Question 183: Electrostats & Capacitance

Question

The nuclear charge (Ze) is non-uniformly distributed within a nucleus of radius R. The charge density $$\rho(r)$$ [charge per unit volume] is dependent only on the radical distance r from the centre of the nucleus as shown in figure. The electric field is only along the radial direction.

Accepted Answer

For a = 0, the graph is as shown, the equation for the graph line is
The charge in the dotted element shown in figure is
$$dq = \sigma  	imes 4\pi {r^2}dr$$
$$	herefore$$ $$dq = \left( {d - {d \over R}r} ight)4\pi {r^2}dr$$
$$ \Rightarrow Ze = \int_0^R {4\pi d{r^2}dr - \int_0^R {{{4\pi d} \over R}{r^3}dr} } $$
$$Ze = 4\pi d{{{R^3}} \over 3} - {{4\pi d} \over R}{{{R^4}} \over 4}$$
$$	herefore$$ $${{Ze} \over {4\pi d{R^3}}} = {1 \over 3} - {1 \over 4} = {1 \over {12}}$$
$$	herefore$$ $$d = {{3Ze} \over {\pi {R^3}}}$$

Answer

$${{3Ze} \over {4\pi {R^3}}}$$

Answer

$${{3Ze} \over {\pi {R^3}}}$$

Answer

$${{4Ze} \over {3\pi {R^3}}}$$

Answer

$${{Ze} \over {3\pi {R^3}}}$$

Electrostats & Capacitance

The nuclear charge (Ze) is non-uniformly distributed within a nucleus of radius R. The charge density $\rho(r)$ [charge per unit volume] is dependent only on the radical distance r from the centre of the nucleus as shown in figure. The electric field is only along the radial direction.

Choose the correct answer: