CUET Preparation Today
Three concentric metallic spherical shells of radii R, 2R, 3R, are given charges Q1, Q2, Q3, respectively. It is found that the surface charge densities on the outer surfaces of the shells are equal. Then what is the ratio of the charges given to the shells, Q1:Q2:Q3, ? |
1:2:3 1:3:5 36:9:4 1:4:15 |
1:3:5 |
The correct answer is Option (2) → 1:3:5 Due to electrostatic induction, the total charge on the outer surface of each shell is the sum of all charges inside it:
Apply Surface Charge Density Condition ($\sigma = \frac{q}{4\pi r^2}$)Since $\sigma_1 = \sigma_2 = \sigma_3$, we set up the ratio using the squares of the radii: $\frac{Q_1}{R^2} = \frac{Q_1 + Q_2}{(2R)^2} = \frac{Q_1 + Q_2 + Q_3}{(3R)^2}$
$\frac{Q_1}{1} = \frac{Q_1 + Q_2}{4} = \frac{Q_1 + Q_2 + Q_3}{9}$
Calculate the Ratios:
For $Q_2$:
$4Q_1 = Q_1 + Q_2$
$Q_2 = 3Q_1$
For $Q_3$:
$9Q_1 = Q_1 + Q_2 + Q_3$
$9Q_1 = Q_1 + 3Q_1 + Q_3$
$Q_3 = 5Q_1$
$Q_1 : Q_2 : Q_3 = 1 : 3 : 5$ |
