This test consists of three parts. Please note that in parts II and III, you can skip one question of those offered.
Possibly useful formulas:
R


C
A
0 e





R

R
0
A
L
2

4
Constants

 1
Circles

R
R
2


10
Resistance
 
0
 1
 




7

19

T

T
0
A
C

T

T
0



Triangles
1
2
BH
Q

Cyclotron
Q
Motion mv

RC Circuits
C



A
V




Spheres

 qRB qB
Q e
1
Part I: Multiple Choice [20 points] m
4
4
3

RC

 t

 e
 t
R
2
R
3


τ
Loops

I A B
U
 
I A B
Cylinders
 
2
V R L
A lat

2

RL
B
Solenoid in


0
L
NI
Field from Straight Wire
B

4


0
I a
B
 cos



2

0
1
I a
 cos

2

B
Biot-Savart


4
0

I  d r

2
V
A lat

F
L
Force
Between
Wires


I I
0 1 2
2
 d
Hall Effect

V
H

IB tnq
Ampere’s Law
  B s  
0
I
1
3
Cones

2
R L
 
R L
2 
R
2
For each question, choose the best answer (2 points each)
1. A loop of current in a constant magnetic field in general will feel
A) A net force, but no net torque
B) A net torque, but no net force
C) Both a net force and a net torque
D) Neither a net force nor a net torque
E) Lonely, isolated, and probably resentful
2. How is a realistic battery different from an ideal battery?
A) Realistic batteries have a maximum voltage; ideal batteries have no voltage limit
B) Realistic batteries have internal resistance
C) Realistic batteries have considerable capacitance
D) Realistic batteries generally take time to produce a voltage; ideal ones work instantly
E) Realistic batteries have unequal current going into/out of their two ends

3. If I am trying to remove 6 C of charge from some object, and I do so with a current of
2 A, how long will it take me to remove the charge?
A) 24 s B) 12 s
C) 3 s
D) 1/3 s E) 1/12 s
4. Which of the following is one of Kirchoff’s rules?
A) The total voltage going into a vertex must be zero
B) The total current around a loop must be zero
C) The total voltage differences across all components in any circuit is zero
D) The total current flowing through a capacitor must be zero
E) The total voltage differences around a loop is zero
5. In the Hall effect, a current flows through a conductor while a magnetic field is present, and a voltage difference appears across the conductor. Which direction is this voltage difference?
A) Parallel to both the current and the magnetic field
B) Parallel to the current and perpendicular to the magnetic field
C) Parallel to the magnetic field and perpendicular to the current
D) Perpendicular to both the current and the magnetic field
E) Neither perpendicular nor parallel to the current or the magnetic field
6. In a velocity selector, there are charged particles in a region of a magnetic field. How come the particles move in a straight line, despite the magnetic field?
A) The particles are moving parallel to the magnetic field, so no force
B) The particles are moving anti-parallel to the magnetic field, so no force
C) The particles move so slowly that the magnetic force is negligible
D) The particles attract electrons that cancel the charge, so no force
E) There is an electric field that exactly balances the magnetic field
7. In which of the following cases is there not a right hand rule?
A) Working out the forces from magnetic fields
B) Working out the magnetic field inside a loop or solenoid
C) Finding the magnetic field from an infinite wire
D) Finding electric forces
E) Actually, there is a right-hand rule for each of these
8. The formulas for loops have the vector A describing the loop. What is this A ?
A) A vector whose magnitude is the area and whose direction is perpendicular to the plane of the loop
B) A vector whose magnitude is the area and whose direction is parallel to the plane of the loop
C) A vector showing the current flowing in the loop, measured in amps, and oriented along the loop
D) A vector showing the current flowing in the loop, measured in amps, and oriented perpendicular to the loop
E) A vector measuring the current density inside the loop of wire

9. When you put two identical resistors in parallel, the resistance is _________, if you put them in series, the resistance is ___________
A) Decreased, increased
B) Increased, decreased
C) Decreased, decreased
D) Increased, increased
E) None of the above
10. A wire with current I going to the right passes through a sphere of radius r . If we calculate the total magnetic flux passing out of the sphere, the result will be
A) Positive
B) Negative
C) Zero
D) Insufficient Information
E) Can I still drop this class?
Part II: Short answer [20 points]
- + answer (1-3 sentences) or brief sketch (10 points each).
11. You have no idea what that curly thing is in this circuit diagram, so you decide to add an ammeter and a voltmeter to the circuit to see what the voltage across it and the current in it are. Add them at appropriate places to the sketch.
I have added them with dotted lines.
V
A
12. Ampere’s Law states   B s  
0
I
. What is the meaning of that funny symbol on the integral? What should be included in the current
I
?
The funny symbol
 
means you integrate around a closed loop, starting and ending at the same point. The current I includes all currents that pass through this loop, added if they go through the loop in a positive sense as governed by the right-hand rule, and subtracted if they go through the opposite way.
13. Suppose I have current flowing through an imperfect conductor, and the charge carriers are electrons. Explain how the direction of the electric field, the drift velocity of the electrons, and the current will be related.
The electrons are being pushed in the direction opposite to the electric field, but the current will always be in the same direction as the electric field.

Part III: Calculation: [60 points]
` Choose of the following four questions and perform the indicated calculations (20 points each)
14. Consider the circuit at right. The capacitor is initially uncharged
(a) [6] What is the effective resistance of the two resistors?
Because the resistors are connected to each other at both ends, they are in parallel, so we have
– +
12 V
40

F
1

1

1

R R
1
R
2
1


1




1
30 k 60 k 60 k 20 k

.
We can replace this pair of resistors with a single one with resistance of R = 20 k

.
(b) [6] What is the time constant for this circuit?
The circuit, once replaced as described, is a simple RC circuit charging from a battery. The time constant is
 
RC


 3 

 
6


0.800 s .
(c) [8] How much charge has accumulated on the capacitor 0.6 s after the switch is closed?
We use the formula
Q
  C

1
 e
 t


 
12.0 V
 

6

 e

0.6 0.8

 
4  

15. A resistor is going to be made in the shape of a cylinder with length
L
= 1.00 cm and unknown radius r
. It will be made from carbon, which has a resistivity of
  
5  at T
0

20 C , and a coefficient of resistance of
  
5.0 10

4  .
(a) [10] At T
0

20 C , it is supposed to have a resistance of exactly R = 470  .
What should be the radius r
of the resistor? be

Since the temperature is the same as the reference temperature, the resistivity will
 
5  .
Using the fact that the area of a circle is A
  r
2
, we use the formula for resistance to find
R


A
L


 r
L
2

2 r R
 
L , r
2 


L
R

,

3.5 10

5 
 
470
 
  r
 
5  

  
10 2
2.37 10 m ,
(b) [10] The resistance is measured on an especially cold day and found to have risen to 480  . What is the temperature?
We use the same formula as before, but this time we solve for the temperature.
1
 

R

R
0
 1
  
T

T
0

T

T
0
 
R
R
0


T

T
0
 
0.0213
480
470



,

1.0213 ,
T
 
0
0.0213


20 C
0.0213

5.00 10

4 

20 C 42.6 C 22.6 C .

4.00 cm
16. A solenoid is in the shape of a cylinder 3.00 m in length and 2.00 cm in radius. The solenoid has 800 turns per centimeter, carries a current of
I
= 0.25 A, and it is wrapped counter-clockwise as viewed from above.
(a) [7] What is the magnetic field and the direction of the field inside the solenoid?
The solenoid is much longer than it is wide, and hence we can approximate it as an infinite solenoid. The ratio N L
   4
800 / cm 8 10 / m , so we have
B in


0
NI
L


4
  
7 


   
0.0251 T .
Curling your fingers in the direction the current is flowing, the magnetic field will point upwards, along the axis of the solenoid.
(b) [8] A proton (mass m

1.672 10

27
kg
is placed in the solenoid and observed to move in a circular orbit. Which direction does it circle? What is the maximum velocity the proton can be given and still stay within the confines of the solenoid?
B v
If the proton is moving to the right, and the magnetic field is pointed out of the page, as sketched at right, then the product v B will be downwards, and so will the force
F  q
  e

, and it
F will turn to the right, as viewed from above. This means it will move in a clockwise circle, as viewed from above, the opposite of the direction of the current flowing in the solenoid.
For simple cyclotron motion, for which we have p
 qRB , so mv
 eRB . If it is to remain in the solenoid, this circle had better be smaller than the radius, so we have v
 eRB m


1.602 10

19
  

27
1.672 10 kg

  4 
48.1 km/s .
T
(c) [5] How long does it take to go in each circle?

Time is distance over velocity, and the distance is 2
 r , so we have
2

R v

2

R m

2
 eRB eB m

2


1.672 10

27
kg


1.602 10

19
  
 
6  

17. A straight infinite wire is carrying a current of 5.00 A to the right. A square loop of side a
= 4.00 cm is carrying a clockwise current of 1.00 A and is located a distance 1.00 cm from the infinite wire.
(a) [10] Find a formula for the magnetic field due to the infinite wire. Find the direction and the magnitude at the top and bottom of the loop.
The magnitude of the magnetic field is
B


2

0
I a


4
  
7 
2
 a
  


6 a
5.00 A
1.00 A
.
In particular, for the top and bottom of the loop, the magnetic field will be
B top


0.01 m

B bottom


0.05 m

The direction of the magnetic field can be found by pointing your thumb along the main wire, and the magnetic field is in the direction of your fingers. It will be into the paper for both the top and the bottom of the loop.
(b) [10] Find the force and the direction of the force on the top and bottom of the loop due to the magnetic field from the wire. Find the total force. Will there be any forces from the other two sides of the loop? Will they contribute to the total force?
The force is given by
F 
I
L B
, which is upwards on the top side of the loop and downwards on the bottom loop. The magnitude of the force is
F top
F bottom


 
 
0.04 m






6
6



 


6

6
Since they are in opposite directions, we subtract them to get a total force of
F
 
6
3.2 10 N , directed upwards.
Although there are forces (outward) on both the left and right side of the loop, they will cancel, because there will be a corresponding point on the other side pulled the opposite direction by exactly the same magnitude.