Capacitance of an isolated conducting sphere | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) The capacitance of an isolated conducting sphere of radius $R_{1}$ is given by $C_{1} = 4 \pi \varepsilon_{0} R_{1}$.When this sphere is enclosed

Vedclass · Accepted Answer

$\frac{n}{n-1}$

Vedclass · Answer

(A) The capacitance of an isolated conducting sphere of radius $R_{1}$ is given by $C_{1} = 4 \pi \varepsilon_{0} R_{1}$.When this sphere is enclosed by a concentric conducting sphere of radius $R_{2}$ which is connected to the earth,the system forms a spherical capacitor. The new capacitance is $C_{2} = \frac{4 \pi \varepsilon_{0} R_{1} R_{2}}{R_{2} - R_{1}}$.According to the problem,the new capacitance is $n$ times the original capacitance: $C_{2} = n C_{1}$.Substituting the expressions: $\frac{4 \pi \varepsilon_{0} R_{1} R_{2}}{R_{2} - R_{1}} = n (4 \pi \varepsilon_{0} R_{1})$.Simplifying the equation: $\frac{R_{2}}{R_{2} - R_{1}} = n$.Dividing the numerator and denominator of the left side by $R_{1}$: $\frac{R_{2}/R_{1}}{(R_{2}/R_{1}) - 1} = n$.Let $x = \frac{R_{2}}{R_{1}}$. Then $\frac{x}{x - 1} = n$.$x = n(x - 1) \Rightarrow x = nx - n \Rightarrow n = nx - x \Rightarrow n = x(n - 1)$.Therefore,$x = \frac{R_{2}}{R_{1}} = \frac{n}{n - 1}$.

Capacitance of an isolated conducting sphere of radius $R_{1}$ becomes $n$ times when it is enclosed by a concentric conducting sphere of radius $R_{2}$ connected to earth. The ratio of their radii $\left(\frac{R_{2}}{R_{1}}\right)$ is:

Similar Questions

The capacitance of an isolated sphere of radius $r_1$ is increased by $5$ times,when it is enclosed by an earthed concentric sphere of radius $r_2$. The ratio of their radii is

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

$A$ conductor is connected to a battery of $5\, V$. It acquires a charge of $50\ \mu C$. Calculate the capacitance of the conductor in $\mu F$.

Between a solid sphere and a hollow sphere of the same radius, which has a greater capacity to hold electric charge?

Eight drops of mercury, each of the same radius and having the same charge, coalesce to form a single larger drop. The capacitance of the larger drop compared to each individual smaller drop is how many times greater?

Vedclass Test Series

Exam Paper Generator

Online Exam Module