Question

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

• A. $ 2 \times 10^{-2} C $
• B. $ 3 \times 10^{-2} C $
• C. $ 4 \times 10^{-2} C $
• D. $ 1 \times 10^{-2} C $

Answer 1

The Correct Option is B

To determine the final charge on the bigger sphere when two metallic spheres are connected by a conducting wire, we need to review the principle of charge redistribution. When two conducting spheres are connected, the charge is distributed according to their radii such that they reach the same potential. The formula for this process involves charge and capacitance.

The potential V on each sphere is given by:

V = \frac{Q}{C}, where C is the capacitance of the sphere.

The capacitance of a sphere is given by C = 4\pi \varepsilon_0 R, where R is the radius.

Given: Radii of spheres are 1 cm and 3 cm.

Initially given charges are:

• Charge on the smaller sphere, Q_1 = -1 \times 10^{-2} \, \text{C}
• Charge on the bigger sphere, Q_2 = 5 \times 10^{-2} \, \text{C}

The total initial charge Q_{\text{total}} is:

Q_{\text{total}} = Q_1 + Q_2 = (-1 \times 10^{-2} \, \text{C}) + (5 \times 10^{-2} \, \text{C}) = 4 \times 10^{-2} \, \text{C}

When connected, the charge will distribute between them according to their capacitance:

Since V (potential) is the same on both:

\frac{Q_1'}{R_1} = \frac{Q_2'}{R_2}

Let Q_1' and Q_2' be the charges on spheres with radii R_1 = 1 \, \text{cm} and R_2 = 3 \, \text{cm}, respectively:

Due to charge conservation:

Q_1' + Q_2' = 4 \times 10^{-2} \, \text{C}

Substituting the potential equality:

Q_2' = 3Q_1'

Plug back into the conservation equation:

Q_1' + 3Q_1' = 4 \times 10^{-2} \, \text{C}

4Q_1' = 4 \times 10^{-2} \, \text{C}

Q_1' = 1 \times 10^{-2} \, \text{C}

Thus, Q_2'

Q_2' = 3 \times 1 \times 10^{-2} \, \text{C} = 3 \times 10^{-2} \, \text{C}

Therefore, the final charge on the bigger sphere is 3 \times 10^{-2} \, \text{C}.