1. (36 points) A voltage of 4000 Volts is applied between concentric shells of a spherical capacitor. Note: Inner radius =a = 0.00060 m = 0.0600 cm and outer radius = b= 0.00120 m= 0.12 cm. Note that the scale used in the diagram may not reflect these dimensions.

(a) ( 20 points) What is the charge Q on the inner shell of radius a ?
(b) (10 points) What is the magnitude E of the electric field at a distance r=0.00100m from the center of the capacitor? ?
(c) (6 points) Suppose the energy in the capacitor is used to heat up a very small piece of metal of specific heat Cx=128.0 J/kg and mass 6.00x10-6 kg. If the initial temperature is exactly 20 o C, then what is the final temperature. Assume heating only.

 

 

  1. (30 points) A resistor has a cross-section in the shape of an equilateral triangle of side a = 0.000100m . It has a circular hole ( of empty space) that runs along the axis. The triangle circunscribes the hole. The resistor has length L= 0.300 m The resisitivity of the material is 1.7x10-8 ohm-m. Suppose a battery of exactly V = 110 volts is connected to the ends of the resistor.

(a) (4 points) What is the area of the equilateral triangle?

(b) (4 points) What is the cross-sectional area of the hole?

(c )22 points) Suppose that the resistor is then submerged in a vat of water of mass m -= 1.00 kg. How long does it take the resistor to heat up the water from exactly 10 degrees to exactly 90 degrees. Cw= 4186 J/oC kg

 3. (25 points) The battery voltage = 10 Volts. Assume: C= 1uF. The resistors are in parallel.

(a) ( 25 points) The switch is closed at t= 0. If the capacitor takes 0.92 seconds to charge to 62% of its maximum value, of charge, then what is R?
(b) EC (4 points) Find the total current at 0.92 seconds.
(c) EC (4 points) Find the current I in each resistor at 0.92 seconds.

4. ( 20 points) A proton moves perpendicular to a uniform magnetic field vector-B. The particle has a speed 1.00 x 107 m/s . The particle experiences an acceleration of magnitude 2.00 x 1013 m/s2 and direction in the (positive)+y-direction. The velocity of the particle is in the (negative) – x direction. The mass of the proton is 1.67x 10-27 kg. The charge is 1.6 x 10 –19 C.

(a) ( 10 points) What is the direction of the magnetic field ?
(b) ( 10 points) What is the magnitude B of the magnetic field?

To get full credit, please sketch the relevant vectors on coordinate axes. Show the directions.

 

 

5. (26 points)

In the figure below, the current in the long straight wire has the value I = 1.00 A and flows downward in the page. The wire lies in the plane of a rectangular loop which carries a clockwise current of value i = 2.00 A. The dimensions are as follows:
c = 0.100 m = distance of left side of loop from long wire; a= 0.200 m = width of loop; and L= 0.500 m = length of loop.

  1. (10 points) What is the direction of the net force on the wire loop?
  2. (10 points) What is the magnitude of the net force on the loop?
  3. (3 points) What is the direction of the magnetic moment of the wire loop, in or out of the page?
  4. (3 points) What is the magnitude of the magnetic moment of the wire loop?

 



6. (20 points)
A circular loop of radius r = 1.0 m is placed in a region where a uniform magnetic field is perpendicular to the plane of the loop. The magnetic field vector points in. The magnitude of the magnetic field is allowed to vary in time according to the equation:

, for
.

Suppose the loop has a resistance R = 2.0x106 . B0 =0.35 T.

(a) (3 points) Is B increasing or decreasing with time t? (Look at the function carefully and make your decision. Try plugging in a few values of times t to test your answer. Be careful !)

(b) ( 7 points) What is the direction of the current in the loop, clockwise or counterclockwise, for ? Please indicate this direction by drawing an arrow at the wire.

(c) (7 points) What is the magnitude of the induced electromotive force at t = 2.0x10-6 seconds?

(d) ( 3 points) What is the magnitude of the current at t = 2.0x10-6 seconds?

(e) Extra Credit (3 points) What is the Rate of Heat generation in the loop at t = 2.0x10-6 seconds?