A long, straight, cylindrical wire of radius R carries a current uniformly distributed over its cross section.

a) At what location is the magnetic field produced by this current equal to third of its largest value? Consider points inside the wire.

r/R=???

b) At what location is the magnetic field produced by this current equal to third of its largest value? Consider points outside the wire.

r/R=???

Explanation:

We shall solve this question with the help of Ampere's circuital law.

Ampere's ,law

∫ B dl = μ₀ I , B is magnetic field at distance x from the axis within wire

we shall find magnetic field at distance x . current enclosed in the area of circle of radius x

=  I x π x²  / π R²

= I x²  /  R²

B x 2π x = μ₀  x current enclosed

B x 2π x = μ₀  x  I x²  /  R²

B =  μ₀   I x  / 2π R²

Maximum magnetic B₀ field  will be when x = R

B₀ = μ₀I   / 2π R

Given

B = B₀ / 3

μ₀   I x  / 2π R² = μ₀I   / 2π R x 3

x = R / 3

b ) The largest value of magnetic field is on the surface of wire

B₀ = μ₀I   / 2π R

At distance x outside , let magnetic field be B

Applying Ampere's circuital law

∫ B dl = μ₀ I

B x 2π x = μ₀ I

B = μ₀ I / 2π x

Given B = B₀ / 3

μ₀ I / 2π x = μ₀I   / 2π R x 3

x = 3R .