A cylindrical conductor has uniform cross-section. Resistivity of its material increase linearly from left end to right end. If a constant current is flowing through it and at a section distance x from left end, magnitude of electric field intensity is E , which of the following graphs is correct

Options

A.
B.
C.
D.

Answer

To solve the problem, we need to analyze how the electric field intensity \( E \) varies along the length of a cylindrical conductor with a linearly increasing resistivity.

### Step-by-Step Solution:

1. **Understand the Resistivity Variation**:
The resistivity \( \rho \) at a distance \( x \) from the left end of the conductor is given by:
\[ \rho(x) = \rho_0 + Ax \]
where \( \rho_0 \) is the resistivity at the left end and \( A \) is a constant that describes how resistivity changes with distance.

2. **Relate Electric Field to Resistivity**:
The electric field intensity \( E \) at a distance \( x \) can be expressed using Ohm's law in terms of current \( I \), resistivity \( \rho \), and cross-sectional area \( A \):
\[ E(x) = \frac{I \cdot \rho(x)}{A} \]

3. **Substitute the Expression for Resistivity**:
Substituting the expression for \( \rho(x) \) into the equation for \( E \):
\[ E(x) = \frac{I}{A} \cdot (\rho_0 + Ax) \]
This can be simplified to:
\[ E(x) = \frac{I}{A} \cdot \rho_0 + \frac{I}{A} \cdot Ax \]

4. **Identify the Form of the Equation**:
The equation can be rewritten as:
\[ E(x) = \left(\frac{I}{A} \cdot \rho_0\right) + \left(\frac{I \cdot A}{A}\right)x \]
which simplifies to:
\[ E(x) = \frac{I}{A} \cdot \rho_0 + \frac{I}{A} \cdot Ax \]
This shows that \( E(x) \) is a linear function of \( x \) with a slope of \( \frac{I}{A} \cdot A \) and a y-intercept of \( \frac{I}{A} \cdot \rho_0 \).

5. **Graphical Representation**:
Since \( E(x) \) is a linear function of \( x \), the graph of \( E \) versus \( x \) will be a straight line. The line will have a positive slope (because \( A \) is positive) and will intersect the y-axis at \( \frac{I}{A} \cdot \rho_0 \).

### Conclusion:
The correct graph representing the relationship between electric field intensity \( E \) and distance \( x \) will be a straight line with a positive slope and a y-intercept at \( \frac{I}{A} \cdot \rho_0 \). Therefore, the correct option is **(b)**.