MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Spring 2014 Final Exam Equation Sheet Force Law: F q = q( E ext + v q B ext ) Poynting Vector: S = ( E B) / µ 0 Force on Current Carrying Wire: F = Id s B ext wire Gauss s Law: closed surface E d A Gauss s Law for Magnetism: B da = 0 closed surface Maxwell-Ampere Law: B d s = µ 0 J ˆn da closed path Faraday s law: E d s = d dt closed fixed path S open surface S = q enc ε 0 + µ 0 ε 0 d dt B ˆn da Volume Current Density: J = ρv Surface Current Density: K = σ v Current: I = J ˆn da closed surface open surface Charge Conservation J ˆn da = d dt volume enclosed S ρdv Volume Energy Density in Fields: 1 2 1 2 u = ε E ; u = B µ E 2 0 B 2 / 0 E ˆn da Source Equations: dq E( r) = k ˆ e r = k 2 e r B( r) = µ o 4π source source Id s ˆr r 2 source = µ o 4π ˆr points from source to field point Electric Potential Difference: b Δ V = Vb Va E d s a E = V Potential Energy: Δ U = qδ V Capacitance: C = Q ΔV U E = 1 2 Q 2 Inductance: C = 1 2 CΔV 2 L = Φ B,Total I dq ( r r ) 3 r r Id s ( r r ) r r 3 source = NΦ B I ε back = LdI / dt U M = 1 LI 2 2 Electric Dipole: p = N i=1 q i ri Torque on a Electric Dipole: τ E = p E ext Magnetic Dipole µ = IAˆn RHR 1
Torque on a Magnetic Dipole τ B = µ B ext Force on a Magnetic Dipole F = ( µ B) F z = µ z B z z Ohm s Law: Δ V = I R J = σ ce where σ c is the conductivity E = ρ J where ρ is the resistivity r Power: P = F v Power from Voltage Source: P souce = IΔV Power Dissipated in Resistor: P Joule = I 2 R = ΔV 2 / R r Differential Equations and Solutions: di ε IR L = 0 dt di IR + L = 0 dt d 2 Q dt 2 + 1 LC Q = 0 I t ε tr / L ( ) = (1 e ) R / I( t) = I0e tr L Q(t) = Q max cos(ω 0 t + φ), ω 0 = 1/ LC Waves: f = 1 T ω = 2π f k = 2π λ c = λ T = λ f = ω k Time Averaging: T sin 2 (ωt + φ) = 1 T sin2 (ωt + φ) dt = 1 2 Radiation Pressure: abs P pressure ref P pressure = 1 c = 2 1 c S Double Slit Interference: 0, perfectly absorbing S, perfectly reflecting. Constructive: d sin θ = mλ ; m = 0, ± 1, ± 2, Destructive: d sinθ = (m + 1/ 2)λ ; m = 0, ± 1, ± 2, Single Slit Diffraction: Destructive: asin θ = nλ ; n = ± 1, ± 2, Constants: c 299,792,458 m s -1 µ 0 4π 10 7 T m A -1 ε 0 1/ µ 0 c 2 8.85 10-12 C 2 N -1 m -2 k e = 1/ 4πε 0 9.0 10 9 N m 2 C -2 2
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Spring 2014 Problem 1 Concept Questions 8.02 Practice Final Exam Spring 2014 Question 1: A magnet has its north pole pointing upward. A conducting circular loop is moving downwards beneath the magnet. The induced current in the coil as seen from above the magnet and the force on the conducting loop due to the magnet are: a) current clockwise and force up. b) current counterclockwise and force up. c) current clockwise and force down. d) current counterclockwise and force down. e) current clockwise and zero force. f) current counter clockwise and zero force. g) zero current and zero force. 3
Question 2: An infinite sheet of positive charge in the y-z plane (see figure) is shaken up and down in the y direction. Which of the following sketches is a possible representation of the electromagnetic wave generated to the right of the sheet as a result of this shaking. a) b) c) d) e) None of the above 4
Question 3: Light from a coherent monochromatic laser of wavelength λ is incident on double slits separated by a distance d. Each slit has width a and the slits are placed a distance L from a screen where the diffraction/interference pattern is recorded. What is the ratio y 2 / y 1 of the distances indicated in the pattern? 1. d /(2 a ). 2. 2 d / a. 3. L / d. 4. 2L / d. 5. L /(2 d ). 6. a / λ. 7. 2 a / λ. 8. a /(2 λ ). 9. d / a. 5
Question 4: A rectangular loop of wire with resistor R is placed next to a straight wire carrying a current that is varying in time according to I( t) = I at ; 0 < t < I / a 0 0 During the time interval 0 < t < I0 / a, the rectangular loop will feel a) a net force to the right. b) a net force to the left. c) a net force down, toward the wire. d) a net force up, away from the wire. e) a net force into the page. f) a net force out of the page. g) no net force. 6
Question 5: In the figure below, the dashed lines show equipotential lines in a region of constant electric field. A charged object is moved directly from point A to point B. The charge on the object is +1 µc. What is the direction of the electric force exerted by the field on the +1 µc charged object when at A and when at B? a) Right at A and right at B. b) Left at A and left at B. c) Left at A and right at B. d) Right at A and left at B. e) No electric force at either. 7
Question 6: The figure below shows the magnetic field lines (without arrows to indicate direction) of the field composed of the superposition of a constant magnetic field in the +y or y direction plus that of a current carrying wire carrying current into or out of the page. Choose the description below that describes the force on the current carrying wire. a) The force is to the left. b) The force is upwards. c) The force is to the right. d) The force is downward. e) More information is needed. 8
Question 7: A dipole antenna is oriented as shown below. The observations points A, B, and C are all located the same distance from the center of the antenna. Let E A be the amplitude (always positive) of the radiation electric field at observation point A, and so on. Which of the following is true a) E A = E C = E B b) E A > E C = E B c) E A = E C >E B d) E A < E C = E B e) None of the above 9
Question 8: A conducting bar is placed across two rails, and a current I is flowing as shown in the figure below. The bar in the center of the figure will experience a) a magnetic force that points to the east b) a magnetic force that points to the west c) no magnetic force d) a magnetic force that points out of the plane of the figure e) a magnetic force that points into the plane of the figure f) a magnetic force that points to the north g) a magnetic force that points to the south 10
Question 9 (5 points): A parallel-plate capacitor has plates with equal and opposite charges ±Q, separated by a distance d, and is not connected to a battery. The plates are pulled apart to a distance D > d. What happens to the stored electrostatic energy and the potential difference V? 1. The stored electrostatic energy increases, V increases. 2. The stored electrostatic energy decreases, V increases. 3. The stored electrostatic energy is the same, V increases. 4. The stored electrostatic energy increases, V is the same. 5. The stored electrostatic energy decreases, V is the same. 6. The stored electrostatic energy is the same, V is the same. 7. The stored electrostatic energy increases, V decreases. 8. The stored electrostatic energy decreases, V decreases. 9. The stored electrostatic energy is the same, V decreases. 11
Question 10: In an LC circuit, the electric and magnetic fields are shown in the figure. At the moment depicted in the figure, the energy in the circuit is stored in 1. the electric field and is decreasing 2. the electric field and is constant 3. the magnetic field and is decreasing 4. the magnetic field and is constant 5. in both the electric and magnetic field and is constant 6. in both the electric and magnetic field and is decreasing Explain your answer. 12
Problem 2 The electric field of a plane electromagnetic wave is described as follows: E = E 0 cos 10 m 1 ( ) x + 3 10 9 s 1 ( )t ˆk. Be sure to include units in your answers to the questions below. a) What is the wavelength λ of the wave? b) What is the period T of the wave? c) In which direction does this wave propagate? Be sure to indicate the direction of propagation with a unit vector ( î, ĵ, or ˆk ) and an appropriate sign (+ or ). Briefly explain why you choose this direction. d) Write an expression for the magnetic field B of the wave in terms of the quantities given and the speed of light c. Be sure to indicate the direction of the magnetic field with a unit vector ( î, ĵ, or ˆk ) and an appropriate sign (+ or ). e) What is the time-dependent Poynting vector associated with this wave? Do not timeaverage. 13
Problem 3: Electromagnetic Plane Wave An electromagnetic plane wave is propagating in vacuum has a magnetic field given by B = B0 f ( ax + bt) j f ( u) = ˆ 1 0 < u < 1 0 otherwise where a and b are positive quantities. The ˆ +k direction is out of the paper. (a) What condition between a and b must be met in order for this wave to satisfy Maxwell s equations? (b) What is the magnitude and direction of the E field of this wave? (c) What is the magnitude and direction of the energy flux (power per unit area) carried by the incoming wave, in terms of B 0 and universal quantities? (d) This wave hits a perfectly conducting sheet and is reflected. What is the pressure (force per unit area) that this wave exerts on the sheet while it is impinging on it? 14
Problem 4 A very long solid conducting cylinder of length l and radius a, carrying a charge + Q is surrounded by a thin conducting cylindrical shell (length l and radius b ) with charge Q. a) What is the direction and magnitude of the electric field E in the three regions below? Show how you obtain your expressions. i) a < r < b, ii) r iii) r < a, > b. What is the electric potential difference between the outer shell and the inner cylinder, Δ V = V ( b) V ( a)? b) What is the capacitance of these coaxial conductors? c) If an additional positive charge 2Q + is placed anywhere on the inner cylinder of radius a, what charge appears on the outside surface of the cylindrical shell of radius b? 15
Problem 5: Bar Sliding down an inclined track A conducting bar of mass m slides down two frictionless conducting rails which make an angle θ with the horizontal, separated by a distance l and connected at the top by a resistor R, as shown in the figure. In addition, a uniform magnetic field B is applied vertically upward. The bar is released from rest and slides down. At time t the bar is moving along the rails at speed v(t). (a) Find the induced current in the bar at time t. Which way does the current flow, from a to b or b to a? (b) Find the terminal speed v T of the bar. (c) After the terminal speed has been reached what is the induced current in the bar? (d) After the terminal speed has been reached, what is the rate at which electrical energy is being dissipated through the resistor? (e) After the terminal speed has been reached, what is the rate of work, F g v, done by the gravitational force on the bar? How does this compare to your answer in part (d)? Explain your reasoning. 16
Problem 6 An N -turn solenoid of radius a and length h has a decreasing current I(t) = I 0 bt in the direction shown in the figure. a) Find the magnitude and direction of the magnetic field B(t) inside the solenoid.. b) Find the magnitude and direction of the electric field E(t) at the point D. c) Find the magnitude and direction of the Poynting vector S(t) at the point D. d) What is the flux of the Poynting vector (power) into/out of the inductor? e) How does this compare to the time derivative of the energy stored in the magnetic field? 17
Problem 7 Particle Orbits in a Uniform Magnetic Field The entire xy -plane to the right of the origin O is filled with a uniform magnetic field B of magnitude B pointing out of the page, as shown. Two charged particles travel along the negative x -axis in the positive x direction, each with velocity v, and enter the magnetic field at the origin O. The two particles have the same mass m, but have different charges, q 1 and q 2. When in the magnetic field, their trajectories both curve in the same direction (see sketch), but describe semi-circles with different radii. The radius of the semi-circle traced out by particle 2 is exactly twice as big as the radius of the semi-circle traced out by particle 1. (a) Are the charges of these particles positive or negative? Explain your reasoning. (b) What is the ratio q2 / q 1? 18
Problem 8: Betatron Induced Electric Fields, and Faraday s Law. An electron of the mass m e, and charge q = e is constrained to move in a circle of radius r by a time changing non-uniform magnetic field. Assume that the magnet has cylindrical symmetry about the central axis passing through the poles and that the plane of the orbit is perpendicular to that symmetry axis. Denote the magnetic field by B(r) = B z (r) ˆk where ˆk points form the north pole to the south pole. Assume that z -component of the magnetic field, B z (r), varies as a function of the distance r from the symmetry axis, that it is symmetric about this axis, and that it is perpendicular to the plane of the orbit of the electron. The magnitude of the average magnetic field over the electron s circular orbit is B ave = 1 πr 2 B da. Find a condition relating the rate of change in time of the average magnetic field db ave / dt, to the rate of change in time of the magnetic field, db / dt, in order for the electron to stay in a circular orbit. disk 19
Problem 9 A stretchable and flexible conducting band in the shape of a circle with radius r(t) has constant resistance R. It sits in a uniform magnetic field B that is directed out of the page (see figure). External agents distributed uniformly over the circumference of the ring exert radial outward forces that cause the ring to expand at a constant speed from radius a to a larger radius b over a time interval 0 t T, where T is a constant with units of seconds. Let v = dr / dt be the constant speed at which the ring expands. Express your answers to the following questions in terms of v, a, b, R, B = B, and T as needed. Note that in this problem R is a resistance, not a radius. a) Give an expression for the induced current I in the ring. Draw the direction of the induced current on the figure above. You may ignore any magnetic field generated by the induced current. b) What is the rate that energy is dissipated (power) as Joule heating during the time interval 0 t T? c) What is the direction and magnitude of the force per unit length that the external agents must apply to overcome the magnetic force per unit length on the conducting band due to the induced current? d) Based on your result for the force per unit length in part c), what power do the external agents provide during the time interval 0 t T? Is this the same as your answer to part b)? If yes, explain why; if no, explain why not. Be sure to give your reasoning. 20
Problem 10: Consider the circuit shown in the figure below. At t = 0, the switch S 1 is closed. The capacitor with capacitance C initially is uncharged, Q(t = 0) = 0. The battery has electromotive force ε. There are three resistors with resistances R 1, R 2, and R 3 as shown in the figure below. What is the current in each branch of the circuit at t = 0 when the switch S 1 is closed? What is the current in each branch of the circuit a very long time ( t = ) after the switch S 1 is closed? Use the loop laws, current conservation, and the relation between charge and current on the capacitor, to derive a differential equation for the charge on one of the plates of the capacitor at a time t after the switch S 1 is closed at t = 0. Express your differential equation in terms of the charge Q(t) on one of the plates, the time derivative of the charge dq(t) / dt, the electromotive force ε, the resistances R 1, R 2, and R 3, and the capacitance C. What is the solution Q(t) of your differential equation? Show that your solution agrees with your results for the current in each branch of the circuit at t = 0 when the switch S 1 is closed, and for the current in each branch of the circuit a very long time ( t = ) after the switch S 1 is closed. 21
Problem 11: RL Circuit Consider the RL circuit with values of the source emf, resistances and inductance shown in the figure below (a) What is the current in the circuit a long time after the switch has been in position a? (b) Now the switch is thrown quickly from a to b. Compute the initial voltage across each resistor, and the back emf ε back due to the inductance. (c) How much time elapses before the back emf ε back drops to 12.0 V? Problem 12: A toroidoil coil has N turns, and an inner radius a, outer radius b, and height h. The coil has a rectangular cross section shown in the figures below. The coil is connected via a switch, S 1, to an ideal voltage source with electromotive force ε. The circuit has total resistance R. Assume all the self-inductance L in the circuit is due to the coil. At time t = 0 switch S 1 is closed and S 2 remains open. 22
a) When a current I is flowing in the circuit, find an expression for the magnitude of the magnetic field inside the coil as a function of distance r from the axis of the coil. b) What is the self-inductance L of the coil? c) What is the current in the circuit a very long time ( t >> L R) after S 1 is closed? d) How much energy is stored in the magnetic field of the coil a very long time ( t >> L R) after S 1 is closed? For the next two parts, assume that a very long time ( t >> L R) after the switch S 1 was closed, the voltage source is disconnected from the circuit by opening S 1, and by simultaneously closing S 2, connecting the toroid to the capacitor of capacitance C. Assume there is negligible resistance in this new circuit. e) What is the maximum amount of charge that will appear on the capacitor? f) How long will it take for the capacitor to first reach a maximal charge after S 2 has been closed? 23
Problem 13: Coaxial Cable and Power Flow A coaxial cable consists of two concentric long hollow cylinders of zero resistance; the inner has radius a, the outer has radius b, and the length of both is l, with l >> b, as shown in the figure. The cable transmits DC power from a battery to a load. The battery provides an electromotive force ε between the two conductors at one end of the cable, and the load is a resistance R connected between the two conductors at the other end of the cable. A current I flows down the inner conductor and back up the outer one. The battery charges the inner conductor to a charge Q and the outer conductor to a charge + Q. (a) Find the direction and magnitude of the electric field E everywhere. (b) Find the direction and magnitude of the magnetic field B everywhere. (c) Calculate the Poynting vector S in the cable. (d) By integrating S over appropriate surface, find the power that flows into the coaxial cable. (e) How does your result in (d) compare to the power dissipated in the resistor? 24
Problem 14: Charging Capacitor A parallel-plate capacitor consists of two circular plates, each with radius R, separated by a distance d. A steady current I is directed towards the lower plate and away from the upper plate, charging the plates. a) What is the direction and magnitude of the electric field E between the plates? You may neglect any fringing fields due to edge effects. b) What is the total energy stored in the electric field of the capacitor? c) What is the time rate of change of the energy stored in the electric field? d) What is the magnitude of the magnetic field B at point P located between the plates at radius r < R (see figure above). As seen from above, is the direction of the magnetic field clockwise or counterclockwise. Explain your answer. e) Make a sketch of the electric and magnetic field inside the capacitor. f) What is the direction and magnitude of the Pointing vector S at a distance r = R from the center of the capacitor? g) By integrating S over an appropriate surface, find the power that flows into the capacitor. h) How does your answer in part g) compare to your answer in part c)? 25
Problem 15 Consider a slab that is infinite in the x and z directions that has thickness d in the y- direction. The slab has a time varying current with the current density as a function of time given by the following expression: J = 0; t 0 (J e t / T ) ˆk; 0 t T J e ˆk; T t, where J e is positive constant with units of amps per square meter and T is a constant with units of seconds. a) Find the direction and magnitude of the magnetic field for the interval 0 t T in the regions: (i) 0 y d / 2 ; (ii) y d / 2. Suppose a square conducting loop with resistance R, and side s is placed in the region y d / 2, at a height h above the top of the slab oriented as shown in the figure below. 26
b) What is the induced current in the square loop for the time interval 0 t T? Draw the direction of the induced current on the figure. c) What is the direction and magnitude of the force due to the induced current on the square loop during the time interval 0 t T? What is the direction and magnitude of the torque due to the induced current on the square loop during the time interval 0 t T? s 27
Problem 15: Another Way to Measure Hair Side View Eye View 1/4 inch 1 inch λ=500 nm Thickness d =? In addition to using hair as a thin object for diffraction, you can also measure its thickness using an interferometer. In fact, you can use this to measure even smaller objects. Its use on a small fiber is pictured at left. The fiber is placed between two glass slides, lifting one at an angle relative to the other. The slides are illuminated with green light from above, and when the set-up is viewed from above, an interference pattern, pictured in the Eye View, appears. What is the thickness d of the fiber? Problem 16 Constructive Interference Coherent light rays of wavelength λ are illuminated on a pair of slits separated by distance d at an angle θ 1, as shown in the figure below. If an interference maximum is formed at an angle θ 2 at a screen far from the slits, find the relationship between θ 1, θ 2, d and λ. 28