The differential magnetic field due to a current element at distance r is correctly written as which expression?

The differential magnetic field due to a current element at distance r is correctly written as which expression?

• A. dB = (μ0/4π) (I dℓ × r̂)/r
• B. dB = (μ0/2π) (I dℓ × r̂)/r^2
• C. dB = (μ0/4π) (I dℓ × r̂)/r^2
• D. dB = (μ0/4π) (I dℓ · r̂)/r^2

Answer: (C)

Magnetic field from a small current element is described by the Biot–Savart law: the differential field at a point is proportional to the current times the element length, directed perpendicular to the plane formed by the element and the line to the field point, and it falls off as 1/r^2. The standard form is dB = (μ0/4π) I (dℓ × r̂)/r^2, where r̂ points from the element to the field point and r is the distance. This cross product fixes the direction via the right-hand rule and ensures the field is perpendicular to both dℓ and r̂, while the 1/r^2 factor captures the correct spatial decay. The same result can be written as dB = (μ0/4π) I dℓ × r / r^3 using the full displacement vector r. The other forms fail because using a dot product instead of a cross product removes the vector nature and proper direction, and using 1/r instead of 1/r^2 or a different constant gives the wrong distance dependence and strength.

Magnetic field from a small current element is described by the Biot–Savart law: the differential field at a point is proportional to the current times the element length, directed perpendicular to the plane formed by the element and the line to the field point, and it falls off as 1/r^2. The standard form is dB = (μ0/4π) I (dℓ × r̂)/r^2, where r̂ points from the element to the field point and r is the distance. This cross product fixes the direction via the right-hand rule and ensures the field is perpendicular to both dℓ and r̂, while the 1/r^2 factor captures the correct spatial decay. The same result can be written as dB = (μ0/4π) I dℓ × r / r^3 using the full displacement vector r. The other forms fail because using a dot product instead of a cross product removes the vector nature and proper direction, and using 1/r instead of 1/r^2 or a different constant gives the wrong distance dependence and strength.