Question

The electric field within the nucleus is generally observed to be linearly dependent on r. This implies:

• A. $a=0$
• B. $a=\frac{R}{2}$
• C. $a=R$
• D. $a=\frac{2R}{3}$

Answer 1

The correct option is C $a=R$

The electric field at distance r is given by

$E=k\frac{q_r}{r^2}$

where $q_r$ is charge enclosed in radius r and $k=\frac{1}{4\pi {ϵ}_o}$.

Note that the electric field due to outside charges will get cancelled.

$q_r$ is given by

$q_r={\int }_0^{r}4\pi x^2\rho \left(x\right)dx$

now for $E\propto r$ we must have $q_r\propto r^3$

thus from the above equation of $q_r$ we must have $\rho \left(x\right)$ independent of $x$.

thus the density must be constant throughout $r$. Or,

$\rho \left(r\right)=Constant$

Hence for the above figure $a$ must be equal to $R$

$$a=R$$