JEE MainPhysicsMagnetic Effects of Current
A long straight thick-walled hollow cylindrical pipe has an inner radius a and an outer radius b . It carries a steady current I which is uniformly distributed across its annular cross-section. The magnitude of the magnetic field B at a distance r from the axis of the pipe, where a < r < b , is given by
Options
- A₀ I (r^2 - a^2) 2 r (b^2 - a^2)
- B₀ I (r - a) 2 r (b - a)
- C₀ I r 2 b^2
- D₀ I (r^2 - a^2) 2 r^2 (b^2 - a^2)
Correct answer
A. ₀ I (r^2 - a^2) 2 r (b^2 - a^2)
Step-by-step solution
The current I is uniformly distributed over the annular cross-sectional area A = (b^2 - a^2) . The uniform current density is: J = I (b^2 - a^2) To find the magnetic field at a distance r ( a I_ enc = J (r^2 - a^2) = I r^2 - a^2 b^2 - a^2 Applying Ampere's circuital law, B d l = ₀ I_ enc : B(2 r) = ₀ I r^2 - a^2 b^2 - a^2 B = ₀ I (r^2 - a^2) 2 r (b^2 - a^2) Answer: ₀ I (r^2 - a^2) 2 r (b^2 - a^2)