What is the Biot–Savart law for the magnetic field due to a current element, and how is its direction determined?

What is the Biot–Savart law for the magnetic field due to a current element, and how is its direction determined?

• A. dB = μ0/(4π) I (dl × r̂) / r^2; direction given by the left-hand rule
• B. dB = μ0/(4π) I dl × r̂ / r^2; direction given by the right-hand rule
• C. dB = μ0/(4π) I (dl × r̂) / r^3; direction given by the right-hand rule
• D. dB = μ0 I dl × r̂ / (4π r^2); direction given by the right-hand rule

Answer: (B)

A current element creates a tiny magnetic field dB that points perpendicular to both the element’s direction and the line to the observation point. The precise form says dB = μ0/(4π) I (dl × r̂) / r^2, where dl is the element’s length vector, r̂ is the unit vector from the element to the field point, and r is the distance between them. This gives the magnitude dB = μ0 I dl sinθ /(4π r^2) with θ the angle between dl and r̂, and the direction set by the cross product dl × r̂. The direction is found with the right-hand rule: align your index finger with dl (the direction of current), point your middle finger along r̂ (from the element to the field point), and your thumb points in the direction of dB. There’s an equivalent but sometimes more convenient form using the vector r: dB = μ0/(4π) I (dl × r) / r^3, since r̂ = r/r and thus (dl × r̂)/r^2 = (dl × r)/r^3.

A current element creates a tiny magnetic field dB that points perpendicular to both the element’s direction and the line to the observation point. The precise form says dB = μ0/(4π) I (dl × r̂) / r^2, where dl is the element’s length vector, r̂ is the unit vector from the element to the field point, and r is the distance between them. This gives the magnitude dB = μ0 I dl sinθ /(4π r^2) with θ the angle between dl and r̂, and the direction set by the cross product dl × r̂.

The direction is found with the right-hand rule: align your index finger with dl (the direction of current), point your middle finger along r̂ (from the element to the field point), and your thumb points in the direction of dB. There’s an equivalent but sometimes more convenient form using the vector r: dB = μ0/(4π) I (dl × r) / r^3, since r̂ = r/r and thus (dl × r̂)/r^2 = (dl × r)/r^3.