Question

Assertion (A) : The work done, in taking a unit charge around a closed loop of an electric circuit involving cells and resistors in the loop, is zero.
Reason (R) : The potential at a point depends on the location of the point in the loop. After completing one round, the charge comes back to the point of start.

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Both Assertion (A) and Reason (R) are false.

Hint:

Kirchhoff's Voltage Law is simply the conservation of energy applied to conservative electric fields: $\oint \vec{E} \cdot d\vec{l} = 0$, meaning work done in any closed path is always zero.\

Answer 1

The Correct Option is A

Concept:
• In electrostatics and steady-state circuit analysis, the electrostatic field is a conservative vector field.
• The line integral of the electrostatic field along any closed loop is identically zero: $\oint \vec{E} \cdot d\vec{l} = 0$.
• Kirchhoff's Voltage Law (Loop Rule) is a direct manifestation of the principle of conservation of energy in an electrical circuit.

Step 1: Analyze the Assertion (A)
Kirchhoff's loop rule states that the algebraic sum of changes in electric potential around any closed circuit loop containing resistors and voltage sources is zero:
\[ \sum \Delta V = 0 \]
The work done in moving a test charge $q$ between two points with potential difference $\Delta V$ is given by $W = q\Delta V$.
For a unit charge ($q = 1\text{ C}$) completing a full traversal around a closed loop, the total change in potential is $\Delta V_{\text{net}} = 0$.
Therefore, the total work done $W = 1 \times 0 = 0$.
Thus, Assertion (A) is true.

Step 2: Analyze the Reason (R)
Electric potential $V(\vec{r})$ is a single-valued scalar function of space (state function) determined entirely by the location of the point in the circuit.
When a charge traverses a complete closed path and returns to its initial starting location, the final electric potential is identical to the initial electric potential ($V_f = V_i$).
The net potential change is $\Delta V = V_f - V_i = 0$, leading to zero net work done.
Therefore, Reason (R) is true and accurately explains why the net work done around any closed loop is zero.

Step 3: Conclusion
Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A). Hence, option (A) is the correct choice.