The expression B = μ0 I / (2π r) describes the magnetic field around which geometry?

The expression B = μ0 I / (2π r) describes the magnetic field around which geometry?

• A. Long straight current-carrying wire
• B. Finite length wire
• C. Ring
• D. Toroid

Answer: (A)

This describes the magnetic field around an infinitely long straight current-carrying wire. Because the wire is effectively infinite, the field has perfect cylindrical symmetry: the magnetic field lines circle the wire in planes perpendicular to it, and the magnitude depends only on the distance r from the wire, decreasing as 1/r. Using Ampère’s law, choose a circular path of radius r around the wire. The magnetic field is tangent to this circle and has the same magnitude at every point on the path, so the line integral is B times the circumference, 2πr. The current enclosed by this loop is I. Ampère’s law gives B(2πr) = μ0 I, so B = μ0 I/(2πr). The direction is given by the right-hand rule: curl the fingers of your right hand around the wire in the direction of the magnetic field when the thumb points in the current’s direction. This form wouldn’t hold for a finite-length wire, where symmetry isn’t the same near the ends, nor would it describe a ring or a toroid, which have different field distributions.

This describes the magnetic field around an infinitely long straight current-carrying wire. Because the wire is effectively infinite, the field has perfect cylindrical symmetry: the magnetic field lines circle the wire in planes perpendicular to it, and the magnitude depends only on the distance r from the wire, decreasing as 1/r.

Using Ampère’s law, choose a circular path of radius r around the wire. The magnetic field is tangent to this circle and has the same magnitude at every point on the path, so the line integral is B times the circumference, 2πr. The current enclosed by this loop is I. Ampère’s law gives B(2πr) = μ0 I, so B = μ0 I/(2πr). The direction is given by the right-hand rule: curl the fingers of your right hand around the wire in the direction of the magnetic field when the thumb points in the current’s direction.

This form wouldn’t hold for a finite-length wire, where symmetry isn’t the same near the ends, nor would it describe a ring or a toroid, which have different field distributions.