A long straight solid wire of radius R carries a steady current uniformly distributed across its cross-section. The magnitude of the magnetic field is maximum at the surface of the wire and is equal to B₀ . At what radial distances from the axis of the wire is the magnitude of the magnetic field equal to B₀ 4 ?

Step-by-step solution

Let the current in the wire be I . The magnetic field is maximum at the surface ( r = R ): B₀ = ₀ I 2 R Case 1: Inside the wire ( r The magnetic field at a distance r is given by B_ in = ₀ I r 2 R^2 = B₀ ( r R ) . Given B_ in = B₀ 4 , we have: B₀ ( r R ) = B₀ 4 r = R 4 Case 2: Outside the wire ( r R ) The magnetic field at a distance r is given by B_ out = ₀ I 2 r = B₀ ( R r ) . Given B_ out = B₀ 4 , we have: B₀ ( R r ) = B₀ 4 r = 4R Thus, the magnetic field is B₀ 4 at radial distances R 4 and 4R . Answer: R 4 and