Three concentric metallic spherical shells of radii , , and are given charges , , and , respectively. It is found that the surface charge densities on the outer surfaces of the shells are equal. Then, the ratio of the charges given to the shells, is

Question

Three concentric metallic spherical shells of radii $R$ , $2 R$ , and $3 R$ are given charges $Q_{1}$ , $Q_{2}$ , and $Q_{3}$ , respectively. It is found that the surface charge densities on the outer surfaces of the shells are equal. Then, the ratio of the charges given to the shells, $Q_{1} : Q_{2} : Q_{3},$ is [2009]

Smart Achievers · Accepted Answer

(2)

The charges on the surfaces of the metallic spheres are shown in the figure. Charge density $σ = \frac{charge}{area}$ and as per question

$\frac{Q_{1}}{4 π R^{2}} = \frac{Q_{1} + Q_{2}}{4 π {(2 R)}^{2}} = \frac{Q_{1} + Q_{2} + Q_{3}}{4 π {(3 R)}^{2}}$

$∴$ $Q_{1} = \frac{Q_{1} + Q_{2}}{4}$ $\Rightarrow Q_{2} = 3 Q_{1}$

$\frac{Q_{1} + Q_{2} + Q_{3}}{9} = Q_{1}$

$\Rightarrow 8 Q_{1} - Q_{2} = Q_{3}$

$\Rightarrow 8 Q_{1} - 3 Q_{1} = Q_{3}$

$\Rightarrow 5 Q_{1} = Q_{3}$

$∴$ $Q_{1} : Q_{2} : Q_{3} = 1 : 3 : 5$

Solution

1 1 : 2 : 3

2 1 : 3 : 5

3 1 : 4 : 9

4 1 : 8 : 18

Want Us to Email you About Special Offers & Updates?