Which expression gives the magnetic field around a long straight wire carrying current I at distance r?

Which expression gives the magnetic field around a long straight wire carrying current I at distance r?

• A. B = μ0 I /(4π r)
• B. B = μ0 I /(π r)
• C. B = μ0 I /(2π r)
• D. B = μ0 I r /(2π)

Answer: (C)

The magnetic field around a long straight current-carrying wire decreases with distance as 1/r. To find its magnitude at a distance r, apply Ampere’s law: ∮ B · dl = μ0 I_enclosed. Choose a circular path of radius r around the wire. The field is tangential and has the same magnitude all along that circle, so the line integral is B(2πr). This must equal μ0 I, so B = μ0 I / (2πr). The direction is tangential to the circle, determined by the right-hand rule: if the current goes in the thumb direction, the magnetic field circulates in the direction your fingers wrap. The other expressions don’t fit because replacing B with μ0 I /(4π r) would give half the correct value, μ0 I /(π r) would give twice the correct value, and μ0 I r /(2π) would imply the field grows with distance, which isn’t true for a long straight wire.

The magnetic field around a long straight current-carrying wire decreases with distance as 1/r. To find its magnitude at a distance r, apply Ampere’s law: ∮ B · dl = μ0 I_enclosed. Choose a circular path of radius r around the wire. The field is tangential and has the same magnitude all along that circle, so the line integral is B(2πr). This must equal μ0 I, so B = μ0 I / (2πr). The direction is tangential to the circle, determined by the right-hand rule: if the current goes in the thumb direction, the magnetic field circulates in the direction your fingers wrap.

The other expressions don’t fit because replacing B with μ0 I /(4π r) would give half the correct value, μ0 I /(π r) would give twice the correct value, and μ0 I r /(2π) would imply the field grows with distance, which isn’t true for a long straight wire.