Problems solved about Electric potential (Purcell Ch.2)

List of Problems solved in Chapter 2

• Problem 2.1: Equivalent statements

  Quick math proof (5/10)

• Problem 2.2: Combining two shells

  Also quick logic exercise on the energy of electric fields

• Problem 2.4: Center vs Corner of the cube

  Very tricky problems for me. It required a construction of a larger cube out of 8 smaller ones, and a dimensional analysis of $\varphi$ [Q/m] except $4\pi {ϵ}_0$ term.

• Problem 2.7: Shell field via direct Integration

  Good exercise that shows how Gauss’s law holds. Did it in two ways: calculating directly $\varphi ,\text{E}$

• Problem 2.8: Cavendish and Maxwell’s experiment

  So far the best problem I solved. It’s hard, but satisfying; it shows what would happen if Coulomb’s law was not perfect inverse square.

• Problem 2.11: $\vec{E}$ of a long wire $\equiv \vec{E}$ infinite wire

  At this point, it’s a usual exercise of finding a good set up for integration. My ‘mistake’ was to leave the answer in $\theta$ instead of rewritting in with $l$.

• Problem 2.13: $\varphi$ at the center of N-gon

  Made a lot of mistakes starting from reading the problem carelessly to confusing notations within a solution. One interesting point - integration of thin wedges.

• Problem 2.16: field of a two disks far away

  Interesting setup. Basically the $\text{E}$ of a thin plate capacitor outside of it can be treated as a field of dipole!

• Problem 2.19: Equipotential surface of a ring

  The part with equipotential surface is rly good because requires some visualization. Turns out that it forms torus near the ring, and a sphere far from the ring (2nd part is obvious). Another essential point for the solution is to recall that $\text{E}$ $\perp$ equipotential surface.

• Problem 2.23:

  Just using Gauss’s law

• Problem 2.24: Energy expressions

  Vector identity proof & two energy expression equality proof. The latter relies on the fact that ${\int }_S\varphi \text{E}\cdot d\text{a}=0$ if S is LARGE sphere. Recall why the integral converges to 0 as $R\to \infty$.

• Problem 2.25 & 2.29

  Good reasoning problems that revolve around essential properties of $\text{E}$. You prove that you can never trap a cation with cations, and that field lines always end up either at $\infty$ or negative charges, but not in empty space (which is a little sloppy for the case where $\text{E}=0$ between two charges ).

List of Exercises from Purcell Chapter 2:

• 2.35: Dimensional Analysis and Superposition of cubes/squares

  The same exact principle as in 2.4

• 2.38: Just calculation

• 2.39: $A{g}_{47}$

  Forgot about the energy conservation. Additionally, acceleration through potential $\varphi$ gives energy $\text{E}=q\cdot \varphi$.

• 2.43: Homework Review - potential of a Rod

  Now, it’s very simple to solve that problem …

• 2.44: Additional part of $\varphi$ of a rod

  Mistake was mathematical - didn’t factorize the answer from the first try, and when factorized missing a minus somewhere … But from physical standpoint, the problem is clear. One more point - when you have a fresh mind, differentiate that integral, pls.

• 2.46: Right triangle

  Interesting problem, especially its result. It also has some connections to multivariable calc.

• 2.47: Potential at the center of a square.

  Relies on the result of 2.46 & superposition. Relatively easy, although almost made a mistake by forgetting a factor of 2. The result is also consistent with the dimensional analysis made in the problem 2.35, 2.04

• 2.57: Four star problem

  Actually, not that hard. Involves a little bit of geometry, but the point is to understand how you integrate over the thin wedges (at the center or off the center). My main mistake was in the finding the leading terms during Taylor expansion. Note that when there is no clear $ϵ\to 0$, just introduce another variable s.t. it will go zero, and then do math (p.2 in b) ).

• 2.63: Dipoles formed by two wires

  Got beaten a little bit by this problem. Some points to remember: calculating the $\varphi$ relative to fixed points (not infinity; remember that the main formula is with respect to infinity), using right approximations for distant points, or rather finding a good set up. The last part of the problem with finding constant potential and constant $\text{E}$ is still not that clear.

• 2.64: TAYLOR EXPANSION !!!

  Used the wrong expansion. Have to redo the problem later …

• 2.65 (Solve after 2.64)

• 2.74: Oscillating $\varphi$

  This is a cool problem, but I still do not understand the part of deducing the equation of field lines by considering the ratio of different $\text{E}$ components. Have to understand 2.7.2 of the Purcell, where they do it. Also, the thing to take away - having a charged surface, the normal component at any height (if charges are only on the surface) will be proportional to the $\sigma$ below it.

• 2.78: Divergence of the curl

  The b) proof is interesting. ‘it boils down to the mathematical fact that a boundary of a boundary is zero.’ $\nabla \cdot (\nabla \times \text{A})=0\forall \text{A}.$

• 2.79: squrl$(\text{F})$

  Fast and easy problem, hint gives away too much.