Electric Field Calculation for Non-uniformly Charged Sphere

[ambuj123]

I am having trouble with this problem
There is a sphere haveing nonuniform volume charge desnity p=ar where a is constant and r is radial distance from centre of sphere.
Radius of sphere is R
we have to find electric field at centre of sphere.
Thanx

 

[HallsofIvy]

Okay, what have you done?

The standard way of determining the electric field due to a distribution of charge is set up the expression for the force due to an infinitesmal volume of the charge and then integrate of the body.

In this case, while the distribution of charge is not uniform it is spherically symmetric and since you are asked for the field strength at the center of the sphere, you don't really need to do the integration! Imagine two infinitesmal volume on opposite sides of the sphere and equally distant from the center. What can you say about the total force on a test charge at the center from those two volumes?

 

[ambuj123]

I understood a bit of your suggestion but i am unable to put it to use to solve my problem
i know for a sphere with uniform charge denisity electric field is zero at centre but for non uniform it beats me.
i first thought about doing it by dividing sphere into think spherical shells then integrating but then i remembered that you cannot integrate as such as electric field due to thin spherical shell at centre is zero

can someone please help me.