Physics 15b Assignment #2

Read sections 2.1-2.12 of Purcell by Monday February 14.

Q&A questions to be answered on the Physics 15b website before 11pm on Monday,February 14:

2QA-1. If φ(r) the potential due to two point charges,Q, fixed at(±a, 0, 0), find∫

dS φ(r) (2QA-1.1)

over a sphere with radiusb < a centered at the origin.

A : 0

B : 2πQb2/a

C : 4πQb2/a

D: 8πQb2/a

E : None of the above.

2QA-2. A flat disk with radiusr with uniform surface charge densityσ is sitting in thex-y planecentered at the origin. The potential at(r, 0, 0) is equal to the potential at a point(0, 0, b). What isb?

A :πr

4

B :(π2 − 4)r

4π

C : r

D: None of the above.

In addition, there are some survey questions and feedback questions.

Problems due at the beginning of class on Thursday, February 17—

2-1. Do problem 2.23 in Purcell:

By means of a van de Graaff generator, protons are accelerated through a potentialdifference of5 × 106 volts. The proton beam then passes through a thin silver foil.The atomic number of silver is 47, and you may assume that a silver nucleus is somassive compared with the proton that its motion may be neglected. What is theclosest possible distance of approach, of any proton, to a silver nucleus? What will bethe strength of the electric field acting on the proton at that position?

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Hint: This one is straightforward. Express your answers in SI units.

2-2. Consider the vector field defined by

~F (r) = (0, x, 0) (2-2.1)

Note that this is just math — you should not assume that this vector field is an electric field. Findthe line integral ∫

d~̀ · ~F (r) (2-2.2)

along a path from~r0 = (d, 0, 0) to ~r1 = (0, d, 0)

a. where the path is described by two straight segments, first from(d, 0, 0) to (0, 0, 0) then from(0, 0, 0) to (0, d, 0).

b. where the path is defined by

~̀(θ) = (d cos θ, d sin θ, 0) (2-2.3)

for θ = 0 to π/2.

2-3. Find the electric field and the charge distribution that go with the following potential:

φ = κ r2 for r2 < a2

φ = −κ a2 +2κ a3

rfor a2 ≤ r2

(2-3.1)

wherer =√

~r · ~r andκ is a constant with units of charge per unit volume.

2-4. Find the charge distribution that produces a potential of the following form

φ(r) = k r−3/2 (2-4.1)

Hint: There is a serious mathematical subtlety in the problem, so you should think carefullyabout the physics. In particular, you may wish to think carefully about why the electric field hasthe direction that it does.

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