This test is 90 minutes long and has a total of 100 points.
[15 points] 1. Multiple choice questions. Each question is worth 3 points. Partial credit may be given for an incorrect answer if work is shown.
(i). A positively-charged insulating rod is brought close to, but not touching, a metal sphere that is initially uncharged. The sphere is grounded briefly, and then the ground connection is removed. The charged rod is then removed. The charge on the sphere is now:
[ ] Positive
[ ] Zero
[ ] Negative
(ii). Two identical charges are separated by a distance of 0.5 m. The force that the first charge exerts on the second is 360 N. The magnitude of each charge is :
[ ] 1.4 x 10-4 C
[ ] 1.0 x 10-4 C
[ ] 1.4 x 10-8 C
[ ] 1.0 x 10-8 C
(iii). Two parallel wires carry currents in opposite directions. The force exerted by wire 1 on wire 2 is:
(iv). If two identical 40 W light bulbs are connected in parallel to a wall socket and left on for a year, the cost would be about $70. If these same lamps are connected in series instead and left on for a year, how much would that cost?
(v). Which of the following diagrams is NOT a possible representation of the electric field lines near a metal sphere?
[5 points] 2. Moving a charge.
[3 points] (a) How much work is done in bringing the charge q from very far away to the point P if q is moved along the dashed line at constant speed?
[2 points] (b) How much work is done to bring the charge from very far away to point P by a more complicated path? Again the charge is moved at constant speed.
[20 points] 3. Model of a hydrogen atom.
One model of a hydrogen atom, which has one proton and one electron, suggests that the proton is at rest, and the electron orbits the proton in a circular orbit of radius r = 5.29 x 10-11 m. The gravitational interaction between the proton and the electron can be ignored.
[8 points] (a) What is the force exerted on the electron by the proton? State the magnitude, and draw an arrow on the diagram to indicate the direction of the force.
[6 points] (b) What is the magnitude of the acceleration of the electron?
[6 points] (c) How long does it take the electron to complete one orbit?
[20 points] 4. Two conducting spheres.
Two conducting hollow spheres share a common center.
The inner sphere has a charge q = +1.5 x10-6 C and a radius of 10 cm.
The outer sphere has a charge Q = -4.0 x10-6 C and a radius of 30 cm.
[6 points] (a) On the diagram, sketch the electric field produced by the charged spheres. Be sure to indicate the direction clearly.
[5 points] (b) What is the electric field at a point 5 cm from the center of the spheres?
[4 points] (c) With a dashed line on the diagram, draw the equipotential line that passes through point P, which is a distance of 40 cm from the center of the spheres.
[ 5 points] (d) What is the electric potential at point P?
[20 points] 5. A multi-loop circuit
[4 points] (a) Apply Kirchoff's loop rule to loop 1. Write down the equation; from your equation, solve for I1. Note that I3 = 0.500 A.
[4 points] (b) Apply Kirchoff's loop rule to loop 2. Write down the equation and solve for I2.
[4 points] (c) Apply Kirchoff's junction rule at one of the junctions. (Again, write down the equation.) You can use this equation to check whether your answers for parts (a) and (b) are correct.
[4 points] (d) What is the magnitude of the potential difference between points A and B in the circuit?
[4 points] (e) How much power is the 5 V battery putting into the circuit?
6. [ 20 points ] A cyclotron.
The first cyclotron was built in 1932; it's a compact device for accelerating charged particles to very high speeds.
A sample cyclotron is shown above. The charges (protons, in this case) are accelerated by applying a potential difference V across the gap. The protons pass through the gap several times, being accelerated each time. The potential difference must be reversed each time the protons cross the gap; otherwise, the speed gained going from left to right would be lost going from right to left.
To swing the protons back into the gap, a uniform magnetic field is present in each of the D-shaped regions, known as dees. The magnetic field is the same in each dee, and is perpendicular to the velocity of the protons.
[5 points] (a) The protons start from rest at point a. If they reach point b with a speed of 3.2 x 105 m/s, what is the potential difference V across the gap? (The gap can be treated as a parallel-plate capacitor.)
[4 points] (b) The width of the gap (the distance between the dees) is 4.4 x 10-3 m . What is the magnitude of the electric field in the gap?
Although the sign of the potential difference across the gap must be changed each time the protons cross it, the direction of the magnetic field in the dees can be left fixed.
[3 points] (c) The protons follow a counter-clockwise path, covering half a circle, in the dees. What direction is the magnetic field in the dees?
[ ] left
[ ] right
[ ] up
[ ] down
[ ] into the page
[ ] out of the page
[4 points] (d) The straight-line distance from b to c is 1.20 cm. If the protons reach point b with v = 3.2 x 105 m/s, what is the magnitude of the magnetic field B in the dees?
[4 points] (e) The time for protons to travel through half a circle in a dee does not depend on the velocity. This is convenient, because the voltage across the gap can then be reversed in the same amount of time each time. It's easy to show that the time for protons to travel half a circle through a dee is:
Calculate the time t for protons in this particular cyclotron.