Minimum speed of the bullet to penetrate a sphere

[Tanya Sharma]

Homework Statement

A bullet of mass m and charge q is fired towards a solid uniformly charged sphere of radius R and total charge + q. If it strikes the surface of sphere with speed u, find the minimum speed u so that it can penetrate through the sphere. (Neglect all resistance forces or friction acting on bullet except electrostatic forces)

Homework Equations

The Attempt at a Solution

I think if the bullet can just reach the center ,then it can penetrate through the sphere because the electric field inside a sphere is radially outwards .As it reaches the center and moves just a little bit outwards radially ,the electric field will push the bullet outwards .

The problem can be approached either by conservation of energy or by work kinetic energy theorem.

1) By conservation of energy

kq²/R+(1/2)mu² = (3/2)kq²/R

2) By work energy theorem

$\int_{R}^{0}\frac{kq^{2}r}{R^3} \hat{r} \cdot dr\hat{r} = \frac{1}{2}mu^2$

Have I approached the problem correctly or is there something more in it ?

 

[TSny]

Both approaches look correct to me.

But in the work energy equation, there might be a sign problem. I think your integral on the left is correct for the work done by the electric force on the bullet as it goes from the surface to the center. But, is the sign correct on the right for the change in KE?

 

[Tanya Sharma]

Yes.. you are right.There should be a minus sign in the LHS .

Thanks TSny...

Would you mind having a look at the other thread :"flux through a circular ring " .

 

[TSny]

Yes.. you are right.There should be a minus sign in the LHS .

Thanks TSny...

Would you mind having a look at the other thread :"flux through a circular ring " .

I'll take a look tomorrow. It's that time again. :zzz:

 

[Tanya Sharma]

Okay