MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Physics
Exam 2 Practice Problems Part 1
Problem 1 Electric Field and Charge Distributions from Electric Potential
An electric potential V ( z ) is described by the function
)(!2V " m -1 )z + 4V ; z > 2.0 m
+
+0 ; 1.0 m < z < 2.0 m
+2
#2
-3 & 3
+ V ! % V " m ( z ; 0 m < z < 1.0 m
3
3
$
'
+
V (z) = *
+ 2 V+ # 2 V " m -3 & z 3 ; ! 1.0 m < z < 0 m
%$ 3
('
+3
+
+0 ; ! 2.0 m < z < ! 1.0 m
+(2V " m -1 )z + 4V ; z < ! 2.0 m
,
The graph below shows the variation of an electric potential V ( z ) as a function of z .
!
a) Give the electric field vector E for each of the six regions in (i) to (vi) below?
b) Make a plot of the z-component of the electric field, Ez , as a function of z . Make
sure you label the axes to indicate the numeric magnitude of the field.
c) Qualitatively describe the distribution of charges that gives rise to this potential
landscape and hence the electric fields you calculated. That is, where are the charges,
what sign are they, what shape are they (plane, slab…)?
Problem 2: Electric Field from Electric Potential
The electric potential V (x) for a planar charge distribution is given by:
)
+
+
+
+
V (x) = *
+
+
+
+,
for x < !d
0
"
x%
!V0 $ 1+ '
# d&
2
for ! d ( x < 0
"
x%
!V0 $ 1+ 2 '
d&
#
for 0 ( x < d
!3V0
for x > d
where !V0 is the potential at the origin and d is a distance.
This function is plotted to the right, with d = 2 cm and V0 = 2 V , the x-axis with units in
cm, the y-axis in units of Volts.
!
(a) What is the electric field E(x) for this problem?
(b) Plot the electric field that you just calculated on the graph below. Be sure to properly
label the y-axis (top and bottom) to indicate the limits of the magnitude of the E field!
Problem 3 Dielectric
A dielectric rectangular slab has length s , width w , thickness d , and dielectric constant
! . The slab is inserted on the right hand side of a parallel-plate capacitor consisting of
two conducting plates of width w , length L , and thickness d . The left hand side of the
capacitor of length L ! s is empty. The capacitor is charged up such that the left hand
side has surface charge densities ±! L on the facing surfaces of the top and bottom plates
respectively and the right hand side has surface charge densities ±! R on the facing
surfaces of the top and bottom plates respectively. The total charge on the entire top and
bottom plates is +Q and !Q respectively. The charging battery is then removed from the
circuit. Neglect all edge effects.
a) Find an expression for the magnitude of the electric field EL on the left hand side
in terms of ! L , ! R , ! , s , w , L , ! 0 , and d as needed.
b) Find an expression for the magnitude of the electric field ER on the right hand
side in terms of ! L , ! R , ! , s , w , L , ! 0 , and d as needed.
c) Find an expression that relates the surface charge densities ! L and ! R in terms of
! , s , w , L , ! 0 , and d as needed.
d) What is the total charge +Q on the entire top plate? Express your answer in terms
of ! L , ! R , ! , s , w , L , ! 0 , and d as needed.
e) What is the capacitance of this system? Express your answer in terms of ! , s ,
w , L , ! 0 , and d as needed.
f) Suppose the dielectric is removed. What is the change in the stored potential
energy of the capacitor? Express your answer in terms of Q , ! , s , w , L , ! 0 ,
and d as needed.
Problem 4: Capacitors and Dielectrics
(a) Consider a plane-parallel capacitor completely filled with a dielectric material of
dielectric constant κ . What is the capacitance of this system?
(b) A parallel-plate capacitor is constructed by filling the space between two square
plates with blocks of three dielectric materials, as in the figure below. You may assume
that ℓ >> d. Find an expression for the capacitance of the device in terms of the plate area
A and d, ! 1 , ! 2 , and ! 3 .
Problem 5: Conductors and Potential
A charge !Q is at the center of a neutral metal (conducting) sphere of radius R1 that is in
turn centered in a larger metal (conducting) sphere of radius R2 , which carries a net
charge of +Q . The potentials of the inner and outer spheres with respect to infinity are
respectively:
Inner
1
1
A. keQ( ! )
R2 R1
Q
B. !ke
R1
C. 0 ,
D. keQ(
Outer
Q
ke
R2
0
keQ(
1
1
! )
R2 R1
Explain your reasoning.
0
1
1
! )
R1 R2
Problem 6: Faraday Cage
Consider
two
nested,
spherical
conducting shells. The first has inner
radius a and outer radius b. The second
has inner radius c and outer radius d.
In the following four situations, determine the total charge on each of the faces of the
conducting spheres (inner and outer for each), as well as the electric field and potential
everywhere in space (as a function of distance r from the center of the spherical shells).
In all cases the shells begin uncharged, and a charge is then instantly introduced
somewhere.
a) Both shells are not connected to any other conductors (floating) – that is, their net
charge will remain fixed. A positive charge +Q is introduced into the center of
the inner spherical shell. Take the zero of potential to be at infinity.
b) The inner shell is floating but the outer shell is grounded – that is, it is fixed at
V = 0 and has whatever charge is necessary on it to maintain this potential. A
negative charge !Q is introduced into the center of the inner spherical shell.
c) The inner shell is grounded but the outer shell is floating. A positive charge +Q
is introduced into the center of the inner spherical shell.
d) Finally, the outer shell is grounded and the inner shell is floating. This time the
positive charge +Q is introduced into the region in between the two shells. In
!
this case the questions “What are E(r) and V (r) ?” cannot be answered
analytically in some regions of space. In the regions where these questions can be
answered analytically, give answers. In the regions where they can’t be answered
analytically, explain why, but try to draw what you think the electric field should
look like and give as much information about the potential as possible (is it
positive or negative, for example).
Problem 7 N-P Semiconductors Revisited Potential Difference
When two slabs of n -type and p -type semiconductors are put in contact, the relative
affinities of the materials cause electrons to migrate out of the n -type material across the
junction to the p -type material. This leaves behind a volume in the n -type material that
is positively charged and creates a negatively charged volume in the p -type material.
Let us model this as two infinite slabs of charge, both of thickness a with the junction
lying on the plane z = 0 . The n -type material lies in the range 0 < z < a and has uniform
charge density + !0 . The adjacent p -type material lies in the range !a < z < 0 and has
uniform charge density ! "0 . Thus:
# + !0
%
!(x, y, z) = !(z) = $" !0
%
&0
0 < z< a
" a< z< 0
z >a
a) Find the electric field everywhere.
b) Find the potential difference between the points P1 and P2. . The point P1. is located
on a plane parallel to the slab a distance z1 > a from the center of the slab. The
point P2. is located on plane parallel to the slab a distance z2 < !a from the center
of the slab.
Problem 8: Conceptual Questions
Part (a) A parallel plate capacitor has capacitance C. It is connected to a battery of
EMFe until fully charged, and then disconnected. The plates are then pulled apart an
extra distance d, during which the measured potential difference between them changed
by a factor of 4.
Below are a series of questions about how other quantities changed. Although they are
related you do not need to rely on the answers to early questions in order to correctly
answer the later ones.
a) Did the potential difference increase or decrease by a factor of 4?
INCREASE
DECREASE
b) By what factor did the electric field change due to this increase in distance?
Make sure that you indicate whether the field increased or decreased.
c) By what factor did the energy stored in the electric field change?
Make sure that you indicate whether the energy increased or decreased.
d) A dielectric of dielectric constant k is now inserted to completely fill the volume
between the plates. Now by what factor does the energy stored in the electric field
change? Does it increase or decrease?
e) What is the volume of the dielectric necessary to fill the region between the plates?
(Make sure that you give your answer only in terms of variables defined in the
statement of this problem, fundamental constants and numbers)
Problem 9: Spherical Capacitor and Stored Energy
A capacitor consists of two concentric spherical shells. The outer radius of the inner shell
is a = 0.1 m and the inner radius of the outer shell is b = 0.2 m .
a) What is the capacitance C of this capacitor?
b) Suppose the maximum possible electric field at the outer surface of the inner shell
before the air starts to ionize is Emax (a) = 3.0 ! 106 V " m -1 . What is the maximum
possible charge on the inner capacitor?
c) What is the maximum amount of energy stored in this capacitor?
d) When E(a) = 3.0 ! 106 V " m -1 what is the potential difference between the shells?
Problem 10: Cylindrical Capacitor
Consider two nested cylindrical conductors of height h and radii a & b respectively. A
charge +Q is evenly distributed on the outer surface of the pail (the inner cylinder), -Q on
the inner surface of the shield (the outer cylinder). You may ignore edge effects.
a)
Calculate the electric field between the two cylinders (a < r < b).
b)
Calculate the potential difference between the two cylinders:
c)
Calculate the capacitance of this system, C = Q/DV
d)
Numerically evaluate the capacitance, given: h ≅ 15 cm, a ≅ 4.75 cm and
b ≅ 7.25 cm.
e)
Find the electric field energy density at any point between the conducting
cylinders. How much energy resides in a cylindrical shell between the conductors of
radius r (with a < r < b ), height h , thickness dr , and volume 2! rh dr ? Integrate your
expression to find the total energy stored in the capacitor and compare your result with
that obtained using U E = (1 / 2)C(!V )2 .
Problem 11 Capacitance of Multiple Plates
A capacitor is made of three sets of parallel plates of area A , with the two outer plates on
the left and the right connected together by a conducting wire. The outer plates are
separated by a distance d . The distance from the middle plate to the left plate is z . The
distance from the inner plate to the right plate is d ! z . You may assume all three plates
are very thin compared to the distances d and z . Neglect edge effects.
a) The positive terminal of a battery is connected to the outer plates. The negative
terminal is connected to the middle plate. The potential difference between the
outer plates and inner plate is !V = V (z = 0) " V (z) . Find the capacitance of this
system.
b) Find the total energy stored in this system.
Problem 12: Capacitance, Stored Energy, and Electrostatic Work
Two flat, square metal plates have sides of length L , and thickness s 2 , are
arranged parallel to each other with a separation of s , where s << L so you may ignore
fringing fields. A charge Q is moved from the upper plate to the lower plate. Now a
force is applied to a third uncharged conducting plate of the same thickness s 2 so that it
lies between the other two plates to a depth x , maintaining the same spacing s 4
between its surface and the surfaces of the other two.
a) What is the capacitance of this system?
b) How much energy is stored in the electric field?
c) If the middle plate is released, it starts to move. Will it move to the right or left?
Hint: If the middle plate moves to the left by a small positive amount !x , the
change in potential energy is approximately !U ! (dU / dx)!x . Will the stored
potential energy increase or decrease?
d) For a small displacement !x in the direction you determined in part c), find the
horizontal force exerted by the charge distribution on the outer plates acting on
the charges on the middle plate that cause it to move.
Problem 13 Energy and Force in a Capacitor
A flat conducting sheet A is suspended by an insulating thread between the surfaces
formed by the bent conducting sheet B as shown in the figure on the left. The sheets are
oppositely charged, the difference in potential, in volts, is !V . This causes a force F , in
addition to the weight of A , pulling A downward.
a) What is the capacitance of this arrangement of conductors as a function of y , the distance
that plate A is inserted between the sides of plate B ?
b) How much energy is needed to increase the inserted distance by !y ?
c) Find an expression for the difference in potential !V in terms of F and relevant
dimensions shown in the figure.