A long cylindrical conductor with large cross section carries an electric current distributed uniformly over its cross-section. Magnetic field due to this current is:
A. maximum at either ends of the conductor and minimum at the midpoint
B. maximum at the axis of the conductor
C. minimum at the surface of the conductor
D. minimum at the axis of the conductor
E. same at all points in the cross-section of the conductor
Choose the correct answer from the options given below :

A long cylindrical conductor carries current uniformly distributed over its cross-section. We need to determine where the magnetic field is minimum.

For a long cylindrical conductor of radius $$R$$ carrying total current $$I$$:

Inside the conductor ($$r < R$$):

$$B(2\pi r) = \mu_0 \cdot \frac{I}{\pi R^2} \cdot \pi r^2 = \mu_0 I \frac{r^2}{R^2}$$

$$B = \frac{\mu_0 I r}{2\pi R^2}$$

The field increases linearly with $$r$$ inside the conductor.

Outside the conductor ($$r > R$$):

$$B = \frac{\mu_0 I}{2\pi r}$$

The field decreases as $$1/r$$ outside.

At the axis ($$r = 0$$): $$B = 0$$. This is the minimum.

At the surface ($$r = R$$): $$B = \frac{\mu_0 I}{2\pi R}$$. This is the maximum.

Statement A: Maximum at ends, minimum at midpoint - FALSE (this describes the axial field, not radial).

Statement B: Maximum at axis - FALSE (B = 0 at axis).

Statement C: Minimum at surface - FALSE (maximum at surface).

Statement D: Minimum at axis - TRUE ($$B = 0$$).

Statement E: Same at all points - FALSE.

The correct answer is Option 3: D Only.