Problems solved about Electrostatics (Purcell Ch. 1)

List of problems to solve in Purcell Chapter 1.

• Problem 1.03 - solve it again to understand why the integral diverges

• Problem 1.08: Oscillating in a ring

• Problem 1.12: Field from a hemispherical shell

  Really good problem on how to do non-trivial integral over a hemisphere. Use angles to express and _Mathematica_to evaluate. Interestingly, you can show that the field just above and below the hemisphere differ by $\sigma /2{ϵ}_0$ )) 8.5/10

• Problem 1.14: Hole in a plane

  Problem that combines E field integration, oscilation, and energy conservation. Manageable. 6/10.

• Problem 1.15: Flux through a circle

  Verifying Gauss’s law by calculating flux in different ways. Similar set-up as in Problem 1.12 with the integral through million thin rings. Result is actually interesting: the flux through a circle can be expressed in terms of the angle only!

• Problem 1.16: Gauss’s law and two point charges

  The problem is about using Taylor expansion to derive some fields, then verifying Gauss’s law to an approximation of negligible field change. 6/10

• Problem 1.19: Sheet on a sphere

  Similarly, to 1.16 key point was using Taylor’s expansion for $\frac{1}{1+ϵ}$ to find a field. 6/10

• Problem 1.21: Field in the end face

  Good thought-provoking problem! The problem relies on imagining an infinite cylinder as a superposition of two half-inifinite cylinders. BTW, What is the field in an infinite hollow cylinder?

• Problem 1.23: Field near a stick

  My mistake was in thinking about the approximation of $E_{||}$ near the stick. Turns out just treat $r$ (distance to P) as $l$ distance to the point below it. Without any angles and heights. Then, the problem is just finding the right Integration limits.

• Problem 1.24: Field in a half-space

  The problem uses an interesting setup to calculate the required work : building up a cylinder layer by layer with thin infinite shells. Requires some thinking to set up two integrals. 7.5/10

• Problem 1.25: Two equal fields

  Did not understand the way they slice the sphere into small rings: the part with $d\theta$ and radius of ring being $l\sin (\theta )$.

• Problem 1.26: Stable equilibrium in electron jelly

  Again, thought experiment type of thing. It is simple to show that equilibrium is stable in radial displacement, but tricky for transverse (involves sin factor). Additionally, compare the jelly to the case where the sphere was replaced by point charge (still in equilibrium). Is it stable? 6/10

• Problem 1.29: Pulling two sheets apart

  I somehow missed that $\text{E}$ outside two sheets is zero!!! This can be shown by a superposition of two parallel sheets. Then recall that force on a sheet of charge is an average of two side fields. 7/10

• Problem 1.31: Decreasing energy

  A good problem on reasoning. 8/10

• Problem 1.33: Deriving the energy density

  The problem was in expressing the dot product of two fields in terms of the polar coordinates: place one on the origin (automatically everything is in $\theta ,r$ for it) and use the law of cosines and some geometric tricks for the second to express its distance and angle with $\theta ,r$. 8/10

• Problem 1.77: Electron jelly

  Easy but an interesting problem! (7/10)