Phys222
Electric Fields for Continuous Charge Distributions
Electric Fields
for
Continuous Charge Distributions
Names:
Grade:
____________
This is a group write-up.
Notes on your report:
You have been given six problems. Follow the write-up criteria shown below. There is no need to wordprocess these problems but each problem should have separate pages, i.e., be sure to start a new page when
you start a new problem.
Work in groups. You should hand in one report but it is not appropriate to divide and conquer entirely.
Each group member must participate in the solution of each problem. Each person will be responsible for
the final write-up of at least one problem. You must understand and be able to reproduce the answers to all
of your groups’s problems.
Write-up Criteria (Be neat and professional.)
1. Restate the problem.
2. Include a drawing of the charge distribution and the point of interest.
3. Include a sample vector diagram, showing the electric field vectors for critical dq’s.
4. Use the diagram in point 2 above to help explain any cancellations due to symmetry.
5. Set up the integrals to determine the non-zero electric field components. Show your
work/reasoning for building the integrals.
6. Solve the integrals. Box or underline your solutions.
7. Assess: is your result reasonable? Do the solutions have the correct behavior far away from the
charge?
Note: It is recommended that you use lots of scratch paper to work through the problems. However, when
you turn in a draft next week or the final version in two weeks, your work must be very clear and neat,
else I won’t be able to assist you (draft) or there will be an automatic deduction (final version). Set extra
time aside for a clean write-up, it does take some time. Think first, then commit to paper. Don’t skip steps
in your algebra: if I don’t see your work there will be a deduction.
Phys222
Electric Fields for Continuous Charge Distributions
Problem 1:
As shown in the figure, a non-conducting rod of length L has charge  q uniformly distributed along its
length.
a. What is the linear charge density of the rod?
b. What is the electric field at point P, a distance a from the end of the rod?
c. If P were very far from the rod compared to L, the rod would look like a point charge. Show that
your answer to part b reduces to the electric field of a point charge for a
L.
Phys222
Electric Fields for Continuous Charge Distributions
Problem 2:
A “semi-infinite” non-conducting rod (that is, infinite in one direction only) has uniform linear charge
density . Show that the electric field at point P makes an angle of 45⁰ with the rod and that this result is
independent of the distance R. (Hint: Separately find the parallel and perpendicular (to the rod) components
of the electric field at P, and then compare those components.)
Phys222
Lab Activity 3: Electric Fields for Continuous Charge Distributions
Problem 3:
A thin glass rod is bent into a semicircle of
radius r. A charge  q is uniformly distributed
along the upper half, and a charge  q is
uniformly distributed along the lower half, as
shown. Find the magnitude and direction of the
electric field
E at P, the center of the semicircle.
Phys222
Lab Activity 3: Electric Fields for Continuous Charge Distributions
Problem 4:
A non conducting hemisphere of radius R centered at the origin has a total charge Q spread uniformly over
its surface. The hemisphere is oriented such that its base is in the (y,z) plane. Find the electric field
anywhere along the x axis for x > 0. Give explicitly the value of the electric field at x = 0.
Hint: consider the hemisphere as a stack of rings.
y
x
z
Phys222
Lab Activity 3: Electric Fields for Continuous Charge Distributions
Phys222
Lab Activity 3: Electric Fields for Continuous Charge Distributions
Problem 5:
A thin, non-conducting rod is in the shape of a
semicircle of radius R. It has a varying charge
per unit length  described by   0 sin 2 ,
where  is defined in the figure.
a. Sketch the charge distribution along the
semicircle.
b. What is the direction of the electric field
c.
E at point ), the center of the
semicircle?
Find the magnitude of the electric field
at point O.
Phys222
Lab Activity 3: Electric Fields for Continuous Charge Distributions
Problem 6:
a.
b.
Consider a uniformly charged thin-walled right circular cylindrical shell having a total charge Q,
radius R, and height h. Determine the electric field at a point a distance d from the right side of the
cylinder as shown in the figure. (Suggestion: treat the cylinder as a collection of ring charges.)
Consider now a solid cylinder with the same dimensions and carrying the same charge, uniformly
distributed through its volume. Find the filed it creates at the same point.