Chapter 29

A flat, rectangular coil consisting of 50 tums measures 25.0 \(\mathrm{cm}\) by 30.0 \(\mathrm{cm}\) . It is in a uniform, \(1.20-\mathrm{T}\) , magnetic field, with the plane of the coil parallel to the field. In 0.222 s, it is rotated so that the plane of the coil is perpendicular to the field. (a) What is the change in the magnetic flux through the coil due to this rotation? (b) Find the magnitude of the average emf induced in the coil during this rotation.

In a physics laboratory experiment, a coil with 200 tums enclosing an area of 12 \(\mathrm{cm}^{2}\) 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 \(6.0 \times 0^{-5} \mathrm{T}\) . (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 (Exercise 29.3) has an area of \(3.20 \mathrm{cm}^{2}, 120\) 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 \(3.56 \times 10^{-5} \mathrm{C}\) . What is the magnitude of the field?

A circular loop of wire with a radius of 12.0 \(\mathrm{cm}\) and oriented in the horizontal \(x y\) -plane is located in a region of uniform magnetic field. A field of 1.5 \(\mathrm{T}\) 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 \(\mathrm{ms}\) , 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 \(\mathrm{cm}\) in radius, containing 500 turns, is placed in a uniform magnetic field that varies with time according to \(B=(0.0120 \mathrm{T} / \mathrm{s}) t+\left(3.00 \times 10^{-5} \mathrm{T} / \mathrm{s}^{4}\right) t^{4} .\) 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 \(t=5.00 \mathrm{s} ?\)

Shrinking Loop. A circular loop of flexible iron wire has an initial circunference of \(165.0 \mathrm{cm},\) but its circunference is decreasing at a constant rate of 12.0 \(\mathrm{cm} / \mathrm{s}\) due to a tangential pull on the wire. The loop is in a constant, viniform magnetic field oriented perpendicular to the plane of the loop and with magnitude 0.500 \(\mathrm{T}\) . (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 rectangle measuring 30.0 \(\mathrm{cm}\) by 40.0 \(\mathrm{cm}\) is located inside a region of a spatially uniform magnetic field of 1.25 \(\mathrm{T}\) , with the field perpendicular to the plane of the coil (Fig. 29.29 ). The coil is pulled out at a steady rate of 2.00 \(\mathrm{cm} / \mathrm{s}\) traveling perpendicular to the field lines. The region of the field ends abruptly as shown. Find the emf induced in this coil when it is (a) all inside the field: (b) partly inside the field; (c) all outside the field.

In a region of space, a magnetic ficld points in the \(+x\) -direction (toward the right). Its magnitude varies with position according to the formula \(B_{x}=B_{0}+b x,\) 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)?

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

Are Motional emfs a Practical Source of Electricity? How fast (in \(\mathrm{m} / \mathrm{s}\) and \(\mathrm{mph}\) ) would a \(5.00-\mathrm{cm}\) copper bar have to move at right angles to a 0.650 - \(T\) magnetic field to generate 1.50 \(\mathrm{V}\) (the same as a AA battery) across its cnds? Does this scem like a practical way to generate electricity?

Motional emfs in Transportation. 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 \(\mathrm{G}\) for the earth's field (a) The French TGV train and the Japanese "bullet train" reach speeds of up to 180 \(\mathrm{mph}\) moving on tracks about 1.5 \(\mathrm{m}\) 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 \(\mathrm{m}\) and a cruising speed of 565 \(\mathrm{mph}\) . 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 \(1.41-\mathrm{m}\) bar moves through a uniform, 1.20 . \(T\) magnetic field with a speed of 2.50 \(\mathrm{m} / \mathrm{s}\) (Fig, 29.40\()\) . In cach case, find the emf induced between the ends of this bar and identify which, if any, end \((a \text { or } b)\) is at the higher potential. The bar moves in the direction of (a) the \(+x\) -axis; (b) the \(-y\) -axis; (c) the \(+z\) -axis. (d) How should this bar move so that the emf across its ends has the greatest possible value with \(b\) at a higher potential than \(a\) , and what is this maximum emf?

A long, thin solenoid has 900 turns per meter and radius \(2.50 \mathrm{cm} .\) The current in the solenoid is increasing at a uniform rate of 60.0 \(\mathrm{A} / \mathrm{s}\) . What is the magnitude of the induced electric field at a point near the center of the solenoid and (a) 0.500 \(\mathrm{cm}\) from the axis of the solenoid; (b) 1.00 \(\mathrm{cm}\) from the axis of the solenoid?

The magnetic field within a long, straight solenoid with a eireular cross section and radius \(R\) is increasing at a rate of \(d B / d t .\) (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 \(r=2 R\) . (e) What is the magnitude of the induced emf in circular turn of radius \(R / 2\) that has its center on the solenoid axis? (f) What is the magnitude of the indnced emf if the radins 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 400 turns per meter and radius 1.10 \(\mathrm{cm}\) . The current in the solenoid is increasing at a uniform rate dildt. The induced electric field at a point near the center of the solenoid and 3.50 \(\mathrm{cm}\) from its axis is \(8.00 \times 10^{-6} \mathrm{V} / \mathrm{m}\) . Calculate dildt.

A metal ring 4.50 \(\mathrm{cm}\) in diameter is placed between the north and south poles of large magnets with the plane of its area perpendicular to the magnetic ficld. 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 \(\mathrm{T} / \mathrm{s}\) (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?

A long, straight solenoid with a cross-sectional area of 8.00 \(\mathrm{cm}^{2}\) is wound with 90 turns of wire per centimeter, and the windings carry a current of 0.350 \(\mathrm{A}\) . A second winding of 12 tums 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 indnced emf in the second winding?

A diclectric of permitivity \(3.5 \times 10^{-11} \mathrm{F} / \mathrm{m}\) completely fills the volume between two capacitor plates. For \(t > 0\) the electric flux through the dielectric is \(\left(8.0 \times 10^{3} \mathrm{V} \cdot \mathrm{m} / \mathrm{s}^{3}\right) t^{3}\) . 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\(\mu \mathrm{A} ?\)

Displacement Current in a Wire. A long, straight, copper wire with a circular cross-scctional area of 2.1 \(\mathrm{mm}^{2}\) carries a current of 16 \(\mathrm{A}\) . The resistivity of the material is \(20 \times 10^{-8} \Omega \cdot \mathrm{m}\) . (a) What is the uniform electric field in the material? (b) If the cur- rent is changing at the rate of 4000 \(\mathrm{A} / \mathrm{s}\) , at what rate is the electric field in the material changing? (c) What is the displacement current density in the material in part (b)? (Hint: Since \(K\) for copper is very close to \(1,\) use \(\epsilon=\epsilon_{0} . )\) (d) If the current is changing as in part (b), what is the magnitude of the magnetic field 6.0 \(\mathrm{cm}\) from the center of the wire? Note that both the conduction current and the displacement current should be included in the calculation of \(B\) . Is the contribution from the displacement current significant?

A long, straight wire made of a type-I superconductor carries a constant current \(I\) along its length. Show that the current cannot be uniformly spread over the wire's cross section but instead must all be at the surface.

The compound \(\mathrm{SiV}_{3}\) is a type-II superconductor. At temperatures near absolute zero the two critical fields are \(B_{c 1}=\) 55.0 \(\mathrm{mT}\) and \(B_{c 2}=15.0 \mathrm{T}\) . The normal phase of \(\mathrm{SiV}_{3}\) has a magnetic susceptibility close to zero. A long, thin \(\mathrm{SiV}_{3}\) cylinder has its axis parallel to an extermal magnetic field \(\vec{B}_{0}\) 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 a temperature near absolute zero the external magnetic field is slowly increased from zero. What are the resultant magnetic field \(\overrightarrow{\boldsymbol{B}}\) and the magnetization \(\vec{M}\) inside the cylinder at points far from its ends (a) just before the magnetic flux begins to penetrate the material, and (b) just after the material becomes completely normal?

A circular wire loop of radius \(a\) and resistance \(R\) initially has a magnetic flux through it due to an external magnetic field. The extermal field then decreases to zero. A current is induced in the loop while the external field is changing; however, this current does not stop at the instant that the external field stops changing. The reason is that the current itself generates a magnetic field, which gives rise to a flux through the loop. If the current changes, the flux through the loop changes as well, and an induced emf appears in the loop to oppose the change. (a) The magnetic field at the center of the loop of radius a produced by a current i in the loop is given by \(B=\mu_{0} i / 2 a\) . If we use the crode approximation that the field has this same value at all points within the loop, what is the flux of this field through the loop? (b) By using Faraday's law, Eq. \((29.3),\) and the relationship \(\mathcal{E}=i R,\) show that after the external field has stopped changing, the current in the loop obeys the differential equation $$ \frac{d i}{d t}=-\left(\frac{2 R}{\pi \mu_{0} a}\right) i $$ (c) If the current has the value \(i_{0}\) at \(t=0\) , the instant that the external field stops changing. snive the equation in part \((b)\) to find \(i\) as a function of time for \(t>0 .\) (Hint: In Section 26.4 we encountered a similar differential equation, Eq. \((26.15),\) for the quantity \(q .\) This equation for \(i\) may be solved in the same way. (d) If the loop has radius \(a=50 \mathrm{cm}\) and resistance \(R=0.10 \Omega,\) how long after the external field stops changing will the current be equal to 0.010 \(\mathrm{o}\) (that is, \(\frac{1}{100}\) of its initial value)? (e) In solving the examples in this chapter, we ignored the effects described in this problem. Explain why this is a good approximation.

Make a Generator? You are shipwrecked on a deserted tropical island. You have some electrical devices that you could uperate using a generalor but you have nu maguels. The eardis magnetic field at your location is horizontal and has magnitude 8.0 \(\times 10^{-5} \mathrm{T}\) , 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 \(\mathrm{V}\) and estimate that you can rotate the coil at 30 \(\mathrm{rpm}\) by turning a crank handle. You also decide that to have an acceptable coil resistance, the maximum mumber 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 this device is feasible? Explain.

Terminal Speed. A bar of length \(L=0.8 \mathrm{m}\) is free to shide without friction on horizontal rails, as shown in Fig. 29.48 . There is a uniform magnetic ficld \(B=1.5 \mathrm{T}\) directed into the plane of the figure. At one end of the rails there is a battery with emf \(\mathcal{E}=12 \mathrm{V}\) and a switch. The bar has mass 0.90 \(\mathrm{kg}\) and resistance \(5.0 \Omega,\) and all other resistance in the circuit can be ignored. The switch is closed at time \(t=0\) . (a) Sketch the speed of the bar as a function of time. (b) Just aner the switch is closed, what is the acceleration of the bar? (c) What is the acceleration of the bar when its speed is 2.0 \(\mathrm{m} / \mathrm{s} ?\) (d) What is the terminal speed of the bar?

Antenna emf. A satellite, orbiting the earth at the equator at an altitude of 400 \(\mathrm{km}\) , has an antenna that can be modeled as a \(2.0-\mathrm{m}-\) long rod. The antenna is oriented perpendicular to the earth's surface. At the cquator, the earth's magnetic field is cssentially horizontal and has a value of \(8.0 \times 10^{-5} \mathrm{T}\) ; ignore any changes in \(B\) with altitude. Assuming the orbit is circular, determine the induced emf between the tips of the antenna.

emf in a Bullet. At the equator, the earth's magnetic field is approximately horizontal, is directed towand the north, and has a value of \(8 \times 10^{-5} \mathrm{T}\) . (a) Estimate the emf induced between the top and bottom of a bullet shot horizontally at a target on the equator if the bullet is shot toward the east. Assume the bullet has a length of 1 \(\mathrm{cm}\) and a diamcter of 0.4 \(\mathrm{cm}\) and is traveling at 300 \(\mathrm{m} / \mathrm{s}\) . Which is at higher potential: the top or boutom of the bullet? (b) What is the emfif the bullet travels south?(c) What is the emf induced between the front and back of the bullet for any horizontal velocity?

A slender rod, 0.240 \(\mathrm{m}\) long, reates with an angular speed of 8.80 \(\mathrm{rad} / \mathrm{s}\) 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 \(\mathrm{T}\) (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 \(\mathrm{rad} / \mathrm{s}\) 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?

An airplane propeller of total length \(L\) rotates around its center with angular spced \(\omega\) in a magnctic ficld that is perpcndicular 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 \(\mathrm{G}\) and the propeller turns at 220 \(\mathrm{rpm}\) and is 2.0 \(\mathrm{m}\) long, what is the potential difference between the middle and either end? It this large enough to be concemed about?

It is impossible to have a uniform electric field that abruptly drops to zero in a region of space in which the magnetic field is constant and in which there are no electric charges. To prove this statement, use the method of contradiction: Assume that such a case is possible and then show that your assumption contradicts a law of nature. (a) In the bottom half of a piece of paper, draw evenly spaced horizontal lines representing a uniform electric field to your right. Use dashed lines to draw a rectangle abcda with horizontal side ab in the electric-field region and horizontal side \(c d\) in the top half of your paper where \(E=0 .\) (b) Show that integration around your rectangle contradicts Faraday's law, Eq. \((29.21) .\)

Falling Square Loop. A vertically oricnted, square loop of copper wire falls from a region where the field \(\overrightarrow{\boldsymbol{B}}\) is horizontal. uniform. where the field is zero. The loop is released from rest and initially is entirely within the magnetic-field region. Let the side length of the loop be \(s\) and let the diameter of the wire be \(d\) . The resistivity of copper is \(\rho_{R}\) and the density of copper is \(\rho_{m}\) . If the loop reaches its terminal speed while its upper segment is still in the magnetic- field region, find an expression for the terminal speed.

A capacitor has two parallel plates with area \(A\) separated by a distance \(d\) . The space between plates is filled with a material having dielectric constant \(K\) . The material is not a perfect insulator but has resistivity \(\rho\) . The capacitor is initially charged with charge of magnitude \(Q_{0}\) on each plate that gradually discharges by conduction through the dielectric. (a) Calculate the conduction current density \(j_{\mathrm{C}}(t)\) in the dielectric. (b) Show that at any instant the dis-placement current density in the diclectric is equal in magnitude to the oonduotion current density but opposite in direction, so the total current density is zero at every instant.

A rod of pure silicon (resistivity \(\rho=2300 \Omega \cdot \mathrm{m} )\) is carry-ing a current. The electric field varies sinusoidally with time according to \(E=E_{0} \sin \omega t,\) where \(E_{0}=0.450 \mathrm{V} / \mathrm{m}, \omega=2 \pi f,\) and the frequency \(f=120 \mathrm{Hz}\) (a) Find the magnitude of the maximum conduction current density in the wire. (b) Assuming \(\epsilon=\epsilon_{0}\) , find the maximum displacement current density in the wire, and compare with the result of part (a). (c) At what frequency \(f\) would the maximum conduction and displacement densitics become equal if \(\epsilon=\epsilon_{0}\) (which is not actually the case)? (d) At the frequency determined in part (c), what is the relative phase of the conduction and displacement currents?