Integrating dq to find that q(r) = Q(1-e^(-r/R))

[jk0921]

Homework Statement

provided with data that
dq = rho(r) *4phi*r^2*dr
rho(r) = [Q*e^(-r/R) / 4phi R *r^2)

I have to show that the charge q(r) enclosed in a sphere of radius r is q(r) = Q(1-e^(-r/R)) by using appropriate integral. how the integral should be?

Homework Equations

The Attempt at a Solution

I've tried to integrate dq = ... but I can't find the final answer that q(r) = Q(1-e^(-r/R))

 

[gabbagabbahey]

I've tried to integrate dq = ... but I can't find the final answer that q(r) = Q(1-e^(-r/R))

It is a pretty straightforward integration. Why not post what you've tried so we can see where you are going wrong?

 

[iceberg123]

never mind

 

[saunderson]

The easiest way is to integrate the charge density in a fitted coordinate system!

Cause you need

$$Q(r) = \int \limits_{\mathcal{V}} \, d^3r \, \rho(r)$$​

of a sphere, the most suitable one is the spherical coordinate system. So you need the volume element

$$d^3r = \rm{?}$$​

and perform the integration!



PS:
In the statement

$$dq = 4\pi \, r^2 \,\rho(r) \cdot dr$$​

two integrations are already perfomed, so the best way to undestand it completley is to do it like I've said above!