It's Gaussy time.

Electricity and Magnetism Level 3

A ball of radius $R$ is charged with a volume charge density which with the distance $r$ from the centre as $\rho = \alpha r^2$ where $\alpha$ is a positive constant. The magnitude of electric field intensity at a distance $r = \dfrac{R}{2}$ is given by:

$\large E = \dfrac{a \alpha R^3}{b \epsilon_0}$

$a$ and $b$ are co-prime.

What is $a+b \ ?$

The answer is 41.

1 solution

Vishwak Srinivasan
May 9, 2015

Consider a thin spherical shell as an element of arbitrary radius $x$ and thickness $dx$ .

Let $dq$ be the charge associated with that shell and is given by

$dq = \rho \alpha x^2 \times 4 \pi x^2 dx$

$\Rightarrow q = \displaystyle \int_{0}^{R/2} dq = q_{enclosed}$

By Gauss' Law,

$E \times 4 \pi (\dfrac{R}{2})^2 = \dfrac{q_{enclosed}}{\epsilon_0}$

$E = \boxed{\dfrac{\alpha R^3}{40\epsilon_0}}$

$a+b = 41$

It's Gaussy time.

The answer is 41.

1 solution

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