Download Calculating Electric Fields of Charge Distribution

Phys222
Lab Activity 3: 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.
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 group’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 with ALL relevant
quantities. If you use symbol  in your math, it should be defined and be in the drawing.
⃗.
3. Include important dq’s , and the resulting electric field vectors 𝒅𝑬
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
Lab Activity 3: Electric Fields for Continuous Charge Distributions
Problem 1:
A thin, non-conducting rod of length L carries a line charge (x) that varies with distance
according to (x)=Ax2 (in SI units) as shown in the figure. Note that A is a positive
constant. Point P is located a distance d from the end of the rod as shown.
a. What are the SI units of the constant A?
b. Find the total charge Q on the rod.
c. Find the electric field at point P due to the line charge. Note that you will not need
the answer to part b to solve this.
y
P
x
L
d
Phys222
Lab Activity 3: Electric Fields for Continuous Charge Distributions
Problem 2:
A non-conducting rod has length 2L. The left half of the rod has a negative uniform linear
charge density -and the right half has a positive uniform linear charge density +
(is a positive constant).
a. Find the electric field at point P (located a distance d from the rod).
b. Suppose the rod is now turned into an infinitely long rod (with two semi-infinite
halves) with the same linear charge density as before. What would the electric field be at
point P?
y
P
d
- - - - - - - - - - - - - - - + + + + + + + + + + + + +
L
L
x
Phys222
Lab Activity 3: Electric Fields for Continuous Charge Distributions
Problem 3:
A uniform positive charge per unit length  exists
along a thin non-conducting rod bent into the
shape of a segment of a circle of radius R,
subtending an angle 20 as shown in the figure.
Find the electric field E at the center of curvature
O. (Hint: consider the field dE due to the charge
dq contained within an element of length
dl  Rd . Use symmetry considerations in
setting up the integral between   0 to
  0 to find the total field E at O.
Phys222
Lab Activity 3: Electric Fields for Continuous Charge Distributions
Problem 4:
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.
(Hint: you might be able to solve this
using your result from problem 3)
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.
Is the charge in the left half of the semicircle
positive or negative?
b. What is the direction of the electric field E at
point ), the center of the semicircle?
c. Find the magnitude of the electric field at point O.
Phys222
Lab Activity 3: Electric Fields for Continuous Charge Distributions
Problem 6a:
A non-conducting thin shell in the shape of a 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.
a1. Find an expression for the surface charge density 𝜂.
a2. Find the electric field at the center of the hemisphere,
i.e. at x=0. Hint: consider the hemisphere as a stack of
rings.
Phys222
Lab Activity 3: Electric Fields for Continuous Charge Distributions
Problem 6b:
Instead of a thin shell we are now considering a solid
hemisphere that has the charge Q distributed uniformly
throughout its entire volume.
b1. Find an expression for the volume charge density 𝜌.
b2. Find the electric field at the center of the hemisphere,
i.e. at x=0. Hint: consider the hemisphere as a stack of thin
shells and use your result from part a.