Question

729 small identical spheres each charged to an electric potential 3V combine to form a bigger sphere. The electric potential of the bigger sphere is

• A. 9 V
• B. 729 V
• C. 81 V
• D. 243 V

Hint:

When N identical spheres of radius $r$ combine to form a single large sphere: - The new radius is $R = N^{1/3}r$. - The new charge is $Q = Nq$. - The new potential is $V_B = N^{2/3}V_s$. This power law relationship is a useful shortcut for such problems.

Answer 1

The Correct Option is D

Step 1: Relate the charge and radius of the small and big spheres.
Let $N$ be the number of small spheres, $N=729$. Let $r$ be the radius of a small sphere and $R$ be the radius of the big sphere. The total volume remains constant when the spheres combine (assuming no loss). \[ \text{Volume of big sphere} = N \times (\text{Volume of small sphere}). \] \[ \frac{4}{3}\pi R^3 = N \left(\frac{4}{3}\pi r^3\right). \] \[ R^3 = N r^3 \implies R = N^{1/3}r. \] For $N=729$, we have $729 = 9^3 = 9 \times 9 \times 9$. So $N^{1/3}=9$. \[ R = 9r. \]

Step 2: Relate the charge of the small and big spheres.
The total charge is conserved. Let $q$ be the charge of a small sphere and $Q$ be the charge of the big sphere. \[ Q = Nq = 729q. \]

Step 3: Relate the potential to charge and radius.
The electric potential $V$ of a charged conducting sphere is $V = \frac{1}{4\pi\epsilon_0} \frac{q'}{r'}$. Potential of the small sphere: $V_s = \frac{1}{4\pi\epsilon_0} \frac{q}{r} = 3 \text{ V}$. Potential of the big sphere: $V_B = \frac{1}{4\pi\epsilon_0} \frac{Q}{R}$.

Step 4: Express the potential of the big sphere in terms of the small sphere's potential.
\[ V_B = \frac{1}{4\pi\epsilon_0} \frac{Nq}{N^{1/3}r} = N^{1 - 1/3} \left(\frac{1}{4\pi\epsilon_0} \frac{q}{r}\right). \] \[ V_B = N^{2/3} V_s. \]

Step 5: Calculate the potential of the big sphere.
Substitute $N=729$ and $V_s=3 \text{ V}$. \[ V_B = (729)^{2/3} \times 3. \] Since $729 = 9^3$, we have $(729)^{2/3} = (9^3)^{2/3} = 9^2 = 81$. \[ V_B = 81 \times 3 = 243 \text{ V}. \] \[ \boxed{V_B = 243 \text{ V}}. \]