Physics 272: Advanced Honors Physics II

Spring 2018

HW2 due in class Monday, January 29, 2018 [DRAFT: more notes and answers to be added]

Reading: Purcell 1.7-1.11 and 1.13

Note: the solution for each problem should be written on a separate page (sorry, trees!) to facilitate the grading.

IN ADDITION, YOU

I have put the answers at the bottom of this page. You can use them to check your own answers -- if they don't agree, you know you need to go back and look for at least one mistake.

1. *Purcell 1.52: Equilateral triangle

The possible tediousness of this problem should inspire you to find ways to group the quantities together for numerical evaluation and also to use symmetry (the face that qB=qC allows you to determine the direction of the force on charge A, and to relate the force on charge B to the force on charge C)

2. **Purcell 1.54(a) Semicircle and wires

The physics in this problem is easy, but the challenge is to draw the diagram correctly, use trigonometry, and Taylor expand to get the length of segment B (you should neglect the width of the wire even though the diagram in the book shows it, so the charge on segment B will be the length of segment B times lambda).

For your convenience, the Taylor expansion of tan (theta+d theta) around theta is tan (theta+d theta) = tan (theta) + d theta /(cos theta)

3. **Purcell 1.55 Field from a finite rod

Use symmetry to determine when a component of the E field is zero (note that point B is under the midpoint of the rod).

More trigonometry for setting up the integral for the field at point B.

4. **Purcell 1.63 Sphere and cones

Compute the potential energy of the charge -q and use conservation of energy. For (b), you have to compute the potential energy of the charge -q at the point of a cone. If you set up the integral right it is very easy (use the coordinate along the cone that varies from 0 to L as the integration variable).

5. *Purcell 1.56 Flux through a cube

This problem uses Gauss law and symmetry. It's a standard problem that all physics students need to know how to solve. For (b), there are two ways. One way is to use their hint about the sphere. A cooler way is to consider the flux through a bigger cube of side 2d centered on the charge.

6. **Purcell 1.69: Carved out sphere

Use superposition!

7. **Purcell 1.76 Hydrogen atom (integrating exponentials)

To evaluate the integral, you could use integration by parts OR you could use a nice trick, based on the fact that d(int exp(- a x) dx)/da = - int x exp(- a x) dx and you know (or should know) how to do int exp(- a x) dx. To get int x

8. **Purcell 1.77 Electron jelly (E field of uniform sphere)

9. **Purcell 1.72 Plane and slab (Application of Gauss' law)

OPTIONAL 10. **Purcell 1.71 Intersecting sheets

This is an absolutely adorable application of Gauss' law. Use what you know about the field of a sheet (that it's constant on each side of the sheet) and symmetry of the hexagonal arrangement.

0. NEVER write only the answer. You need to show where it came from and why it is right.

1. Check that you have drawn at least one diagram and labelled it clearly. This might involve copying the diagram from the problem statement and adding extra labels, or making a diagram from scratch. Make it pretty large to leave room for labels and make it readable. Quantities from the statement of the problem should appear in the diagram.

2. Check that the meanings of any symbols you have introduced yourself are clear. First, if the symbol is a label in the diagram, make sure the diagram makes its meaning clear. If not, or if it is not in the diagram, write a short phrase to explain what the symbol represents.

3. Check that all vector quantities are written so that their vector character is clear. In typesetting, boldface indicates a vector quantity. In a diagram, if you draw an arrow to indicate the direction and write the magnitude next to the arrow (this is how we draw force diagrams). In handwriting, put an arrow over the quantity to show it is a vector. Velocity is a vector, so write v with an arrow over it. The x component of the velocity is not a vector, so do not put an arrow over it. In a sum, you cannot add a vector to a number. In an equation, you cannot have a vector on one side and a number on the other side.

4. Check that each statement clearly follows from the previous statement. (This is in the same spirit as Checklist Item #0 - you can't state things without showing where they came from).

5. Check that you have stated the answer at the end of the solution. The answer should be an equation in the form

desired quantity = expression in terms of given quantities

You might be tempted to save writing by just writing the right hand side, but it is best practice to write the whole thing (this acts as a final check that your expression is indeed for the quantity the problem asked for).

To make it clear that it is the answer, PUT A BOX AROUND IT.

1. a) FA = 2.34 N, FB = FC = 1.96 N (really should be only 1 sig fig); b) 6.75 x 10

2. proof

3. EA = 1.85 x 10

4. a) v = Sqrt[(2R sigma q)/(epsilon0 m)], b) v = Sqrt[(2R sigma q)/(epsilon0 m)]

5. a) q/(6 epsilon0), b) q/(24 epsilon0)

6. EA = a rho / (6 epsilon0) upward, EB = 17 a rho/(54 epsilon0) downward

7. 0.323 e, 3.5 x 10

8. r = a/2

9. (sigma - rho d + 2 rho x)/(2 epsilon0) for 0<x<d, (sigma + rho d)/(2 epsilon0) for x>d, -(sigma + rho d)/(2 epsilon0) for x<d

10. Check your answer by adding up the fields from each of the three sheets in each of the six regions