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Physics 202
Spring 1999
Solution to Mid-term 1
• This is FORM A; the questions in the other forms are identical,
except for the order in which they appear.
• It is strongly advised that you go through this detailed solution and
understand where you may have erred. If you find conceptual
difficulties with some of the questions/solutions, consult an instructor
to have this cleared up.
1
1. The figure below shows 4 point charges located on a circle centered about the origin.
The exact locations of the charges on the circle are not given. All 4 charges have the
same magnitude, but two of them POSITIVE whilst the other two are NEGATIVE.
What can you say about the TOTAL electric field created by these charges at the
center of the circle?
Y
-Q
+Q
X
+Q
-Q
In order to figure out the net E, you have to carry out a vector sum of 4 vectors here.
Although you know that all these vectors have the same magnitude (since the charges
equidistant from the origin, you DON'T KNOW their DIRECTIONS. So, there is no way
you can precisely calculate the vector sum.
(a)
(b)
(c)
(d)
(e)
Since the charges are equidistant from the origin, the field is 0.
The field points along the x-axis.
The field points along the y-axis.
The field makes a 450 angle with the x-axis in the first quadrant.
Without knowing the positions of the charges, we cannot be precise about the
magnitude or direction of the field.
2
2. The figures below shows two positively charged spheres of equal mass, each
suspended by a string of equal length. Sphere A has twice as much charge as B.
Which of the following figures correctly depicts the entire system in static
equilibrium?
From a free body diagram for any particular sphere, you know that the horizontal force F
exerted by the other charged sphere is F = mgtanθ, where θ is the angle between the
string and the vertical. (e.g see HW 1 + also reinforced in lab 1.) But, by Newton's 3rd
Law, the spheres exert equal and opposite force on each other. Since their masses are the
same, the angle θ must be the same.
(a)
(b)
B
(c)
A
B
A
B
(d)
A
(e)
A
B
A
B
3
3. You are given four identical steel marbles A, B, C and D that carry an excess charge
of +Q, +5Q, -10Q, and -20Q, respectively. All four spheres are connected together
by a conducting wire. The wire is then removed. What is the final charge on the
spheres?
The charge is equally shared since all the marbles are identical.
(a)
(b)
(c)
(d)
(e)
A: +Q, B:+5Q, C: -10Q, D: -20Q
A: +3Q, B: +3Q, C: -15Q, D:-15 Q.
They each have a charge of +6Q.
They each have a charge of -9Q.
They each have a charge of -6Q.
4. A point charge of -5 C and mass 5 kg experiences an electric field of 5 N/C directed
along the positive x axis when it is placed at some point in space. What is the
instantaneous acceleration experienced by the charge?
The charge is negative, so the force is opposite to E i.e. along negative x. The
acceleration = force/mass = (charge.E)/mass = 5 m/s2
(a)
(b)
(c)
(d)
(e)
5 m/s2 directed along the positive x-axis.
5 m/s2 directed along the negative x-axis.
25 m/s2 directed along the positive x-axis.
125 m/s2 directed along the negative x-axis.
125 m/s2 directed along the positive x-axis.
5. An electric dipole that is completely free to move (both rotation and translation) is
placed in a region of uniform electric field. The dipole moment makes an angle of 300
to the direction of the electric field. Which of the following statements is correct?
The total force on a dipole in an uniform electric field = 0 since the charges are equal in
magnitude but opposite in sign. The TORQUE however is non-zero = pEsinθ, where p is
the dipole moment.
(a) The total torque on the dipole is zero, but the total force on the dipole is non-zero.
(b) The total torque on the dipole is zero.
(c) Both the total force and the total torque on the dipole are non-zero.
(d) The total force on the dipole is zero, but the total torque is non-zero.
(e) Both the total torque and total force on the dipole are zero.
4
6. A charge of +2C is placed 3m away from a charge of +8C. Both charges are free to
move. Describe how you would arrange a third charge on the line between the two
charges such that all 3 charges are in static equilibrium.
Clearly the charge needs to be negative. Suppose you put this charge of -q at a distance x
from the +2 C. Then, balance out the force on this charge: k(2)(q)/x2 = k(8)(q)/(3-x)2.
Solving this will give you x = 1m. Now, balance out the force on the +2C charge:
k(2)(q)/(12) = k(8)(2)/(32). Solve this to get (magnitude) q =8/9 C.
(a)
(b)
(c)
(d)
(e)
Place a charge of -8/9 C at distance of 1 m from the +2C charge.
Place a charge of -8/9 C at distance of 2 m from the +2C charge.
Place a charge of +8/9 C at distance of 2 m from the +2C charge.
Place a charge of -8/3 C at distance of 1 m from the +2C charge.
This cannot be done.
7. Two point charges of equal magnitude but opposite sign are placed 1 cm away from
each other to form an electric dipole. At a point on the dipole axis 10 m away from
the center of the dipole, the electric field is 0.054 N/C. What is the electric field on
the dipole axis at a spot 30 m away from the dipole center?
In the lecture and lab, you calculated the electric field of a dipole on the dipole axis and
also on the dipole bisector. In both cases, when you are very far away from the dipole, the
field varies inversely with the CUBE of the distance. (This is actually true for any point
in space far from a dipole. Prove this: any takers??) So, tripling the distance reduces the
field by a factor of 27.
(a) There isn’t enough information given to answer this.
(b) 0.027 N/C.
(c) 0.018 N/C.
(d) 0.002 N/C
(e) 0.006 N/C.
5
8. The electric field lines in some region of space are shown in the figure below. If an
proton is released from rest at point A, how will it behave?
A
Positive charges move in the direction of E. The field lines get closer, so the force is
greater, hence acceleration is greater.
(a) It will move towards the right with increasing acceleration.
(b) It will move towards the right with increasing speed but constant acceleration.
(c) It will move towards the left with increasing speed.
(d) It will move towards the left with increasing acceleration.
(e) It will move towards the left with decreasing speed.
9. A point charge placed 1 mm away from an infinitely large sheet of charge
experiences a force of 10 N. What is the force experienced by this charge if you move
it to a point that is 5 mm away from the sheet?
The electric field of an infinitely large sheet of charge is UNIFORM i.e. does not change
with distance.
(a) 0.4 N.
(b) Finite, but extremely small.
(c) 50 N.
(d) 2 N.
(e) 10 N.
6
10. A point charge +Q of mass m is moving with uniform speed V in the positive x
direction. At t = 0, it enters a region of uniform electric field E, where E is pointing in
the positive y-direction. What is the speed of the charge after a time t?
The component of the velocity along x does not change. The component along y changes
with uniform acceleration i.e. vy = at = Qet/m. The speed is then the magnitude of the
vector with these two components.
(a) Vt +
1 QE 2
t
2 m
(b) V
QE
t
(c)
m
QE
t
(d) V +
m
2
QEt 
(e) V 2 + 
m 
7
11. The figure below shows a thin, straight rod of length L over which a charge Q is
uniformly distributed. Which of the expressions below gives the magnitude of the
electric field at point P located a distance Y away from the rod on a line that is
collinear with the rod?
This problem was done in the lecture. Choose the origin at the extreme left of the rod.
The charge per unit length is Q/L. For a differential element of the rod of length dx, the
contribution to the field is (kQ/L)/(L-x+Y)2. Integrating this from x=0 to x=L gives the
net field.
P
L
Y
L
(a)
Q
dx
∫
4πε0 L 0 ( L − x + Y ) 2
(b)
Q
dx
∫
4πε0 L 0 ( x + L) 2
L
Q
(c)
4πε0 L
L+ Y
dx
Y )2
∫ (x +
0
L
(d)
Q
dx
∫
4πε0 L 0 ( x + Y )
L
Q
dx
(e)
∫
4πε0 L 0 ( x + L + Y ) 2
12. A sphere of radius 2 m contains a total charge 0.008 C uniformly distributed over its
entire volume. Consider any spherical volume of radius 1 m inside this sphere. How
much charge is contained within this volume?
For a sphere, volume goes as radius cubed. So, charge enclosed also goes as radius
cubed. Reducing the radius by half then reduces the charge by 1/8.
(a) 0.008 C.
(b) 0.006 C.
(c) 0.004 C.
(d) 0.002 C.
(e) 0.001 C.
8
13. The figure below shows two uniformly charged arcs of radius R. The arc on the left
has a net negative charge of -Q, whilst the arc on the right has a net negative charge
of -2Q. If the magnitude of the electric field at the origin due to the arc on the left is
E, what is the magnitude and direction of the electric field at the origin due to the arc
on the right?
y
y
x
x
See diagram above. You get the direction of E for the case on the left simply by
symmetry. Then for the case on the right, you need to do a vector sum of two such
vectors that are at 900 to each other.
(a) 2E, along the positive x-axis.
(b) 2E along the negative x-axis.
(c) 1.414E along the negative x-axis.
(d) 1.414E along the positive x-axis.
(e) 1.414E at an angle of 450 to the x-axis, pointing upwards and to the right.
9
14. The figure below shows 3 positive point charges, each of magnitude +Q, arranged at
the corners of an equilateral triangle of side L. What is the electric field at the
geometric center C of the triangle?
+Q
C
+Q
+Q
You want to sum 3 vectors of equal magnitude making angles of 1200 with each other.
The sum is clearly 0. (Easy to see if you put the vectors end to end as shown above.)
(a) 0
1.73Q
(b)
4πε0 L2
3Q
(c)
4πε0 L2
5.41Q
(d)
4πε0 L2
3.62Q
(e)
4πε0 L2
10
15. The figure below shows an edge-on view of three extremely large, parallel insulating
sheets positively charged with uniform and identical surface charge densities. In
which of the regions shown is the electric field a maximum?
I
II
III
IV
(I think everyone got this one!) Just use superposition. Clearly, the fields all add up only
in regions I and IV. In all the other regions, the field from 2 of the plates opposes that
from the remaining plate.
(a) I
(b) II
(c) III
(d) IV
(e) I & IV
16. You are told that the total electric flux through the 6 faces of a cube is +1 Nm2/C.
What can you conclude from this information?
Use Gauss' Law. Q = Flux x 8.85 x 10-12 = 8.85 x 10-12 C. (by the way, (a) is wrong
because the problem makes no mention of symmetry.)
(a) Each face of the cube must have a flux of +0.167 Nm2/C.
(b) The cube probably contains some negative charge, but we can't say how much.
(c) The cube probably contains some positive charge, but we can't say how much.
(d) The cube contains +1 C of charge.
(e) The cube contains +8.85 x 10-12 C of charge.
11
17. You are located at distance of a 5 m from the axis of an extremely long uniformly
charged cylinder of radius 1 mm. If you double your distance from the cylinder, what
happens to the electric field?
(See HW 3 problem 24.27P)
E field from a line of charge drops off as 1/r -- a cylinder acts like a line of charge just a
sphere acts like a point charge. Recall how to calculate the field at a distance r from a line
of charge. Draw a gaussian cylinder of radius r and length L concentric with the charge.
The flux is (2πrL)E. The enclosed charge is λL, where λ= linear charge density. Gauss
Law then yields:
(2πrL)E = λL/ε0, so that E = λ/(2 πε0r).
(a)
(b)
(c)
(d)
(e)
Remains the same.
Decreases by a factor of 2.
Decreases by a factor of 4.
Decreases by a factor of 8.
Increases by a factor of 4.
18. A hollow metal spherical shell contains a point charge placed somewhere inside it.
Which of the following affect the magnitude of the electric field outside the sphere?
(See sample problem 24.4)
The charge placed inside induces an equal and opposite charge on the inside surface. The
remaining charge on the shell goes to the outside surface. Moving the charge around
inside only changes the charge distribution on this inside surface, following Gauss' Law
so that the field in the metal shell = 0. Outside the sphere, all you see is a charged sphere
with a uniform distribution of charge on the surface. The magnitude/sign of this charge
determines the E-field on the outside (acts like a point charge at the center of the sphere.)
The magnitude/sign of this charge is affected by the amount of charge inside. But, this is
NOT affected by where you place the charge inside, nor by the size of the sphere!
(a) The radius of the sphere.
(b) The exact location of the point charge inside the sphere.
(c) The magnitude of the charge.
(d) Both the exact location of the point charge inside the sphere and the magnitude of the
charge.
(e) All of the factors in (a), (b) and (c).
12
19. A total charge +Q is uniformly distributed throughout the volume of a very long
cylinder of radius R and length L. What is the magnitude of the electric field at a
distance r from the cylinder axis, where r < R?
This is analogous to the problem you did with spherical charge distribution (HW3,
problem 24.53P), but requires you to extend this argument to cylindrical symmetry.
Congratulations to those who got it!
Keywords: "very long cylinder" -- the field is directed radially outwards, perpendicular to
the cylinder axis. Choose a cylindrical Gaussian surface of radius r and length L'.
What's the flux thru this surface? = (2πrL')E.
What's the enclosed charge? = Q(πr2L')/ (πr2L) = (Qr2L')/ (R2L).
Gauss' Law says flux = charge enclosed /ε0. So:
(2πrL')E=(Qr2L')/ (ε0R2L).
or, E=(Qr)/ ((2πε0R2L).
(a)
Q
4πε0 r 2
(b)
Qr
2πε0 R 2 L
QR
4πε0 r 2 L
Q
(d)
2πε0 r
Qr
(e)
2πε0 R 2
(c)
13
20. The figure below shows a spherical conducting shell surrounding a concentric
conducting sphere that carries an excess charge of +Q. If an excess charge of -5Q is
placed on the conducting shell, how does this charge distribute itself on the inner and
outer surfaces of the shell?
Inside sphere contains
excess charge of +Q.
Shell carries excess
charge of -5Q.
The field inside the metallic region must be 0. Hence, Gauss' Law dictates that the
charge on the inside surface = -4Q. (Take a spherical surface inside the metallic shell.)
So, this leaves -Q on the outside.
(a)
(b)
(c)
(d)
(e)
0 on the outside, -5Q on the inside.
-2.5Q on the outside, -2.5 Q on the inside.
+Q on the outside, -Q on the inside.
-4Q on the outside, -Q on the inside.
+Q on the outside, -6Q on the inside.
14