Question 522: Electrostatics & Capacitance

Question

Two spherical conductors of capacities $$3 \mu \mathrm{F}$$ and $$2 \mu \mathrm{F}$$ are charged to same potential having radii $$3 \mathrm{~cm}$$ and $$2 \mathrm{~cm}$$ respectively. If '$$\sigma_1$$' and '$$\sigma_2$$' represent surface density of charge on respective conductors then $$\frac{\sigma_1}{\sigma_2}$$ is

Accepted Answer

We know, $$\mathrm{C}=\frac{\mathrm{Q}}{\mathrm{V}}$$
As both the charged spheres are at the same potential, the charge on both spheres is
$$\begin{aligned}
& \mathrm{Q}_1=\mathrm{C}_1 \mathrm{~V} \
& \mathrm{Q}_2=\mathrm{C}_2 \mathrm{~V}
\end{aligned}$$
The charge densities of both spheres are
$$\sigma_1=\frac{\mathrm{Q}_1}{\mathrm{~A}_1}=\frac{\mathrm{C}_1 \mathrm{~V}}{4 \pi \mathrm{r}_1^2}$$
Similiarly,
$$\sigma_2=\frac{\mathrm{Q}_2}{\mathrm{~A}_2}=\frac{\mathrm{C}_2 \mathrm{~V}}{4 \pi \mathrm{r}_2^2}$$
Taking the ratios,
$$\begin{aligned}
\frac{\sigma_2}{\sigma_1} & =\frac{C_2 r_1^2}{C_1 r_2^2} \
\frac{\sigma_2}{\sigma_1} & =\frac{2 	imes 10^{-6} 	imes(0.03)^2}{3 	imes 10^{-6} 	imes(0.02)^2}=\frac{3}{2} \
	herefore \quad \frac{\sigma_1}{\sigma_2} & =\frac{2}{3}
\end{aligned}$$

Answer

$$\frac{1}{3}$$

Answer

$$\frac{1}{2}$$

Answer

$$\frac{2}{3}$$

Answer

$$\frac{3}{4}$$

Electrostatics & Capacitance

Two spherical conductors of capacities $3 \mu \mathrm{F} and 2 \mu \mathrm{F} are charged to same potential having radii 3 \mathrm{~cm} and 2 \mathrm{~cm} respectively. If '\sigma_1' and '\sigma_2' represent surface density of charge on respective conductors then \frac{\sigma_1}{\sigma_2}$ is

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