Calculating Volume Charge Density in a Sphere of Charge

[psingh]

A sphere of radius R has total charge Q. the volume chrage density within the sphere is

p=p_0(1-r/R)
the charge density decreases linearly from p_0 at the center to zero at the edge of the sphere.

1) show that p_0= 3Q/pi*R^3

to find this i used p*volume=Q
Q/volume=p

p_0(1-r/R)=Q/volume and the volume of a sphere is 4/3pi*r^3

is this right??

 

[Astronuc]

Q = the volume integral of the charge density over the volume of the sphere.

Q = $\int_V\,\rho(r)\,dV$. One needs to find the expression for dV in spherical coordinates, and since the distribution is dependent only one r, the integral can be simplified with respect to the polar and azimuthal angles.

 

[kyle8921]

Yes, your approach is correct. To find the volume charge density, we can use the formula p = Q/V, where Q is the total charge and V is the volume of the sphere. In this case, the volume of the sphere is 4/3pi*R^3. Plugging this into the formula, we get p = Q/(4/3pi*R^3). Simplifying this, we get p = 3Q/pi*R^3. Therefore, p_0 = 3Q/pi*R^3, which is the volume charge density at the center of the sphere. This shows that the charge density decreases linearly from p_0 at the center to zero at the edge of the sphere, as mentioned in the given content.