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PHY 217: Electricity and Magnetism I
Prof. S. Teitel stte@pas.rochester.edu ---- Fall 2016

## Problem Set 3

Due Friday, September 23, by 4pm in the homework locker

• Problem 1 [10 points]

Griffiths Problem 1.21

• Problem 2 [10 points]

Although the line integral of a gradient is independent of the integration path, the same is not true for an arbitrary vector function. For the function v = y2 + 2x(y+1), compute the integral ∫ab v·d along path (1), and then along path (2), as shown in the figure below.

Now compute ∮ v·d along the closed loop that consists of going out from point a along parth (1) and returning back to point a along path (2). What do you get?

• Problem 3 [10 points]

This problem is meant to help you understand the notion of the flux of a vector field through a surface.

a) Suppose water flows with speed v down a rectangular pipe of crossectional lengths s and ℓ, as shown in the figure below. What is the rate Φ = volume per unit time, at which water accumulates in the bucket?

b) Now we slice the end of the pipe off at some angle θ , as in the figure below. This doesn't change Φ of course. Express your formula for Φ in terms of the dimensions s and ℓ' of the end, and the angle of tilt θ.

c) Write teh formulat for Φ more compactly, in terms of the vector area a of the end and the vector velocity v, as shown in the figure below.

d) What if the velocity varied (in magnitude and/or direction) from point to point, and the end of the pipe were cut in some arbitary way, as in the figure below. How would you then compute Φ in this case?

• Problem 4 [10 points]

Griffiths Problem 1.54

• Problem 5 [10 points]

Griffiths Problem 2.10