Physics 220 Homework Problems, Spring 2012

1-1. A cat slides down a rubber rod and falls from the rodinto a metal pail A resting on a non-conducting shelfwith two other metal pails, B and C, which are incontact, but neither is in contact with A. The shelfbreaks when the cat lands in A, transferring charge toA, and all pails fall separated to the non-conductingfloor. The cat then runs away.(a) At the end of this process the charge on pail A1. is positive.2. is negative.3. is zero.(b) At the end of this process the charge on pail B1. is positive.2. is negative.3. is zero.4. has the same sign as pail A.5. has the same sign as pail C.(c) At the end of this process the charge on pail C1. is positive.2. is negative.3. is zero.4. has the same sign as pail B.5. both (1) and (4) are correct.

1-2. Consider vectors R = (2.10, y = [01] , 1.00) and S = (3.30, 4.00, 0.90).(a) Calculate the magnitude of R. (b) Calculate the z component of unit vector R.(c) Calculate the angle between vectors R and S. (d) Calculate the z component ofR× S. [(a) 3.00, 5.00 (b) 0.200, 0.400 (c) 90.0, 110.0 (d) 10.0, 30.0]

1-3. There are identical Q = [02] µC charges located at three positions: (0,−1, 2),(1, 2, 0), and (−2, 0,−1). Coordinates are listed in units of meters. (a) What is themagnitude of the force that a charge of −1.00 µC feels at the origin? (b) What is theangle between this force and the positive x axis? [(a) 5.00× 10−3, 9.00× 10−3 N(b) 120.0, 130.0]

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1-4. A charged particle with a charge of −7.5 pC is placed at the origin where the electricfield (SI units) is E = (3.75 i− 2.90 j). This force is directed toward which quadrant oraxis of the xy plane?1. I2. II3. III4. IV5. +x6. −x7. +y8. −y

1-5. The sketch in the square frame represents two negative pointcharges and one positive point charge, all of the same magnitude.The letters “a” and “b” simply designate two positions within theframe. Note that point “a” is down and to the right from thepositive charge. We label several directions as follows: (1) ↑, (2) ,(3) →, (4) , (5) ↓, (6) , (7) ←, (8) , (9) magnitude is zero,(10) none of the above.

A. What is the approximate direction of the electric field at position “a”?B. What is the approximate direction of the electric field at position “b”?

1-6. In the lab, an object having a net charge of Q = [03] µC is placed in auniform electric field of 500 N/C that is directed vertically. What is the mass of thisobject if it “floats” in the field? [0.100, 0.300 g]

2-1. Find the area of region bound by the curve y = b− x2 and the x axis, whereb = [01] . [1.00, 7.00]

2-2. A long chain lying along the x axis has linear charge density λ = λa sin2(x) + λb cos2(x),where λa = [02] C/m and λb = [03] C/m. What is the averagecharge density of the chain? Hint: λ = 1

T

∫ T0λ(t). [0.50, 1.50 C/m]

2-3. Consider a paraboloid “drinking cup”, as shown. The height ofthe cup is L, and the radius of the cup at the top is a. What isan appropriate differential volume for determining the totalvolume of the cup? The radius r of the cup varies with theheight z according to (r/a)2 = z/L.1. πa2 dz2. 1

2πLa dr3. πa2z dz/L4. 1

2πa2 dL

5. πa2r dz/L

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2-4. Consider a cone of height L and base radius a. What is anappropriate differential area for determining the total outersurface area?1. 2πa

√a2 + L2z dz/L2

2. 2πaz dz/L3. 2πa dz/L4. πa2 dz/L2

5. π√a2 + L2 dz

3-1. Two charged particles with charges of ±q = [01] pC are separated by adistance of a = 0.820 nm. (a) What is the dipole moment of this charge pair? (b) Usingthe dipole approximation (d a), what is magnitude of the electric field at a positionalong the dipole axis which is a distance of d = 0.915 cm away from the charge pair?[(a) 5.00× 10−19, 8.00× 10−19 C·m (b) 0.0100, 0.0200 N/C]

3-2. A uniformly charged ring of radius 11.6 cm has a total charge of Q = [02] pC.What is the magnitude of the electric field on the axis of the ring at a distance of 4.81 cmfrom the center of the ring? [1.00, 1.60 N/C]

3-3. A charged hemispherical bowl with radius 13.7 cm and chargedensity σ = [03] nC/m2 sits on the xy plane asshown. Determine the magnitude of (a) the x component,(b) the y component, and (c) the z component of the vectorelectric field at the origin. Hint: Try using spherical coordinates.[(a) 0, 200 N/C (b) 0, 200 N/C (c) 0, 200 N/C]

4-1. Complete this problem on a separate sheet of paper and submit it with your CID#prominently displayed.(a) Two conducting spheres of the same radius r, carrying equal but opposite charges,are separated by a center-to-center distance of 4r. Sketch the pattern of electric fieldlines in a plane that includes the centers of the two spheres.(b) A negatively charged rod of finite length has a uniform charge per unit length.Sketch the pattern of electric field lines in a plane that includes the rod.

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4-2. Consider the pattern of electric field lines in the figure.(a) By counting field lines, rank the left-hand (L) andright-hand (R) charges in order of decreasing magnitude.1. L > R2. R > L3. L = R

(b) Rank the points in the figure according to decreasingelectric field magnitude.1. A = B > C2. C > A = B3. A > C = B4. C > B > A5. A > B > C

(c) The direction of the electric field at point C is1. up2. down3. left4. right5. no direction because magnitude is zero

4-3. An electron is projected from the ground at an angle of 30 above the horizontal at aspeed of v = [01] m/s in a region where an upward electric field has a uniformmagnitude of 400 N/C. Neglecting the effects of gravity, find (a) the time it takes theelectron to return to the ground, (b) the maximum height it reaches along its trajectory,and (c) its horizontal distance between the launching and landing points.[(a) 0.100, 0.150 µs (b) 5.0, 15.0 cm (c) 60, 110 cm]

5-1. A uniform electric field E = Exi + Eyj where Ex = [01] N/C andEy = [02] N/C, intersects a surface with area of 2.70 m2. What is themagnitude of the flux through this area if the surface lies (a) in the yz plane? (b) in thexz plane? (c) in the xy plane? [(a) 0.00, 9.50 N·m2/C (b) 0.00, 9.50 N·m2/C(c) 0.00, 9.50 N·m2/C]

5-2. Four closed surfaces, S1 through S4, are drawn together with threecharges, −2Q, +Q, and −Q. Rank the four surfaces according to theamount (not magnitude, consider the sign) of electric flux exitingeach one. In answering this problem we are asking about NET flux.[Exiting flux (field lines going out) is canceled by field lines comingin.] That is, summing the net charge enclosed is important.1. S4 > S2 > S1 > S3

2. S3 > S1 > S2 > S4

3. S4 = S2 > S1 > S3

4. S3 > S1 > S2 = S4

5. S2 = S3 > S1 = S4

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5-3. A [03] nC point charge is located on the z axis adistance 0.800 m above the circular end cap of theparaboloidal cup shown in the figure. If L = 2.00 m anda = 0.510 m, calculate the magnitude of the total electric fluxdue to the point charge (a) through the circular end cap and(b) through the paraboloidal surface. [(a) 20.0, 40.0 N·m2/C(b) 20.0, 40.0 N·m2/C]

5-4. A point charge of q = [04] pC is placed at the center ofa regular triangular pyramid with an edge dimension of a = 1 cm.Determine the total electric flux exiting the pyramid.[0.500, 0.900 N·m2/C]

6-1. The charge per unit length on a long, straight filament is λ = [01] µC/m.(a) Determine the electric field at a distance of 2.50 cm from the filament. Here, define +to mean outward and − to mean inward. (b) Repeat for a distance of 25.0 cm from thefilament. [(a) 400, 600 kN/C (b) 40.0, 60.0 kN/C]

6-2. A square plate of copper with 52.6-cm sides has no net charge and is placed in a uniformE = [02] kN/C electric field directed perpendicular to the plate. (a) Find themagnitude of the charge density on each face of the plate. (b) Find the magnitude of thetotal charge on each face of the plate. [(a) 10.0, 40.0 nC/m2 (b) 4.00, 9.90 nC]

6-3. A thick conducting shell contains a second conducting shellas well as three conducting balls with charges of −3 nC, +2nC, and q = [03] nC, as shown. The conductingshells have zero net charge. The outer shell has outer radius5.50 m. (a) Determine the magnitude of the electric field atpoint A. (b) Determine the total charge on the inner surfaceof the thick shell. (c) Determine the magnitude anddirection of the electric field just outside its outer surface.Here, + means outward and − means inward.[(a) 0.00, 3.00 N/C (b) 3.0, 8.0 nC (c) −2.50, 2.50 N/C]

7-1. An electron is placed half way between two parallel plates (A and B). Plate A is held at0 V and plate B is held at 100 V. The electron will:1. Hit plate A with 0 J of energy.2. Hit plate B with 0 J of energy.3. Hit plate A with 8× 10−18 J of energy.4. Hit plate B with 8× 10−18 J of energy.5. Hit plate A with 1.6× 10−17 J of energy.6. Hit plate B with 1.6× 10−17 J of energy.

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7-2. An electron is released from rest in a uniform electric field of magnitudeE = [01] V/m. (a) Through what potential difference will it have passed aftermoving 1.24 cm? (b) How fast will the electron be moving after having traveled that1.24 cm? [(a) 40.0, 90.0 V (b) 4.00× 106, 6.00× 106 m/s]

7-3. A charge of +q is at the origin and a charge of [02] q is at x = 2.000 m. (a) Forwhat finite positive values of x is the electric potential zero? (b) If q = 1.50 nC, what isthe magnitude of the electric field at this point? [(a) 0.300, 0.700 m (b) 40, 120 N/C]

8-1. Calculate the electric potential at a point x = 0.489 malong the axis of the annulus as shown. The annulushas a uniform charge density of σ = 1.35 µC/m2, anouter radius of b = 1.13 m and an inner radius ofa = [01] m. [20.0, 60.0 kV]

8-2. Another application of Gauss’s law to charged conductors.In the figure, each of the dots represent a point charge ofq1 = [02] µC. The three conducting shells arerepresented by circles and carry a net charge of −1.00 µC,−2.00 µC, and −3.00 µC on the small, medium, and largeshells, respectively. Find the charge on the outer surfaceof the largest shell. [0.0, 20.0 µC]

8-3. A set of equipotential lines are shown in thefigure. Their potential values are shown. Anumber of locations are labeled with dots. Wealso label several directions as follows: (1) ↑,(2) , (3) →, (4) , (5) ↓, (6) , (7) ←,(8) .(a) Which point has the highest electric field?(b) What direction is that highest electric fieldpointing?(c) Which point has the lowest electric field?(d) What direction is that lowest electric fieldpointing?

9-1. A uniformly-charged rod of length L = 2.00 m and charge density λ = 2.65× 10−9 C/mlies along the x axis with its left end at the origin. Calculate the electric potential at thepoint located a distance d = [01] m beyond the end of the rod along the −xaxis. [10.0, 50.0 V]

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9-2. A uniformly charged insulating rod of length 60.0 cm is bent into the shape of asemicircle. If the rod has a total charge of Q = [02] pC. Find the electricpotential at the center of the semicircle. [−2.50, 2.50 V]

9-3. A hollow spherical metallic shell of radius of R = 25 cm holds a net surface charge ofQ = [03] pC. (a) Calculate the electric potential at a distance of 2R from thecenter of the sphere. (b) Calculate the electric potential at the surface of the sphere.(c) Calculate the electric potential at the center of the sphere. [(a) −1.00, 1.00 V(b) −1.00, 1.00 V (c) −1.00, 1.00 V]

10-1. An air filled capacitor consists of two parallel plates each with an area of 7.60 cm2,separated by a distance of [01] mm. If a 20-V potential difference is appliedto these plates, calculate (a) the electric field between the plates, (b) the capacitance,(c) the charge on each plate, and (d) the surface charge density. [(a) 9.0, 12.0 kV/m(b) 3.00, 4.00 pF (c) 60.0, 80.0 pC (d) 8.00× 10−8, 9.90× 10−8 C/m2]

10-2. An air filled spherical capacitor is constructed with inner and outer shell radii of 7.0 cmand [02] cm, respectively. (a) Calculate the capacitance of the device.(b) What potential difference between the spheres results in a charge of 4.00 µC on thecapacitor? [(a) 10.0, 30.0 pF (b) 100, 400 kV]

10-3. In the following capacitance network,C1 = [03] µF, C2 = 10.0 µF,and C3 = 15.0 µF. (a) What is theequivalent capacitance between points aand b? (b) If a potential difference of15 V is applied between points a and b,what charge is stored on C3?[(a) 9.0, 12.0 µF (b) 100, 120 µC]

10-4. You have a capacitor connected across a battery. If you wish to increase the total chargedrawn from this battery, which of the following options will work? Choose all of thecorrect answers.1. Add a larger capacitor in series with the first.2. Add a smaller capacitor in series with the first.3. Add a larger capacitor in parallel with the first.4. Add a smaller capacitor in parallel with the first.

11-1. Two capacitors C1 = 25.0 µF and C2 = [01] µF are connected in parallel andcharged with a 100 V power supply. (a) Calculate the total energy stored in the twocapacitors. (b) If the same two capacitors were connected in series, what potentialdifference would be required to store [02] mJ of energy? [(a) 0.150, 0.250 J(b) 50, 150 V]

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11-2. A parallel plate air gap capacitor is connected across a 12.0 V potential. At this point itstores [03] µC of charge. It is then disconnected from the source while stillcharged. (a) What is the capacitance of the capacitor? (b) A piece of Teflon is insertedbetween the plates. What is the new capacitance? (c) What is the voltage on thecapacitor? (d) What is the charge on the capacitor? [(a) 2.00, 6.00 µF (b) 5.0, 15.0 µF(c) 5.0, 30.0 V (d) 20.0, 70.0 µC]

11-3. A small rigid object carries positive and negative [04] nC charges. It isoriented so that the positive charge is at the point (−1.20 mm, 1.10 mm) and thenegative charge is at the point (1.40 mm,−1.30 mm). The object is placed in an electricfield E = (7800 ı− 4900 ) N/C. (a) What is the magnitude of the electric dipole momentof the object? (b) What is the magnitude of the torque acting on the object? (c) What isthe potential energy of the object in this orientation? (d) If the orientation of the objectcan change, what is the difference between its maximum and its minimum potentialenergies? [(a) 1.00× 10−11, 3.00× 10−11 C·m (b) 2.00× 10−8, 5.00× 10−8 N·m(c) 1.00× 10−7, 3.00× 10−7 J (d) 2.00× 10−7, 5.00× 10−7 J]

11-4. You have a square parallel plate capacitor (edge length a and separation d). It howeverdoes not fit in the assigned volume of space. You plan to make a second configuration ofequal capacitance. Which of the following options would work?1. half the edge length, and half the separation.2. half the edge length, half the separation, and add a dielectric of constant 2.3. half the edge length, and add a dielectric with κ = 2.5.4. one fourth the edge length and four times the separation.5. one fourth the edge length, twice the separation, and add a dielectric with κ = 8.0.

12-1. A uniform metallic rod, with a cross-sectional area of 1.83 cm2 and a length of 7.08 m,contains 6.24× 1028 conduction electrons per cubic meter of material, which have a meancollision time of [01] femtoseconds. (a) Determine the resistivity of the rod.When the rod experiences a potential difference of 2.52 mV from end to end, determine(b) the drift velocity of the electrons and (c) the current density in the rod.[(a) 1.50× 10−8, 3.00× 10−8 Ω·m (b) 1.00× 10−6, 2.00× 10−6 m/s(c) 10000, 20000± 100 A/m2]

12-2. Suppose that the current through a conductor decreases exponentially with timeaccording to the expression I(t) = I0e

−t/τ , where I0 is the initial current equal to1.321 mA and τ is a constant equal to [02] s. Consider a piece of theconductor. (a) How much charge passes through this piece between t = 0 and t = τ?(b) How much charge passes through this piece between t = 0 and t = 4τ? (c) How muchcharge passes through this piece between t = 0 and t =∞? [(a) 1.00, 4.00 mC(b) 1.00, 4.00 mC (c) 1.00, 4.00 mC]

12-3. A resistor is constructed of a carbon rod that has a uniform cross sectional area of 5.00mm2. When a potential difference of 15.0 V is applied across the ends of the rod, there isa current of [03] mA in the rod. What is (a) the resistance of the rod and(b) the rod’s length? [(a) 2.00× 103, 6.00× 103 Ω (b) 400, 800 m]

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13-1. When the voltage across a certain conducting filament is doubled, the current flowingthrough it is observed to increase by a factor greater than two. What type of materialcould the conductor be made of? Hint: Consider the effects of heating.1. copper2. quartz3. lead4. silicon

13-2. The resistance of a platinum wire is to be calibrated for low-temperature measurements.A platinum wire with a resistance of [01] Ω at 20C is immersed in liquidnitrogen at 77 K (−196C). If the temperature response of the platinum wire is linear,what is the expected resistance of the platinum wire in the liquid nitrogen?(αplatinum = 3.92× 10−3/C) [0.100, 0.400 Ω]

13-3. A toaster is rated at [02] W when connected to a 120-V source. (a) Whatcurrent does the toaster carry? (b) What is its resistance? [(a) 4.00, 7.00 A(b) 10.0, 30.0 Ω]

13-4. An electric car is designed to run off a 12.0-V battery with a total energy storage of[03] J. (a) If the electric motor draws 8.00 kW, what is the current deliveredto the motor? (b) If the electric motor draws 8.00 kW as the car moves at a steady speedof 20.0 m/s, how far will the car travel before it is “out of juice”? [(a) 500, 900 A(b) 30.0, 60.0 km]

14-1. A battery has an emf of 15.00 V. The terminal voltage of the battery is [01] Vwhen it is delivering 20.00 W of power to an external load resistor R. (a) What is thevalue of R? (b) What is the internal resistance of the battery? [(a) 6.00, 9.00 Ω(b) 1.00, 3.00 Ω]

14-2. Consider the circuit shown. R1 = 5.0 Ω,R2 = 10.0 Ω, R3 = [02] Ω, andE = 25.0 V. (a) What is the current in R3?(b) What is Vb − Va? [(a) 0.100, 0.300 A(b) 5.00, 6.00 V]

14-3. You have a resistor connected across a battery. If you wish to increase the current drawnfrom the battery, which of the following options will work? Choose all of the correctanswers.1. Add a larger resistor in series with the first.2. Add a smaller resistor in series with the first.3. Add a larger resistor in parallel with the first.4. Add a smaller resistor in parallel with the first.

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14-4. A resistor is constructed by shaping a material of resistivity ρ into a hollow cylinder oflength L and with inner and outer radii ra and rb, respectively. The resistivityρ = 3.52× 105 Ω·m, L is 4.00 cm, ra = 0.50 cm and rb = [03] cm. (a) Theapplication of a potential difference between the ends of the cylinder produces a currentparallel to the axis. What is the resistance in this configuration? (b) If the potentialdifference is now applied between the inner and outer surfaces, what is the resistance?[(a) 10.0, 50.0 MΩ (b) 1.00, 2.00 MΩ]

15-1. The current in a circuit is tripled by connecting a [01] -Ω resistor in parallelwith the resistance of the circuit. What is the resistance of the circuit in the absence ofthe additional resistor? [400, 900 Ω]

15-2. In the following circuit, R1 = 5.00 Ω, R2 = [02] Ω,R3 = 25.00 Ω, E1 = 25.00 V, E2 = 15.00 V, and E3 = 5.00 V.(a) What is I1? (b) What is I2? (c) What is I3? (d) What is thepotential difference across R3? [(a) 0.70, 1.10 A(b) −0.10, −0.50 A (c) −0.50, −0.70 A (d) 14.0, 17.0 V]

15-3. From the diagram, which of the following are true?1. I1 + I2 + I3 = 02. I1 + I2 = I33. I2 + I3 = I14. I1R+ E + I2R = 05. I1R− I3R = 06. −I2R+ E− I3R = 07. I1R+ I3R = 0

16-1. The circuit shown has been connected for along time. R1 = 1.00 Ω,R2 = [01] Ω, R3 = 4.00 Ω,R4 = 2.00 Ω, E = 20.0 V, and C = 1.00 µF.(a) What is the voltage across thecapacitor? (b) If the battery isdisconnected, how long does it take thecapacitor to discharge to one-tenth its initialvoltage? [(a) 5.0, 14.0 V(b) 4.00× 10−6, 9.90× 10−6 s]

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16-2. A [02] -ft extension cord has two 18-Gauge copper wires, each with a diameterof 1.024 mm. What is the I2R loss in this cord when it carries a current of (a) 1.00 A?(b) 10.0 A? Note: Because current flows up one wire and down the other, the length ofthe current path is twice that of the wire. [(a) 0.050, 0.200 W (b) 5.0, 20.0 W]

16-3. Consider the circuit in the figure.(a) If at some instant the capacitor in this circuit has no charge,what is the current in the resistors?1. 02. E/2R3. E/R4. 2E/R

(b) If at some instant the capacitor in this circuit has charge Q = CE, what is thecurrent in the resistors?1. 02. E/2R3. E/R4. 2E/R

(c) If at some instant the capacitor in this circuit has charge Q = 2CE, what is thecurrent in the resistors?1. 02. E/2R3. E/R4. 2E/R

17-1. The magnetic field of the earth can be reasonably approximated by assuming theexistence of a point dipole at the center of the earth with a dipole moment ofm = 8.00× 1022 A·m2. Using the equation for the magnetic field of a point dipole,B = (µ0/4π)[3(m · r)r−m]/r3, determine the magnitude and direction of the magneticfield on the surface of the earth at (a) the equator and (b) the geographic north pole. Forthe direction, use + to indicate geographic north and − to indicate geographic south.Don’t worry about the geomagnetic angle of declination – assume that it is zero. Use6.37× 106 m as the radius of the earth. [(a) −70.0, 70.0 µT (b) −70.0, 70.0 µT]

17-2. Consider an electron moving near the earth’s equator. It experiences a Lorentz force dueto the earth’s magnetic field. Possible directions for this force include (1) verticallyupward away from the center of the earth, (2) vertically downward towards the center ofthe earth, (3) east, (4) west, (5) north, (6) south, and (7) zero force. What will be thedirection of the force if the electron is moving(a) vertically upward?(b) east?(c) north?

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17-3. An electron is projected at a speed of 3.70× 106 m/s in the i + j + k direction into auniform magnetic field B = 6.43 i +By j− 8.29 k (Tesla), where By = [01] T.Calculate (a) the x component, (b) the y component, and (c) the z component of theresulting vector magnetic force on the electron. [(a) 3.00, 4.00 pN (b) −4.00, −6.00 pN(c) 1.00, 2.00 pN]

18-1. A thin, horizontal copper rod is 1.29 m long and has a mass of 52.6 g. What is theminimum current in the rod that can cause it to float in a horizontal magnetic field of[01] T? [0.100, 0.500 A]

18-2. Assume that in Atlanta, Georgia, the Earth’s magnetic field points northward anddownward at 60 below the horizontal, with a field strength of 52.0 µT. A tube in a neonsign carries a current of 35.0 mA between two diagonally-opposite corners of a shopwindow, which lies in a north-south vertical plane. The current enters the tube at thebottom south corner and exits at the opposite corner which is [02] m farthernorth and 0.85 m higher up. Between these two points, the tube spells out the wordDONUTS. Determine (a) the x component, (b) the y component, and (c) thez component of the total vector magnetic force on the neon tube. Define coordinate axesso that the x axis points east, the y axis points north, and the z axis points up.[(a) −4.00, 4.00 µN (b) −4.00, 4.00 µN (c) −4.00, 4.00 µN]

18-3. A rectangular loop consisting of N = 100 closelywrapped turns of wire has dimensions a = 0.400 mand b = 0.300 m and is oriented in a vertical plane soas to make an angle of 30 with the x axis, as shown.The loop carries a current I = 1.20 A andexperiences a uniform B = [03] Tmagnetic field directed along the +x axis.(a) Calculate the magnitude of the magnetic momentof the current-carrying loop. (b) Calculate themagnitude of the magnetic torque experienced by theloop. (c) Calculate the magnetic potential energy ofthe loop in the field. [(a) 10.0, 20.0 A·m2

(b) 1.00, 4.00 J (c) −1.00, −4.00 J]

19-1. A cosmic-ray proton traveling at [01] c is heading directly toward the center ofthe Earth in the plane of Earth’s equator. Assuming that the Earth’s magnetic field hasa uniform magnitude of B = 50.0 µT in the equitorial plane, determine the radius ofmotion of the cosmic-ray proton. Neglect relativistic effects if you know about them.[4.00, 6.00 km]

19-2. At the equator, assume that the earth’s magnetic field is directed northward with amagnitude of 50 µT and that there is an electric field of 100 N/C directed radiallyinward. The Earth’s radius is roughly 6.37× 106 m. A hypothetical charged particle isorbiting the earth in the equatorial plane and near the earth’s surface at[02] m/s in an easterly direction under these conditions. What is thishypothetical particle’s charge to mass ratio (watch the sign)? [−20.0, −40.0 kC/kg]

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19-3. A long metallic conductor oriented along the z axis has anoblong cross section in the xy plane as shown and carries currentin the −z direction (a right-handed coordinate system directs −zinto the page). There is a uniform external magnetic fieldpresent with field lines lying parallel to the xy plane. We labelseveral directions as follows: (1) ↑, (2) , (3) →, (4) , (5) ↓,(6) , (7) ←, (8) , (9) magnitude is zero, (10) none of theabove. What is the direction of the external magnetic field if themost negative potential occurs at point A?

20-1. A loop of wire of length L = 10.8 cm is stretched into the shape of a square and carries acurrent of I = [01] A. Determine the magnitude of the magnetic field at thecenter of the loop due to the current-carrying wire. [10.0, 20.0 µT]

20-2. A conductor consisting of a circular loop ofradius R = [02] m and two straight,long sections, carries a current of I = 7.00 A. Inthe figure, the loop is viewed from the +zdirection. Determine the z component of theresulting magnetic field at the center of the loop.[−1.00, −3.00 µT]

20-3. Two long parallel wires, each having mass densityλ = [03] g/m, are supported in thehorizontal plane by strings 6.00 cm long, as shown.When both wires carry the same current I in oppositedirections, the wires repel each other so that the angleθ between the supporting strings is 16.0. Determinethe magnitude of the current. [10.0, 30.0 A]

21-1. Two square current-carrying loops and two closed integration paths, one dashed and onesolid, are arranged as shown. If the positive current direction is chosen to be clockwise,the current in the loop on the left is +10.0 A. Defining ξ =

∮B · ds for a given path, we

find that the ratio ξdashed/ξsolid = [01] . Determine the current (magnitudeand sign) in the right-hand loop. Hint: Draw a top-view diagram of the figure, whichshould make the looped paths and current directions more apparent. [−90.0, 90.0 A]

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21-2. In the cross-sectional view of a coaxial cable below, thecenter conductor is surrounded by a rubber layer,which is surrounded by an outer conductor, which issurrounded by another rubber layer. In a particularapplication, the current in the inner conductor isIinner = [02] mA, directed out of the page,while the current in the outer conductor isIouter = [03] mA, directed into the page.Determine magnitude and sign of the vertical (up = +)component of (a) the magnetic field at point a and(b) the magnetic field at point b. [(a) −40.0, 40.0 µT(b) −40.0, 40.0 µT]

21-3. A superconducting solenoid with 2000 turns/m is meant to generate a magnetic field of[04] T. (a) Calculate the current required. (b) Determine the force per unitlength exerted on the windings by this magnetic field. Note that while an individualcurrent-carrying wire segment experiences no force due to the B-field that it creates, thatwire segment does experience a force due to the collective field produced by all of thecurrent-carrying coils around the solenoid. [(a) 3.00, 6.00 kA (b) 30.0, 90.0 kN/m]

22-1. An ideal solenoid 7.20 cm in diameter and 38.0 cm long hasN = [01] turns and carries 12.0 A of current.Calculate the magnetic flux through the surface of a disk ofradius 5.00 cm that is positioned perpendicular to and centeredon the axis of the solenoid. [1.00, 5.00 mT·m2]

22-2. Consider the hemispherical closed surface with radiusR = 3.00 cm shown below, which is in a uniform magnetic fieldof 0.250 T that makes an angle θ = [02] with thevertical. (a) Calculate the magnetic flux entering the circularface of the closed surface. (b) Calculate the magnetic fluxentering through the hemispherical surface.[(a) −0.700, 0.700 mT·m2 (b) −0.700, 0.700 mT·m2]

22-3. The magnetic field due to a magnetic dipole located at the origin, can be described as afunction of position ~r = rr, via the expression

B =µ0

4π3( ~m · r)r− ~m

r3,

where m is the dipole moment vector. Let a dipole moment of [03] A·m2 beoriented along the +z direction and imagine a Gaussian sphere of radius 0.750 mcentered on the dipole. (a) Calculate the magnetic flux exiting the upper (+z)hemisphere. (b) What is the magnetic flux exiting the lower (−z) hemisphere? Hint: Tryusing spherical coordinates, and don’t panic. The math is easy. [(a) −50.0, 50.0 µT·m2

(b) −50.0, 50.0 µT·m2]

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23-1. When 1.00 g of an unknown material is placed in a north-pointing 50-µT magnetic field,it is found to exhibit a magnetic moment that points roughly 25 east of north. Howmany of the following statements are true of this material? (1) The material isdiamagnetic. (2) The moment will grow with decreasing temperature. (3) The materialexhibits magnetic hysteresis. (4) The material is paramagnetic. (5) Removing theapplied field would eliminate the moment. (6) The moment is permanent. (7) Themoment is induced by the present field. (8) The material is ferromagnetic. (9) Thematerial has a magnetic domain structure. (10) The material would be weakly repelledby a strong magnet. (11) The material would be weakly attracted by a strong magnet.(12) The material could be strongly attracted to a strong magnet. (13) The materialwould be attracted to a steel paper clip. (14) The material would not be attracted to asteel paper clip. (15) χ < 0. (16) µ/µ0 > 1. (17) This is impossible behavior for anyknown material.

23-2. A toroid with N = 500 turns, a mean radius of R = 20.0 cm, a coil radius of r = 1.00 cm,and a powdered steel core with susceptibility χ = [01] , is carrying 2.55 A ofcurrent. Assume that the field is uniform inside the toroid (i.e. R r). (a) Calculate themagnetic permeability µ of the steel core. (b) Calculate the magnetic field strength µ0Hinside the toroid. (The factor µ0 makes the units the same as those of B.) (c) Calculatethe magnetization µ0M inside the toroid. (d) Calculate the magnetic field B inside thetoroid. [(a) 0.100, 0.300 mT·m/A (b) 1.00, 2.00 mT (c) 0.100, 0.300 T(d) 0.100, 0.300 T]

23-3. At room temperature (T = 300 K), a paramagnetic gas of densityρ = 5.00× 1019 molecules/cm3 is subjected to a [02] T magnetic field. Thegas responds with a magnetic moment of 0.01µB per molecule. (a) Determine themagnetization (magnetic moment per unit volume) of the gas. (b) Determine themagnetic susceptibility of the gas. (c) Determine the Curie constant of the gas.[(a) 3.00, 6.00 A/m (b) 1.00× 10−6, 3.00× 10−6 (c) 300, 700 A·K/T·m]

24-1. A uniform magnetic field oscillates in time as B = B0 cos(ωt),where B0 = [01] T, within a circular region of radiusa = 2.50 cm. A loop of wire containing a single 1.20 V light bulbsurrounds the field-containing region. Determine the oscillationfrequency needed to light the bulb (i.e. to match the emf amplitudewith the light bulb voltage specification). Note: do NOT use themore appropriate rms quantities if you know about them.[400, 990 Hz]

24-2. A coil of N2 = 15 turns and radius a = 10.0 cm surrounds a long solenoid of radiusr = 2.00 cm and N1 = 1000 turns/m. If the current in the solenoid varies asI = I0 cos(ωt), where I0 = [02] A and ω = 120 s−1, determine the maximuminduced emf in the coil. [5.00, 9.00 mV]

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24-3. A square coil that consists of 100 turns of wire rotates about avertical axis at 1500 rev/s. The horizontal component of theEarth’s magnetic field at the location of the coil is[03] µT. Determine the maximum emf induced in thecoil by this field. [30.0, 80.0 mV]

25-1. Use Lenz’s law and Figures (a)–(d) below to answer the following questions concerningthe direction of induced currents.(a) What is the direction of the induced current in resistor R in Fig. (a) when the barmagnet is moved to the left? (1) left (2) right (3) zero current(b) What is the direction of the current induced in the resistor R right after the switch Sin Fig. (b) is closed? (1) left (2) right (3) zero current(c) What is the direction of the induced current in R when the current I in Fig. (c)decreases rapidly to zero? (1) left (2) right (3) zero current(d) A copper bar is moved to the right while its axis is maintained in a directionperpendicular to a magnetic field, as shown in Fig. (d). If the top of the bar becomespositive relative to the bottom, what is the direction of the magnetic field? (1) left(2) right (3) up (4) down (5) into page (6) out of page.

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25-2. The square loop shown is made of wires with totalseries resistance of 12.6 Ω. It is placed in a uniform0.117-T magnetic field directed into the plane of thepaper. The loop, which is hinged at each corner, ispulled as shown until the separation between points Aand B is 3.00 m. If this process takest = [01] seconds, what is the averagecurrent generated in the loop (magnitude and sign).Let “+” indicate clockwise current and “−” indicatecounterclockwise current. [2.00, 3.00 mA]

25-3. A circular loop of wire is moved at constant speed through regions where uniformmagnetic fields of the same magnitude are directed into or out of the paper, as indicated.The instantaneous location of the loop, as it moves to the right, is indicated at sevenpositions.(a) At how many of the seven positions will there be no induced emf in the loop?(b) At how many of the seven positions will there be a CW induced emf in the loop?(c) At how many of the seven positions will there be a CCW induced emf in the loop?(d) At which position will the magnitude of the induced emf be maximum?

25-4. An electric motor consists of a rectangular coil (2.50 cm × 4.00 cm) with 80 turns of wirethat draws I = [02] A of current as it rotates at 3600 rev/min in a uniformB = 0.800 T magnetic field. (a) Determine the maximum torque delivered by the motor.(b) Determine the peak power produced by the motor. [(a) 0.100, 0.300 J(b) 40.0, 99.0 W]

26-1. A helicopter has blades with a length of 3.00 m extending outward from a central huband rotating at f = [01] rev/s. If the vertical component of the Earth’smagnetic field is 50.0 µT, what is the emf induced between the blade tip and the centerhub? [1.00, 3.00 mV]

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26-2. A conducting axle with mass m = [02] kg and length L = 1.50 m long rolls atconstant velocity along a pair of conducting rails that are inclined 30 from thehorizontal. A resistive load R = 100 Ω connects the rails, which are immersed in auniform 0.800 T magnetic field that points downward. (a) Determine the magnitude ofthe magnetic force on the axle. (b) Determine the speed of the rolling axle. Hints: Drawa free-body diagram. Separate forces into components, parallel and perpendicular to therail. Take the inclination angle into account when computing the induced emf.[(a) 50.0, 99.0 N (b) 4.00, 8.00 km/s]

26-3. A closed rectangular wire loop has dimensionsw = 0.80 m, ` = 1.50 m, mass m = [03] g,and resistance R = 0.750 Ω. The rectangle is allowed tofall through a region of uniform magnetic field, directedout of the page as shown, and accelerates downward asit approaches a terminal speed of 2.00 m/s with its topnot yet in the region of the field. Calculate themagnitude of the magnetic field. [0.400, 0.700 T]

27-1. A 10.0-mH inductor carries a current of I = Imax sinωt with Imax = 5.00 A andω/2π = 60.0 Hz. What is the magnitude of the back emf at t = [01] s?[0.0, +20.0 V]

27-2. For the RL circuit shown, let L = 3.00 H, R = 8.00 Ω, andE = [02] V. The switch is closed at t = 0.(a) Calculate the ratio of the potential difference across theresistor to that across the inductor when I = 2.00 A.(b) Calculate the voltage across the inductor [03] safter the switch is closed. [(a) 0.60, 1.20 (b) 0.100, 0.800 V]

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27-3. Two coils, held in fixed positions, have a mutual inductance of 130 µH. What is the peakvoltage in one when a sinusoidal current given by I(t) = Imax sin(ωt) flows in the other?Imax = 12.0 A and ω = [04] s−1. [1.00, 1.50 V]

27-4. A [05] -V battery, a 5.00-Ω resistor, and a 12.0-H inductor are connected inseries. After the current in the circuit has reached its maximum value, calculate (a) thepower being supplied by the battery, (b) the power being delivered to the resistor, (c) thepower being delivered to the inductor, and (d) the energy stored in the magnetic field ofthe inductor. [(a) 20.0, 99.0 W (b) 20.0, 99.0 W (c) 0.0, 99.0 W (d) 20.0, 99.0 J]

28-1. The switch in the circuit shown is connected topoint a for a long time. R = 14.0 Ω,L = 0.110 H, C = [01] µF, andE = 12 V. After the switch is thrown to point b,what are (a) the frequency of oscillation of theLC circuit, (b) the maximum charge thatappears on the capacitor, (c) the maximumcurrent in the inductor, and (d) the totalenergy the circuit possesses at t = 3.00 s?[(a) 400, 500 Hz (b) 10.0, 20.0 µC(c) 30.0, 50.0 mA(d) 7.00× 10−5, 9.90× 10−5 J]

28-2. In the figure, let R = 7.60 Ω, L = [02] mH, andC = 1.80 µF. (a) Calculate the frequency of the dampedoscillation of the circuit. (b) What is the critical resistance?[(a) 2.00× 103, 3.00× 103 Hz (b) 60.0, 90.0 Ω]

28-3. The switch in the figure is thrown closed at t = 0.R = 75 Ω, E = [03] V, C = 1.80 µF, andL = 2.20 mH. Before the switch is closed, the capacitor isuncharged and all currents are zero. The instant after theswitch is closed, determine the currents in (a) L, (b) C,and (c) R. Also determine the potential differences across(d) L, (e) C, and (f) R. A long time after the switch isclosed, determine the potential differences across (g) L,(h) C, and (i) R. [(a) 0.000, 0.500 A (b) 0.000, 0.500 A(c) 0.000, 0.500 A (d) 0.0, 40.0 V (e) 0.0, 40.0 V(f) 0.0, 40.0 V (g) 0.0, 40.0 V (h) 0.0, 40.0 V(i) 0.0, 40.0 V]

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29-1. An inductor is connected to a 20.0-Hz power supply that produces a 50.0-V peak voltage.What inductance is needed to keep the instantaneous current in the circuit below[01] mA? [4.00, 7.00 H]

29-2. What maximum current flows through a [02] -µF capacitor when it isconnected across (a) a North American outlet having Vrms = 120 V and f = 60.0 Hz?(b) a European outlet having Vrms = 240 V and f = 50.0 Hz? [(a) 0.100, 0.400 A(b) 0.100, 0.400 A]

29-3. (a) Draw to scale a phasor diagram showing Z, XL, XC , and φ for an AC series circuitwith R = [03] Ω, C = 11.0 µF, L = 0.200 H, and f = (500/π) Hz. Submitthis part of the problem to the 220 homework bins on a single sheet of paper before class.Include your CID!!! (b) What is Z? (c) What is the phase angle φ? [(b) 200, 500 Ω(c) 10.0, 30.0]

30-1. A series RLC circuit is used in a radio to tune in to an FM station broadcasting at99.7 MHz. The resistance in the circuit is 12.0 Ω, and the inductance is[01] µH. What capacitance should be used? [1.00, 3.00 pF]

30-2. In a certain series RLC circuit, Irms = 9.00 A, Vrms = 180 V, and the current leads thevoltage by [02] . (a) What is the resistance R of the circuit? (b) What is thereactance of the circuit (XL −XC)? [(a) 5.0, 20.0 Ω (b) 5.0, 20.0 Ω]

30-3. In a series RLC circuit, R = [03] Ω, XC = 150 Ω, XL = 100 Ω,E = 100 V (rms), and f = [04] Hz. (a) Find L. (b) Find C. (c) Find the rmscurrent flowing in the circuit. (d) Find the phase shift φ. (e) Find the power dissipated.(f) Find the ratio of VCmax/VRmax . [(a) 0.100, 0.400 H (b) 10.0, 25.0 µF(c) 0.300, 0.700 A (d) −90.0, +90.0 (e) 30.0, 70.0 W (f) 0.50, 2.00]

31-1. A step down transformer is used for recharging the batteries of a portable device such asa tape player. The turns ratio inside the transformer is 13:1, and it is used with 120 V(rms) household service. If a particular ideal transformer draws [01] A (rms)from the house outlet, (a) what (rms) voltage is supplied to the tape player from thetransformer? (b) What (rms) current is supplied to the tape player from the transformer?(c) How much power is delivered? [(a) 5.00, 9.50 V (b) 3.00, 6.00 A (c) 30.0, 50.0 W]

32-1. An air-filled circular parallel plate capacitor with radius a = 5.00 cm and plateseparation d = 2.00 mm, is driven by a 60 Hz alternating voltage with amplitudeV = [01] V. Naturally, the magnitude of the current is greatest at the instantwhen the voltage is zero. At such an instant, determine the magnitude of the (a) rate ofchange of electric flux in the capacitor, (b) displacement current in the capacitor, and(c) magnetic field near the edge of the capacitor. [(a) 100, 300 kV·m/s (b) 1.00, 3.00 µA(c) 5.00, 9.99 pT]

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32-2. Which of the following laws or principles are required to solve the problems describedbelow. In each case, choose only one answer. If more than one response seemsappropriate, choose the one most fundamental to the problem at hand. Possibleresponses are: (1) Gauss’s law of electrostatics, (2) Gauss’s law of magnetism,(3) Faraday’s law, (4) Ampere-Maxwell law, (5) Lorentz force law.(a) Determine the magnetic field near a current carrying wire.(b) Determine the trajectory of a proton in a uniform magnetic field.(c) Determine the electric field inside a charged capacitor.(d) Determine the magnetic field inside a charging capacitor.(e) Determine the power delivered by a wind-turbine generator.(f) Determine the electric field near the surface of a conductor.(g) Determine the voltage difference between the ends of a metal bar moving in amagnetic field.(h) Determine the total magnetic flux through a closed surface.(i) Determine the voltage in the secondary winding of a transformer.(j) The magnetic field produced by a moving charged particle.

32-3. Determine the validity of each of the following statements. Possible responses are(1) True or (2) False.(a) Ampere’s law is physically equivalent to the Lorentz force law.(b) Gauss’s law of electrostatics is physically equivalent to Gauss’s law of magnetism.(c) Coulomb’s law is physically equivalent to Gauss’s law of electrostatics.(d) The Biot-Savart law is physically equivalent to Faraday’s law.(e) Lenz’s law is a corollary of Faraday’s law.(f) Gauss’s law of electrostatics relates electric charge to electric flux.(g) Gauss’s law of magnetism relates magnetic charge to magnetic flux.(h) The Ampere-Maxwell law relates magnetic circulation to changing electric flux.(i) The Ampere-Maxwell law relates magnetic circulation to electric current.(j) Faraday’s law relates electric charge to changing magnetic flux.

32-4. Complete this problem on a separate sheet of paper and submit it with your CID#prominently displayed.Name and state each of Maxwell’s equations and the Lorentz force law in plain Englishwith no reference to symbols or acronyms.

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33-1. A transverse wave on a string is described by the wave function y = y0 sin(kx+ ωt),where y0 = [01] m, k = [02] m−1, and ω = [03] s−1. Attime t = 0.200 s and string position x = 1.6 m, determine the following quantities:(a) Wavelength.(b) Wavenumber.(c) Wave frequency (cyclic).(d) Wave period. (e) Wave velocity (magnitude and sign).(f) Transverse string position (magnitude and sign).(g) Peak transverse string position (magnitude).(h) Average transverse string position (magnitude).(i) Transverse string velocity (magnitude and sign).(j) Transverse string acceleration (magnitude and sign).[ (a) 10.0, 25.0 m (b) 0.200, 0.600 m−1 (c) 1.00, 3.00 Hz (d) 0.400, 0.700 s(e) −60.0, +60.0 m/s (f) −0.150, 0.150 m (g) 0.000, 0.150 m (h) −0.150, 0.150 m(i) −3.00, +3.00 m/s (j) −40.0, +40.0 m/s2 ]

33-2. The speed of an electromagnetic wave traveling in a transparent nonmagnetic substanceis 1/

√µ0κε0, where κ is the dielectric constant of the substance, which depends on

frequency. Determine the speed of light in a liquid with a dielectric constant ofκ = [04] at optical frequencies. FYI, water has a dielectric constant of 1.78 inthis range. [2.00× 108, 3.00× 108 m/s]

33-3. A standing-wave interference pattern is set up by radio waves between two metal sheetsd = [05] m apart. This is the shortest distance between the plates that willproduce a standing-wave pattern. What is the fundamental frequency of the radio waves?[50.0, 95.0 MHz]

33-4. Match the following object sizes to the wavelength of the appropriate electromagneticradiation. Possible responses are (1) gamma rays, (2) x-rays, (3) ultraviolet rays,(4) visible light, (5) infrared, (6) microwaves, (7) FM radio waves, (8) AM radio wave,(9) long-wavelength radiation.(a) An atom.(b) Your finger.(c) Your height.(d) The thickness of a human hair.(e) A bacterium(f) A virus.(g) An atomic nucleus.(h) Your campus.(i) Your world (which may also be your campus).

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34-1. Neldon the Nerd went to see Star Wars and was fascinated by the red light pulses fromthe laser blasters. He decides to make such a weapon. He chooses a pulsed laser with awavelength of 580 nm so that the light will be red.(a) The pulses in the movie appeared to be L = [01] m long and lastedroughly 0.2 seconds. But Neldon is annoyed to discover that a light pulse this long musthave a temporal duration of only .(b) Neldon can’t make such a pulse, but does manage to build a gun with aT = [02] µs pulse. At t = 0, when the pulse begins, E is exactly zero at themuzzle of the gun. During the length of the pulse, E will be zero again moretimes.(c) Neldon decides to blast the white clock on the wall across the room since its whitereflective surface resembles that of a storm trooper uniform. A short time after the smallspot of light strikes near the center of the clock face, the electric field points towardM = [03] minutes after 12 o’clock. At this same instant, the magnetic fieldpoints toward minutes after 12 o’clock.[(a) 1.00, 4.00 ns (b) 1.00× 109, 3.00× 109 (c) 0.0, 59.9 min]

34-2. Neldon then adjusts the pulse length of his laser blaster to 2.00 µs and the beamdiameter to D = [04] mm. Though he finds that his laser pulse has animpressive total energy of E = [05] kJ, he is again annoyed when the clockdoesn’t shatter. (a) So he computes the momentum delivered by the pulse assumingcomplete reflection, and finds it to be a mere . He then calculates (b) the averagebeam intensity, (c) the peak beam intensity, (d) the peak E-field magnitude, and (e) thepeak B-field magnitude. [(a) 2.00× 10−5, 4.00× 10−5 kg·m/s (b) 5.00× 1014, 9.00× 1014

W/m2 (c) 1.00× 1015, 2.00× 1015 W/m2 (d) 6.00× 108, 9.00× 108 N/C (e) 2.00, 3.00 T]

35-1. An AM radio station broadcasts isotropically (equally in all directions) with an averagepower of 4.00 kW. An optimally-oriented λ/2 dipole antenna 65.0 cm long is locatedd = [01] km from the transmitter. (a) Compute the maximum E-field at thereceiving antenna. (b) Compute the maximum B-field at the receiving antenna.Compare this to the magnetic field of the earth, which is roughly 50 µT. (c) Compute themaximum emf induced by this signal between the two ends of receiving antenna.[(a) 0.300, 0.500 N/C (b) 1.00, 2.00 nT (c) 0.200, 0.400 V]

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