looking down on it from above, is the induced current in the loop clockwise or counterclockwise?

A coil 4.00 cm in radius, containing 500 turns, is placed in a uniform magnetic field that varies with time according to \(B =\) (0.0120 T/s)\(t\) + (3.00 \(\times\) 10\(^{-5}\) T/s\(^4)t^4\). The coil is connected to a 600-\(\Omega\) resistor, and its plane is perpendicular to the magnetic field. You can ignore the resistance of the coil. (a) Find the magnitude of the induced emf in the coil as a function of time. (b) What is the current in the resistor at time \(t =\) 5.00 s?

A circular loop of flexible iron wire has an initial circumference of 165.0 cm, but its circumference is decreasing at a constant rate of 12.0 cm/s due to a tangential pull on the wire. The loop is in a constant, uniform magnetic field oriented perpendicular to the plane of the loop and with magnitude 0.500 T. (a) Find the emf induced in the loop at the instant when 9.0 s have passed. (b) Find the direction of the induced current in the loop as viewed looking along the direction of the magnetic field.

A closely wound rectangular coil of 80 turns has dimensions of \(25.0 \mathrm{~cm}\) by \(40.0 \mathrm{~cm} .\) The plane of the coil is rotated from a position where it makes an angle of \(37.0^{\circ}\) with a magnetic field of \(1.70 \mathrm{~T}\) to a position perpendicular to the field. The rotation takes \(0.0600 \mathrm{~s}\). What is the average emf induced in the coil?

In a region of space, a magnetic field points in the +\(x\)-direction (toward the right). Its magnitude varies with position according to the formula \(B_x = B_0 + bx\), where \(B_0\) and \(b\) are positive constants, for \(x \geq\) 0. A flat coil of area \(A\) moves with uniform speed \(v\) from right to left with the plane of its area always perpendicular to this field. (a) What is the emf induced in this coil while it is to the right of the origin? (b) As viewed from the origin, what is the direction (clockwise or counterclockwise) of the current induced in the coil? (c) If instead the coil moved from left to right, what would be the answers to parts (a) and (b)?

In many magnetic resonance imaging (MRI) systems, the magnetic field is produced by a superconducting magnet that must be kept cooled below the superconducting transition temperature. If the cryogenic cooling system fails, the magnet coils may lose their superconductivity and the strength of the magnetic field will rapidly decrease, or \(quench\). The dissipation of energy as heat in the now-nonsuperconducting magnet coils can cause a rapid boil-off of the cryogenic liquid (usually liquid helium) that is used for cooling. Consider a superconducting MRI magnet for which the magnetic field decreases from 8.0 T to nearly 0 in 20 s. What is the average emf induced in a circular wedding ring of diameter 2.2 cm if the ring is at the center of the MRI magnet coils and the original magnetic field is perpendicular to the plane that is encircled by the ring?

The armature of a small generator consists of a flat, square coil with 120 turns and sides with a length of 1.60 cm. The coil rotates in a magnetic field of 0.0750 T. What is the angular speed of the coil if the maximum emf produced is 24.0 mV?

How fast (in m/s and mph) would a 5.00-cm copper bar have to move at right angles to a 0.650-T magnetic field to generate 1.50 V (the same as a AA battery) across its ends? Does this seem like a practical way to generate electricity?

Airplanes and trains move through the earth's magnetic field at rather high speeds, so it is reasonable to wonder whether this field can have a substantial effect on them. We shall use a typical value of 0.50 G for the earth's field. (a) The French TGV train and the Japanese "bullet train" reach speeds of up to 180 mph moving on tracks about 1.5 m apart. At top speed moving perpendicular to the earth's magnetic field, what potential difference is induced across the tracks as the wheels roll? Does this seem large enough to produce noticeable effects? (b) The Boeing 747-400 aircraft has a wingspan of 64.4 m and a cruising speed of 565 mph. If there is no wind blowing (so that this is also their speed relative to the ground), what is the maximum potential difference that could be induced between the opposite tips of the wings? Does this seem large enough to cause problems with the plane?

A 0.650-m-long metal bar is pulled to the right at a steady 5.0 m/s perpendicular to a uniform, 0.750 T magnetic field. The bar rides on parallel metal rails connected through a 25.0-\(\Omega\) resistor (\(\textbf{Fig. E29.30}\)), so the apparatus makes a complete circuit. Ignore the resistance of the bar and the rails. (a) Calculate the magnitude of the emf induced in the circuit. (b) Find the direction of the current induced in the circuit by using (i) the magnetic force on the charges in the moving bar; (ii) Faraday's law; (iii) Lenz's law. (c) Calculate the current through the resistor.

A metal ring 4.50 cm in diameter is placed between the north and south poles of large magnets with the plane of its area perpendicular to the magnetic field. These magnets produce an initial uniform field of 1.12 T between them but are gradually pulled apart, causing this field to remain uniform but decrease steadily at 0.250 T/s. (a) What is the magnitude of the electric field induced in the ring? (b) In which direction (clockwise or counterclockwise) does the current flow as viewed by someone on the south pole of the magnet?

The magnetic field within a long, straight solenoid with a circular cross section and radius \(R\) is increasing at a rate of \(dB/dt\). (a) What is the rate of change of flux through a circle with radius \(r_1\) inside the solenoid, normal to the axis of the solenoid, and with center on the solenoid axis? (b) Find the magnitude of the induced electric field inside the solenoid, at a distance \(r_1\) from its axis. Show the direction of this field in a diagram. (c) What is the magnitude of the induced electric field \(outside\) the solenoid, at a distance \(r_2\) from the axis? (d) Graph the magnitude of the induced electric field as a function of the distance \(r\) from the axis from \(r =\) 0 to \(r = 2R\). (e) What is the magnitude of the induced emf in a circular turn of radius R/2 that has its center on the solenoid axis? (f) What is the magnitude of the induced emf if the radius in part (e) is \(R\)? (g) What is the induced emf if the radius in part (e) is 2\(R\)?

A long, thin solenoid has 900 turns per meter and radius 2.50 cm. The current in the solenoid is increasing at a uniform rate of 36.0 A/s. What is the magnitude of the induced electric field at a point near the center of the solenoid and (a) 0.500 cm from the axis of the solenoid; (b) 1.00 cm from the axis of the solenoid?

A long, thin solenoid has 400 turns per meter and radius 1.10 cm. The current in the solenoid is increasing at a uniform rate \(di/dt\). The induced electric field at a point near the center of the solenoid and 3.50 cm from its axis is 8.00 \(\times\) 10\(^{-6}\) V/m. Calculate \(di/dt\).

A long, straight solenoid with a cross-sectional area of 8.00 cm\(^2\) is wound with 90 turns of wire per centimeter, and the windings carry a current of 0.350 A. A second winding of 12 turns encircles the solenoid at its center. The current in the solenoid is turned off such that the magnetic field of the solenoid becomes zero in 0.0400 s. What is the average induced emf in the second winding?

At temperatures near absolute zero, \(B_c\) approaches 0.142 T for vanadium, a type-I superconductor. The normal phase of vanadium has a magnetic susceptibility close to zero. Consider a long, thin vanadium cylinder with its axis parallel to an external magnetic field \(\overrightarrow{B}_0\) in the +\(x\)-direction. At points far from the ends of the cylinder, by symmetry, all the magnetic vectors are parallel to the x-axis. At temperatures near absolute zero, what are the resultant magnetic field \(\overrightarrow{B}\) and the magnetization \(\overrightarrow{M}\) inside and outside the cylinder (far from the ends) for (a) \(\overrightarrow{B}_0\) = (0.130 T)\(\hat{\imath}\) and (b) \(\overrightarrow{B}_0\) = (0.260 T)\(\hat{\imath}\) ?

A very long, rectangular loop of wire can slide without friction on a horizontal surface. Initially the loop has part of its area in a region of uniform magnetic field that has magnitude \(B =\) 2.90 T and is perpendicular to the plane of the loop. The loop has dimensions 4.00 cm by 60.0 cm, mass 24.0 g, and resistance \(R =\) 5.00 \(\times\) 10\(^{-3} \Omega\). The loop is initially at rest; then a constant force \(F_{ext}\) = 0.180 N is applied to the loop to pull it out of the field (Fig. P29.46). (a) What is the acceleration of the loop when \(v =\) 3.00 cm/s? (b) What are the loop's terminal speed and acceleration when the loop is moving at that terminal speed? (c) What is the acceleration of the loop when it is completely out of the magnetic field?

A very long, straight solenoid with a crosssectional area of 2.00 cm\(^2\) is wound with 90.0 turns of wire per centimeter. Starting at t = 0, the current in the solenoid is increasing according to \(i(t) = (0.160 A/s^2)t^2\). A secondary winding of 5 turns encircles the solenoid at its center, such that the secondary winding has the same cross-sectional area as the solenoid. What is the magnitude of the emf induced in the secondary winding at the instant that the current in the solenoid is 3.20 A?

You are shipwrecked on a deserted tropical island. You have some electrical devices that you could operate using a generator but you have no magnets. The earth's magnetic field at your location is horizontal and has magnitude 8.0 \(\times\) 10\(^{-5}\) T, and you decide to try to use this field for a generator by rotating a large circular coil of wire at a high rate. You need to produce a peak emf of 9.0 V and estimate that you can rotate the coil at 30 rpm by turning a crank handle. You also decide that to have an acceptable coil resistance, the maximum number of turns the coil can have is 2000. (a) What area must the coil have? (b) If the coil is circular, what is the maximum translational speed of a point on the coil as it rotates? Do you think this device is feasible? Explain.

A circular conducting ring with radius \(r_0 =\) 0.0420 m lies in the xy-plane in a region of uniform magnetic field \(\overrightarrow{B} = B_0 [1 - 3(t/t_0)^2 + 2(t/t_0)^3]\hat{k}\). In this expression, \(t_0 =\) 0.0100 s and is constant, \(t\) is time, \(\hat{k}\) is the unit vector in the +\(z\)-direction, and \(B_0\) = 0.0800 T and is constant. At points \(a\) and \(b\) (Fig. P29.58) there is a small gap in the ring with wires leading to an external circuit of resistance \(R =\) 12.0 \(\Omega\). There is no magnetic field at the location of the external circuit. (a) Derive an expression, as a function of time, for the total magnetic flux \(\Phi_B\) through the ring. (b) Determine the emf induced in the ring at time \(t =\) 5.00 \(\times\) 10\(^{-3}\) s. What is the polarity of the emf? (c) Because of the internal resistance of the ring, the current through \(R\) at the time given in part (b) is only 3.00 mA. Determine the internal resistance of the ring. (d) Determine the emf in the ring at a time \(t =\) 1.21 \(\times\) 10\(^{-2}\) s. What is the polarity of the emf? (e) Determine the time at which the current through \(R\) reverses its direction.

A slender rod, 0.240 m long, rotates with an angular speed of 8.80 rad/s about an axis through one end and perpendicular to the rod. The plane of rotation of the rod is perpendicular to a uniform magnetic field with a magnitude of 0.650 T. (a) What is the induced emf in the rod? (b) What is the potential difference between its ends? (c) Suppose instead the rod rotates at 8.80 rad/s about an axis through its center and perpendicular to the rod. In this case, what is the potential difference between the ends of the rod? Between the center of the rod and one end?

A 25.0-cm-long metal rod lies in the \(xy\)-plane and makes an angle of 36.9\(^\circ\) with the positive \(x\)-axis and an angle of 53.1\(^\circ\) with the positive \(y\)-axis. The rod is moving in the \(+x\)-direction with a speed of 6.80 m/s. The rod is in a uniform magnetic field \(\overrightarrow{B} =\) (0.120 T)\(\hat{\imath}\) - (0.220 T)\(\hat{\jmath}\) - (0.0900 T)\(\hat{k}\). (a) What is the magnitude of the emf induced in the rod? (b) Indicate in a sketch which end of the rod is at higher potential.

An airplane propeller of total length \(L\) rotates around its center with angular speed \(\omega\) in a magnetic field that is perpendicular to the plane of rotation. Modeling the propeller as a thin, uniform bar, find the potential difference between (a) the center and either end of the propeller and (b) the two ends. (c) If the field is the earth's field of 0.50 G and the propeller turns at 220 rpm and is 2.0 m long, what is the potential difference between the middle and either end? It this large enough to be concerned about?

A dielectric of permittivity 3.5 \(\times\) 10\(^{-11}\) F/m completely fills the volume between two capacitor plates. For t 7 0 the electric flux through the dielectric is (8.0 \(\times\) 10\(^3\) V \(\cdot\) m\(/s^3)t^3\). The dielectric is ideal and nonmagnetic; the conduction current in the dielectric is zero. At what time does the displacement current in the dielectric equal 21 \(\mu\)A ?

A square, conducting, wire loop of side L, total mass m, and total resistance R initially lies in the horizontal xy-plane, with corners at (\(x, y, z\)) = (0, 0, 0), (0, \(L\), 0), (\(L\), 0, 0), and (\(L, L\), 0). There is a uniform, upward magnetic field \(\overrightarrow{B}\) = B\(\hat{k}\) in the space within and around the loop. The side of the loop that extends from (0, 0, 0) to (\(L\), 0, 0) is held in place on the \(x\)-axis; the rest of the loop is free to pivot around this axis. When the loop is released, it begins to rotate due to the gravitational torque. (a) Find the \(net\) torque (magnitude and direction) that acts on the loop when it has rotated through an angle \(\phi\) from its original orientation and is rotating downward at an angular speed \(\omega\). (b) Find the angular acceleration of the loop at the instant described in part (a). (c) Compared to the case with zero magnetic field, does it take the loop a longer or shorter time to rotate through 90\(^\circ\) ? Explain. (d) Is mechanical energy conserved as the loop rotates downward? Explain.