With the help potentiometer, we can determine the value of emf of a given cell. The sensitivity of the potentiometer is
(A) directly proportional to the length of the potentiometer wire
(B) directly proportional to the potential gradient of the wire
(C) inversely proportional to the potential gradient of the wire
(D) inversely proportional to the length of the potentiometer wire

Options

A. B and D only
B. A and C only
C. A only
D. C only

Answer

To determine the sensitivity of a potentiometer, we need to understand the relationship between the sensitivity, the potential gradient, and the length of the potentiometer wire.

### Step-by-step Solution:

1. **Understanding Sensitivity**:
The sensitivity of a potentiometer is defined as the ability to detect small changes in voltage (or EMF). It can be mathematically expressed in terms of the potential difference (V) across the potentiometer wire and its length (L).

2. **Potential Gradient**:
The potential gradient (k) of the potentiometer wire is defined as the potential difference per unit length of the wire. It can be expressed as:
\[ k = \frac{V}{L} \]
where V is the total potential difference across the wire and L is the length of the wire.

3. **Relationship Between Sensitivity and Potential Gradient**:
The sensitivity (S) of the potentiometer can be expressed as:
\[ S \propto \frac{1}{k} \]
Substituting the expression for potential gradient, we get:
\[ S \propto \frac{L}{V} \]
This indicates that the sensitivity is inversely proportional to the potential gradient.

4. **Conclusion**:
From the above relationship, we can conclude that:
- The sensitivity of the potentiometer is inversely proportional to the potential gradient of the wire.
- Therefore, as the potential gradient increases (for a fixed length), the sensitivity decreases.

### Final Answer:
The correct option is **(C) inversely proportional to the potential gradient of the wire**.