1. ## Ring of charge

Problem 9.
The electric field vanishes inside a uniform spherical shell of charge because the shell has exactly the right geometry to make the 1/r2 field produced by opposite sides of the shell cancel according to the intuition we developed from our derivation of Gauss’s Law. It isn’t a general result for arbitrary symmetries, however.
Consider a ring of charge of radius R and linear charge density +λ. Pick a point P that is in the plane of the ring but not at the center. (a) Write an expression the field produced by the small pieces of arc subtended by opposed small angles with vertex P, along the line that bisects this small angle. (b) Does this field point towards the nearest arc of the ring or the farthest arc of the ring? (c) Suppose a charge −q is placed at the center of the ring (at equilibrium). Is this equilibrium stable4? d) Suppose the electric field dropped off like 1/r instead of 1/r2. Would you expect the electric field to vanish in the plane inside of the ring?

I dont know how to do this.

you can set up |dE|=(k dq)/R^2

you can do dq=λ dx but i dont think you can do dx= (distance) d(theta) because p is not at the center.

any help?

2. Originally Posted by BrianMath
Problem 9.
The electric field vanishes inside a uniform spherical shell of charge because the shell has exactly the right geometry to make the 1/r2 field produced by opposite sides of the shell cancel according to the intuition we developed from our derivation of Gauss’s Law. It isn’t a general result for arbitrary symmetries, however.
Consider a ring of charge of radius R and linear charge density +λ. Pick a point P that is in the plane of the ring but not at the center. (a) Write an expression the field produced by the small pieces of arc subtended by opposed small angles with vertex P, along the line that bisects this small angle. (b) Does this field point towards the nearest arc of the ring or the farthest arc of the ring? (c) Suppose a charge −q is placed at the center of the ring (at equilibrium). Is this equilibrium stable4? d) Suppose the electric field dropped off like 1/r instead of 1/r2. Would you expect the electric field to vanish in the plane inside of the ring?

I dont know how to do this.

you can set up |dE|=(k dq)/R^2

you can do dq=λ dx but i dont think you can do dx= (distance) d(theta) because p is not at the center.

yes you can ...
let $P$ be the point on the diameter, not in the center.

let $R$ = farther distance from $P$ to the ring along the diameter

let $r$ = closer distance from P to the ring along the diameter

$d\theta$ = pair of vertical angles on both sides of P bisected by the diameter

since $ds = r \, d\theta$ ...

$dq_R = \lambda ds_R = \lambda R \, d\theta$

$dq_r = \lambda ds_r = \lambda r \, d\theta$

$dE_P = \dfrac{k \lambda r}{r^2} d\theta - \dfrac{k \lambda R}{R^2} d\theta$

$dE_P = k\lambda \left(\dfrac{1}{r} - \dfrac{1}{R}\right) d\theta$

note that if the field were a function of 1/r instead of 1/r^2 , then the field anywhere in the ring would equal 0.