Calculate the total charge on a sphere

[nosmas]

Homework Statement

Using integration calculate the total charge on the sphere

radius R
Volume charge density at the surface of the sphere p0

p = p0r/R



I started with dq = 4*pi*r^2*dr*(p0r/R)

but i am not sure how to integrate (in terms of what variable I would assume r=0 to r=R) but i am not sure i set up the question right

 

[SammyS]

Homework Statement

Using integration calculate the total charge on the sphere

radius R
Volume charge density at the surface of the sphere p0

p = p0r/R

I started with dq = 4*pi*r^2*dr*(p0r/R)

but i am not sure how to integrate (in terms of what variable I would assume r=0 to r=R) but i am not sure i set up the question right

I presume that you mean ρ = ρ₀(r/R) is the volume charge density for a sphere of radius, R, where ρ₀ is the volume charge density at the surface of the sphere.

The volume element is dV = 4πr²dr.

So that dq = 4πr²(ρ)dr = 4πr²(ρ₀/R)r dr .

4, π, ρ₀, and R are all constants.

Integrate that over the entire sphere. → r goes from 0 to R .

 

[nosmas]

That makes sense so if I was asked to find the E field using gauss's law for r<=R would I just use E=q/(area*epsilon) but how would I know what q enclosed is?

 

[SammyS]

That makes sense so if I was asked to find the E field using gauss's law for r<=R would I just use E=q/(area*epsilon) but how would I know what q enclosed is?

To find the charge enclosed in a sphere of radius, r, integrate from 0 to r .

 

[asteriatic]

I would like to clarify that the total charge on a sphere can only be calculated if we know the total volume charge density, not just the density at the surface. The volume charge density, p, is defined as the charge per unit volume, therefore, we need to know the distribution of charge within the sphere in order to calculate the total charge.

Assuming we have the total volume charge density, p, we can integrate it over the entire volume of the sphere to find the total charge. The integral would be set up as follows:

Q = ∫∫∫p dV

Where Q is the total charge, p is the volume charge density, and dV is the infinitesimal volume element. The limits of integration would be from 0 to R for r, 0 to 2π for φ, and 0 to π for θ, as we are integrating over a spherical coordinate system.

However, since the volume charge density is given as p0 at the surface of the sphere, we can use this information to calculate the total charge by integrating over the surface of the sphere. The integral would be set up as follows:

Q = ∫∫p0 dA

Where Q is the total charge, p0 is the surface charge density, and dA is the infinitesimal area element on the surface of the sphere. The limits of integration would be from 0 to 2π for φ and 0 to π for θ, as we are integrating over a spherical coordinate system.

I hope this clarifies the approach to calculating the total charge on a sphere using integration.