MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Physics
8.02
Spring 2013
Problem Set 2
Due: Tuesday, February 19 at 9 pm.
Hand in your problem set in your section slot in the boxes outside the door of 32-082 or 26-152
depending on which is your classroom. Make sure you clearly write your name and section on
your problem set.
Text: Dourmashkin, Belcher, and Liao; Introduction to E & M MIT 8.02 Course Notes
Revised.
Week Two: Gauss’s Law
Problem Set 1 Due Tuesday Feb 12 at 9 pm
W02D1 M/T Feb 11/12
Reading
Electric Dipoles and Continuous Charge Distributions
Course Notes: Sections 2.9-2.13, 2.14.5-2.14.6
W02D2 W/R Feb 13/14
Reading
Gauss’s Law
Course Notes: Sections 3.1-3.2, 3.6, 3.7, 3.10
W02D3 F Feb 15
Reading
PS02: Gauss’s Law
Course Notes: Sections 3.6, 3.7, 3.10
Week Three:
Electric Potential
Problem Set 2 Due Tuesday Feb 19 at 9 pm
W03D1 T Feb 19
Monday Schedule: Faraday Law Exploration
W03D2 W/R Feb 20/21
Reading
Electric Potential, Discrete and Continuous Charges;
Configuration Energy
Course Notes: Sections Sections 4.1-4.3
W03D3 F Feb 22
Reading
PS03: Electric Potential
Course Notes: Sections Sections 4.7-4.10
Problem 1
(a) Flux Through a Face of a Cube Consider a cube with each side of length a . A pointlike object with charge Q is placed at one corner of a cube shared by three faces. What is
the flux of the electric field emerging from each of the other three square faces of the
cube?
(b) Estimation: Excess Charge on a Thundercloud Estimate the magnitude of the excess
charge on a thundercloud just before a lightening strike? (HINT: Dry air breaks down at
an electric field strength of about 3  106 N  C1 ). Explain your reasoning.
Problem 2 Electric Field of a Washer
A very thin washer has outer radius b and inner radius a . The washer is uniformly charged with
a positive surface charge density  . What is the magnitude and direction of the electric field at
an arbitrary point P along the positive z -axis, a distance z from the origin?
Problem 3 Charged Rods
(a) A very thin rod of length L lies along the x-axis with its left end at the origin. The rod has a
non-uniform charge density    x , where  is a positive constant and x is the distance from
the origin.
Calculate the magnitude and direction of the electric field at the point P, shown in the figure
above. Take the limit d ? L . What does the electric field look like in this limit? Is this what
you expect? Explain. Hint: the following mathematical facts may be useful:
x dx
 (x  a)
2

a
 ln(x  a)
xa
ln(1  x)  x 
x2
L
2
(for small x)
(b) Two thin rods of length L carry equal charges Q uniformly distributed over their lengths. The
rods are aligned end to end, with their nearest ends separated by a distance d. What is the
magnitude of the force that one rod exerts on the other?
Problem 4 Non-uniformly Charged Cylinder A solid very long cylinder of radius R has a
charge density   0 (r / R) where  0 is a constant and r is the radial distance from the center
axis of the cylinder. Find the direction and magnitude of the electric field everywhere (both
inside and outside the cylinder).
Problem 5: Consider a spherically symmetric charge distribution given by
 0 (1  r / R) ; r  R
0 ; r  R
 (r)  
where  0 is a negative constant. Embedded in the center of the distribution is a point-like
positively charge object with charge Q  0 such that the electric field in the region r  R is
zero.
a) Find an expression for the charge Q of the point-like charged object.
b) Find a vector expression for the electric field in the region r  R .
Problem 6: Gauss’s Law for Gravitation
The gravitational force on a point-like object 1 with mass m1 due to the interaction with a second
point-like object 2 of mass m2 that are separated by a distance r12 is given by the expression
r
Gm1m2
F21  
rφ
r212 21
where rφ21 is the unit vector located at object 1 and pointing from object 2 to object 1.
r
a) In your own words, define the gravitational field g of a point-like object of mass m .
Then find a vector expression for the gravitational field at a distance r from the pointlike object.
b) Find an expression for a gravitational version of Gauss’s law.
c) Suppose you are given a spherically symmetric distribution of matter with uniform mass
density  and radius R ? Determine a vector expression for the gravitational field in the
regions (i) r  R and (ii) r  R ?
d) Consider an infinite universe with a uniform mass density  . Use your gravitational
version of Gauss’s Law to show that the gravitational field must be zero everywhere.
Hint: Consider any point P in space. Choose two spherical Gaussian surfaces centered
about different points such that the point P lies on the surface of either Gaussian surface.
What is the gravitational field at the point P ?
Problem 7: Infinite Uniformly Charged Slabs
Three infinite uniformly charged thin sheets are shown in the figure below. The sheet on the left
at x  d is positively charged with charge per unit area of 2 , The sheet in the middle at
x  0 is negatively charged with charge per unit area of  , and the sheet on the right at x  d
is positively
charged with charge per unit area of 2 . Determine an expression for the electric
r
field E in each of the regions 1, 2, 3, and 4.
Problem 8 N-P Junction
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’s model the N-P junction 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 a positive uniform
charge density  0 . The adjacent P-type material lies in the range a  z  0 and has a negative
uniform charge density  0 . Thus:
 
 0
 (z)   0

0
0  z a
 a z 0
z a
Find an expression for the electric field in the regions (i) z  a , (ii) a  z  0 , (iii) 0  z  a ,
and (iv) z  a .