Ben Palmer
01372065
Due Thursday, February 1, 11:59pm
PHYS 2220 Spring 2024- Dr. De Grandi
Homework 3 - Part 2
Instructions: To guarantee full credit show all your steps, if you are using some additional equations
that are not the ones we discussed in class (not on the formula sheet) you need to be able to derive
them. If you have any questions post them on the Homework 3- Questions discussion board or come to
Study-hall.
Problem 1: Electric field of a flat ring with considerable thickness
In class, in Lecture 5, we discussed how to calculate the Electric field of a charged ring with negligible
thickness (i.e. we considered it one-dimensional). In this problem instead, you’ll calculate the case of a
charged ring that has a non-negligible thickness; we’ll call a the inner radius of the ring, and b the outer
radius of the ring (see the figure below). The ring has an overall charge Q and is flat, therefore can be
described as a two-dimensional object.
a) Find an expression for the surface charge density of the charged ring. Make sure your answer is in
terms of the given quantities only (a, b, Q and any given constant like k, ϵ0 , π, e, etc.).
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b) Review how in Lecture 5 (find slides and notes under Modules: Week 3), we calculated the Electric
field created by a charged disk at a point P at a distance z from the center of the disk, along an axis
perpendicular to the plane of the disk. No work is needed to be shown here, just make sure you are
comfortable with the calculation for the disk before moving to the next part.
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c) Prove that magnitude of the electric field due to the ring (see figure below) at a point P at a distance
z from the center of the ring, along an axis perpendicular to the plane of the ring is given by:
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2Q
z
z
√
√
EP = Edisk (z) = k 2
−
b − a2
z 2 + a2
z 2 + b2
Show all the steps of your calculation. Note: you want to make sure to replace the surface charge
density from part a) in terms of the given quantities (σ is NOT a given quantity in this problem), the
given quantities are: a, b, Q, z and any given constant like k, ϵ0 , π, e, etc.
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d) Show that in the limit of z → ∞ the Electric field of the ring reduces to Electric field of a point charge
with magnitude
Q
E = k 2.
z
Hints: you need to use the following Taylor series1 :
√
1
1
≈1− x
n
r
2
1+x
I
where x is a small quantity (in your case x will be either az or zb which are small quantities when
z → ∞).
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Problem 2: Who is right?
Hints!
Review the content of Lecture 5 that you can find under Modules: Week 3.
Two parallel, infinite planes of charge have charge densities +2σ and −σ as shown in Figure 1. They are
at a distance L0 from each other. Three students are asked to determine the electric field at points A
and B, at a distance L above the top plane, and at a distance L below the bottom plane, respectively, as
shown in the figure. Each student comes up with a different answer and a different explanation. Read
carefully their statements below.
The students are reminded that the electric field magnitude of an infinite plane of charge with charge
density σ is E = 2ϵσ0 .
Figure 1: Two equivalent views of the two infinite parallel planes
• Student 1 draws the sketch in Fig. 2 [top] (on the next page) and says: “Not sure how to explain it
but I think from the outside we can see it as a single infinite plane with charge density +2σ − σ = σ,
so then we know the electric field must be σ/(2ϵ0 ) pointing upwards at point A and also σ/(2ϵ0 )
pointing downwards at B”.
• Student 2 draws the sketch in Fig. 2 [middle] (on the next page) and says “I don’t agree with you, I
think you need to consider the effect of each single plane, so at point A there will be a field upwards
due to the top plane with magnitude 2σ/(2ϵ0 ) = σ/ϵ0 , and at point B there will be a field upwards
due to the bottom plane with magnitude σ/(2ϵ0 )”.
• Student 3 draws the sketch in Fig. 2 [bottom] (on the next page) and says “Do we just add the
electric fields together? I’m going to add up the field from each plane at both points. So at A I
get a field upward of magnitude σ/ϵ0 + σ/(2ϵ0 ) = 3σ/(2ϵ0 ), and at point B I get the same 3σ/(2ϵ0 )
but now pointing downward”.
Evaluate each explanation and decide whether it is correct or not. If an explanation contains
incorrect reasoning, explain how the argument is flawed.
Figure 2: Students’ sketches.
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Problem 3: Show your steps for Problem 8: Part A of Homework Part 1
After you have finished Problem 8 of Part 1 (on MasteringPhysics) The Trajectory of a Charge in an
Electric Field, show here the steps of your work for Part A (not Part B): make a clear sketch of the
situation including an x-y plane, list what are the given and wanted quantities, show the steps and
reasoning you took to get to the final answer.
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