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:

Solution

Physics sir · 3 steps

Draw an Amperian circle inside the wire: it encloses less current the closer it gets to the axis.

1. Inside the conductor. Take a circle of radius r < R about the axis. With uniform current density, it encloses only the fraction r²/R² of the total current I.

   B(2π r) = μ₀ I r²/R² ⇒ B = (μ₀ I r)/(2π R²)

2. Read the result. Inside, B ∝ r. It is zero at the axis (r = 0) and rises to its largest value (μ₀ I)/(2π R) at the surface. Outside, B ∝ 1/r and falls off again.

   Statement	True or false	Why
   A: max at ends, min at midpoint	False	A long wire looks the same all along its length, so B does not change from end to middle
   B: maximum at the axis	False	B = 0 at the axis
   C: minimum at the surface	False	The surface is where B is largest
   D: minimum at the axis	True	B = 0 at r = 0
   E: same at all points	False	B grows linearly with r inside

3. So. Only statement D is correct, which is option (d).

The idea to keep. For a uniformly filled wire, B rises as r inside, peaks at the surface and falls as 1/r outside. Zero at the axis, maximum at the surface.

Answer from the NTA final answer key. Solution by PrepWiser.