Question

Two metal spheres of radii $r_1$ and $r_2$ ($> r_1$) having charges $q_1$ and $q_2$ respectively kept in air, are brought in contact. Which of the following statements is not correct ?

• A. The total charge of the two spheres is conserved.
• B. Both spheres attain the same potential.
• C. The final potential of the system equals $\frac{1}{4\pi\epsilon_0} \frac{(q_1 + q_2)}{(r_1 + r_2)}$
• D. The final potential of the system equals $\frac{1}{4\pi\epsilon_0} \frac{(q_1 + q_2) (r_1 + r_2)}{r_1 r_2}$

Hint:

When tackling "Which is NOT correct" questions, always systematically verify the basic physical conservation laws (like charge and energy) first, as they quickly eliminate the definitely true statements.

Answer 1

The Correct Option is D

Concept:
• When two solid conductive metal spheres are physically brought into electrical contact (or seamlessly connected via a conductive wire), charge will immediately flow aggressively between them.

• This rapid redistribution of charge firmly obeys the universal law of conservation of charge, completely ensuring no net charge is ever lost or created.

• The transient flow of electrons strictly ceases only when the entire connected system achieves perfect electrostatic equilibrium, meaning both spheres securely attain the exact same common electrostatic potential.

• The total capacitance of this newly combined system determines the final numerical value of this shared common potential.

Step 1: Analyze Option (A) - Charge Conservation
The foundational principle of electrostatics dictates that for any isolated system, the net total electrical charge remains forever constant.
When the two spheres make physical contact, they effectively merge into a single isolated conductive system.
Thus, the total initial charge perfectly equals the total final charge: $Q_{total} = q_1 + q_2$.
Therefore, statement (A) is absolutely correct.

Step 2: Analyze Option (B) - Common Potential
By fundamental definition, a conductive material in stable equilibrium constitutes a complete equipotential volume.
When the two distinct spheres connect, they form one unified conductor. Charge will dynamically shift until the potential difference strictly becomes zero.
Hence, both spheres must inevitably reach the exact same common potential $V$.
Therefore, statement (B) is absolutely correct.

Step 3: Analyze Option (C) - Derivation of Common Potential
We assume the spheres are sufficiently far apart such that their mutual electrostatic influence is negligible, acting purely as parallel capacitors.
The intrinsic capacitance of an isolated spherical conductor of radius $r$ is formally given by $C = 4\pi\epsilon_0 r$.
The total capacitance of the combined connected system is the sum of their individual capacitances:
\[ C_{total} = C_1 + C_2 = 4\pi\epsilon_0 r_1 + 4\pi\epsilon_0 r_2 = 4\pi\epsilon_0 (r_1 + r_2) \]
The final common potential $V$ is mathematically derived by dividing the conserved total charge by the total system capacitance:
\[ V = \frac{Q_{total}}{C_{total}} = \frac{q_1 + q_2}{4\pi\epsilon_0 (r_1 + r_2)} \]
Therefore, statement (C) is mathematically correct.

Step 4: Analyze Option (D) - Dimensional Analysis
Let's rigorously examine the algebraic expression presented in statement (D):
\[ V' = \frac{1}{4\pi\epsilon_0} \frac{(q_1 + q_2) (r_1 + r_2)}{r_1 r_2} \]
In the numerator, we have physical dimensions of $[Charge] \times [Length]$.
In the denominator, we have $[Length]^2$.
This simplifies dimensionally to $[Charge] / [Length]$, which is standard for potential. Let us check the full expression.
The correct potential derived in Step 3 is $\frac{1}{4\pi\epsilon_0} \frac{Q}{R}$, which represents $[Charge] / [Length]$.
The expression in (D) evaluates to $\frac{1}{4\pi\epsilon_0} \frac{Q \cdot L}{L^2} = \frac{1}{4\pi\epsilon_0} \frac{Q}{L}$. Dimensionally it is not obviously wrong, but mathematically it is completely incorrect based on our rigorous derivation in Step 3.
Since there can be only one correct specific mathematical expression for the final common potential, and (C) is the proven correct one, (D) must undeniably be the incorrect statement.

Step 5: Conclusion
The question explicitly asks to identify the statement that is NOT correct. Based on our detailed derivation, statement (D) provides an entirely flawed formula for the final system potential.