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**1. Homework Statement**

Consider a sphere of radius ##R##

*,*with a charge density ##\rho(r)=\frac{\alpha} {r^2},## with ##\alpha## a constant. Use Gauss' law to calculate the electric field outside the sphere at a distance ##r## from the sphere's centre (ie. ##(r > R)## and inside the sphere (ie. ##(r > R).## Plot the magnitude of the electric field for both inside and outside the sphere.

**2. Homework Equations**

Gauss' Law:

##\int\vec E.d\vec A=\frac {q_{encl}} {\epsilon_0}.##

**3. The Attempt at a Solution**

For the first part

*outside*the sphere ##(r > R),##

I calculated the enclosed charge as follows:

##\frac {\alpha} {R^2}\times \frac {4} {3} \pi R^3.##

Now using a Gaussian surface (sphere) of radius ##r## enclosing this charge,

## \Rightarrow E = \frac{R\alpha} {3\epsilon_0 r^2}. ##

For the second part

*inside*the sphere ##(r < R),##

The enclosed charge now is:

##\frac {\alpha} {r^2}\times \frac {4} {3} \pi r^3.##

Using another Gaussian surface (sphere) inside, with radius ##r##,

## \Rightarrow E = \frac{\alpha} {3\epsilon_0 r}. ##

When I draw the graph, the magnitude of the E-field continues to decay with distance, but increasingly outside the sphere. Is this correct?