Motion of an electron inside the cavity of a charged sphere

[Zayan]

Homework Statement

A cavity of radius r is made inside a solid sphere. The volume charge density of the remaining sphere is p. An electron (charge e, mass m) is released inside the cavity from a point P as shown in figure. The center of sphere and center of cavity are separated by a distance a. The time after which the electron again touches the sphere is

Relevant Equations

Non conducting Sphere electric field = ρr/3ε0

I know the field I don't know whether the field will be uniform inside the cavity or not. If it is, I don't understand how or why the electron will move. I got the force(considering uniform field inside the cavity) as epr/3E0. But then again I don't understand how the electron will move. If I get the trajectory I can apply kinematic equation to get the time.

 

[haruspex]

considering uniform field inside the cavity

Why would it be?

The usual trick is to represent the charge distribution as the sum of two distributions, one for the complete sphere (no cavity) and a complementary one in the cavity.
Can you write down the two resulting forces on the electron when at $(r, \theta)$ relative to the centre of the cavity?

 

[Zayan]

Why would it be?

The usual trick is to represent the charge distribution as the sum of two distributions, one for the complete sphere (no cavity) and a complementary one in the cavity.
Can you write down the two resulting forces on the electron when at $(r, \theta)$ relative to the centre of the cavity?

But the "remaining charge distribution" is already given?

 

[haruspex]

But the "remaining charge distribution" is already given?

Yes, so? You must have misunderstood me.
You have a uniform charge density p in what remains of the sphere. We can represent that as the sum of a uniform charge of density p over the complete sphere (cavity filled in) and a uniform charge density -p over the spherical cavity.

 

[Zayan]

Yes, so? You must have misunderstood me.
You have a uniform charge density p in what remains of the sphere. We can represent that as the sum of a uniform charge of density p over the complete sphere (cavity filled in) and a uniform charge density -p over the spherical cavity.

Oh got it. I calculated the total force on both and turns out one cancels. And it is uniform inside. Thanks. But tell me one thing does the now calculated force have the radial direction(towards the bigger sphere's radius)?

 

[haruspex]

does the now calculated force have the radial direction(towards the bigger sphere's radius)?

It depends where the electron is.
I'll call the centre of the sphere O and the centre of the cavity C.
If the electron is at E, define the vector $\vec q$ as OE and the vector $\vec s$ as CE.
- what is the force on it from the (complete) sphere centred on O?
- what is the force on it due to the $-\rho$ charge in the cavity?
- what is the vector sum of those forces ?