If the resistivity of a potentiometer wire be `rho` and area of cross-section be A , then what will be potential gradient along the wire

Options

A. IA
B. `lA//p`
C. `Ip//A`
D. `IAp`

Answer

To find the potential gradient along a potentiometer wire, we can follow these steps:

### Step 1: Understand the Concept of Potential Gradient
The potential gradient (K) is defined as the potential difference (V) per unit length (L) of the wire. Mathematically, it is expressed as:
\[ K = \frac{V}{L} \]

### Step 2: Relate Potential Difference to Current and Resistance
According to Ohm's law, the potential difference (V) across a resistor is given by:
\[ V = I \cdot R \]
where \( I \) is the current flowing through the wire and \( R \) is the resistance of the wire.

### Step 3: Express Resistance in Terms of Resistivity
The resistance (R) of a wire can be expressed in terms of its resistivity (\( \rho \)), length (L), and cross-sectional area (A) as:
\[ R = \frac{\rho L}{A} \]

### Step 4: Substitute Resistance into the Potential Difference Equation
Substituting the expression for resistance into the equation for potential difference, we get:
\[ V = I \cdot \frac{\rho L}{A} \]

### Step 5: Substitute V into the Potential Gradient Equation
Now, substituting this expression for V into the potential gradient equation:
\[ K = \frac{V}{L} = \frac{I \cdot \frac{\rho L}{A}}{L} \]

### Step 6: Simplify the Equation
The length (L) cancels out:
\[ K = \frac{I \cdot \rho}{A} \]

### Conclusion
Thus, the potential gradient along the wire is given by:
\[ K = \frac{I \cdot \rho}{A} \]