Two metallic spheres of radii 1 cm and 3 cm are given charges of −1 × $10^{−2}$C and 5 × $10^{−2}$C, respectively. If these are connected by a conducting wire, the final charge on the bigger sphere is ?

Two metallic spheres of radii 1 cm and 3 cm  are given charges of −1 × 10−2C and 5 × 10−2C, respectively. If these are connected by a conducting wire, the final  charge on the bigger sphere is ? 

(a) \(2 \times 10^{-2} \, \text{C}\)

(b) \(3 \times 10^{-2} \, \text{C}\)

(c) \(4 \times 10^{-2} \, \text{C}\)

(d) \(1 \times 10^{-2} \, \text{C}\)

Answer : (b) \(3 \times 10^{-2} \, \text{C}\)

Concept: Redistribution of charges. Charges flow until the voltage is equalized. Total Charge is conserved. (Learn the water tank analogy. Water flows from higher level to lower level until the levels are equal. Amount of water before and after pulling the plugs are same.)

Total charges conserved: 

\( q_1 + q_2 = q_1' + q_2' \)

\( -1 \times 10^{-2} + 5 \times 10^{-2} = q_1' + q_2' \)

Voltages are equal: 

\(\frac{kq_1'}{r_1}=\frac{kq_2'}{r_2}\)

\( \frac{k q_1'}{1\, \text{cm}} = \frac{k q_2'}{3\, \text{cm}} \)

\( 3q_1' = q_2' \)

\( 4 \times 10^{-2} = q_1' + q_2' \)

\(4 \times 10^{-2} = q_1' + 3q_1' \)

\(4 \times 10^{-2}= 4q_1' \)

\(q_1' = 1 \times 10^{-2} \, \text{C} \)

\( q_2' = 3 \times 10^{-2} \, \text{C} \)