Click here for Aut. 01 final

From Autumn 2000

Student Performance

1. (15 points) Extra Credit.
Suppose a proton with charge +e is located at an infinite distance from a negative fixed point charge -Q.

The proton is released with an initial speed v0 at infinity. See Fig. 1. The mass of the proton is m. Please answer this part using the given symbols and other relevant constants. What will be the speed of the proton when it reaches a distance 3R from the point charge -Q? See Fig. 2 . Use symbols!

Solution:
0 + mvo2/2 = -ekQ/3R + mvf 2/2

Solve for vf

 

2. (20 points)

The heating element of a coffee maker operates at a voltage V =
120 Volts. It has a resistance R = 40.0 ohms. Find the time that it takes the coffee maker to heat water of mass m = 0.500 kg from Ti = 23.00C to Tf = 90.0 0C ? The specific heat of water is C = 4186 J/Kg0C. Please solve first for using symbols and find the numerical answer by plugging in the values given at the last step.
Solution:

 

3. (25 points)

The capacitor begins to discharge at t = 0. The initial charge is Q0 = 1x10-6 C. R1 = 1.0 ohm, R2 = 2.0 ohm and R3 = 1.0 ohm. C = 1x10-6 C.

The capacitor is made of parallel plates with A =1x10-1 m2 and a distance between the plates d = 1x10-6 m. Note: =8.85x10-12 C2/Nm2

a. (5 points) What is the capacitance C of the capacitor? For full credit, you should derive the formula, then plug in the numbers.

b. (5 points) Find the charge q on the capacitor after a time period of 1.0x10-6 seconds.

c.(5 points) What is the total current i at t = 1.0x10-6 seconds?

d.(5 points) What is the current i1 in R1 at t = 1.0x10-6 seconds?

e.(5 points) What is the current i2 in R2 at t = 1.0x10-6 seconds?
Solution:
(a)

(b)q = Qoe-t/RC

R = 1 +1*2/(1+2) = 5/3

q = 5.08x10-7 C

(c) i = dq/dt = -(Qo/RC)*e-t/RC

| i | = (Qo/RC)*e-t/RC = 0.344 A

(d)

i1 = iRp/R1 = 0.344*(2/3)/1= 0.23 A

(e)

i2 = iRp/R2 = 0.344*(2/3)/2= 0.115 A

 

4. ( 25 points)
A singly charged positive ion of charge q = 1.0x10-19 C and mass m = 1.00x10-26 kg is accelerated through a potential difference of V =1000 Volts. The ion then enters a region perpendicular to a magnetic field that has magnitude B = 1.000 T. As indicated below, in this region the magnetic field vector points inward, perpendicular to the plane of the page. The circular orbit of the particle is parallel to the plane of the page. Now focus your attention on the instantaneous position of the ion when it is at the bottom of the circle.

 

a.(5 points) Since the ion is moving in a circle, what must be the direction of the force vector at the location of the ion at the bottom? Please indicate the direction of the force at that point by drawing an arrow.

b.(8 points) What is the direction of the velocity vector at the bottom? Please indicate the direction of the velocity by drawing an arrow.

c.(6 points) Calculate the speed v of the ion.

d.(6 points) Calculate the radius r of the orbit of the ion.
Solution:
(a) Up, toward center of circle

(b) To the right

(c) mv2/2 = qV so that v = 1.4x105 m/s

(d) r = mv/qB = 0.014 m

 

5. (30 points)

 

The current in the long straight wire is I 1 and goes down. The wire lies in the plane of the paper. The loop carries a current I2 and also lies in the plane of the paper. The dimension of the loop are shown. The distance from the long wire to the left side of the loop is also shown.

(a) (5 points) What is the magnitude of the force on the left side of the loop labeled "left" where the current I2 goes down. Use symbols!

(b) (5 points) What is the direction of the force on the left side. Show the direction of the force by drawing an arrow.

(c) (5 points) What is the magnitude of the force on the right side of the loop labeled "right" where the current I2 goes up. Use symbols

(d) (5 points) What is the direction of the force on the right side? Show the direction of the force by drawing an arrow.

(e) (5 points) What is the magnitude of the net force on the loop? Use symbols

(f) (5 points) What is the direction of the net force on the loop? Show the direction of the net force by drawing an arrow.

Solution:
(a)

(b) To the left

(c)

(d) To the right

(e)

(f) To the left

 

6. (24 points) A circular loop of wire with 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-B points in. The magnitude of the magnetic field is allowed to increase in time according to the equation:

where t is in seconds.

Suppose the loop has a resistance R = 2.0 ohms.

(a) ( 10 points) Find the direction of the current i in the loop, either clockwise or counterclockwise, for t > 0. Please indicate this direction by drawing an arrow at the wire.

(b) (8 points) What is the magnitude of the electromotive force i at t = 2.0 seconds?
(c) (2 points) What is the magnitude of the current i at t = 2.0 seconds?

(d) (2 points) What is the direction of the magnetic moment caused by i for t > 0 ?

(e) (2 points) What is the magnitude of the magnetic moment caused by i at
t = 2.0 seconds?

 

 

Solution:
(a) Counterclockwise

(b)

(c)

(d)

Out of the page from RHR

(e)