Biot-Savart law for the magnetic field due to a current element I dℓ is written as which expression?

Biot-Savart law for the magnetic field due to a current element I dℓ is written as which expression?

• A. dB = (μ0/4π) (I dℓ × r̂)/r^2
• B. dB = (ε0/4π) (I dℓ × r̂)/r^2
• C. dB = (μ0/4π) (I dℓ · r̂)/r^2
• D. dB = (μ0/4π) (I dℓ × r̂) r^2

Answer: (A)

The field from a tiny current element is a vector that wraps around the element in a way set by the cross product. The magnetic field contribution at a point due to a differential element I dℓ is proportional to the cross product of the element’s direction (dℓ) with the unit vector pointing from that element to the field point (r̂), and it falls off like 1/r^2. The constant μ0/(4π) sets the overall strength in vacuum. So dB = μ0/(4π) (I dℓ × r̂) / r^2. Geometrically, the magnitude is |dB| = μ0/(4π) I dℓ sinθ / r^2, where θ is the angle between dℓ and r̂. The direction is perpendicular to the plane formed by dℓ and r̂, determined by the right-hand rule. The other forms would misrepresent the physics: using ε0 instead of μ0 changes the units; using a dot product would give a scalar component rather than a transverse field; and placing r^2 outside the fraction or omitting the division would give an incorrect dependence on distance.

The field from a tiny current element is a vector that wraps around the element in a way set by the cross product. The magnetic field contribution at a point due to a differential element I dℓ is proportional to the cross product of the element’s direction (dℓ) with the unit vector pointing from that element to the field point (r̂), and it falls off like 1/r^2. The constant μ0/(4π) sets the overall strength in vacuum. So dB = μ0/(4π) (I dℓ × r̂) / r^2.

Geometrically, the magnitude is |dB| = μ0/(4π) I dℓ sinθ / r^2, where θ is the angle between dℓ and r̂. The direction is perpendicular to the plane formed by dℓ and r̂, determined by the right-hand rule.

The other forms would misrepresent the physics: using ε0 instead of μ0 changes the units; using a dot product would give a scalar component rather than a transverse field; and placing r^2 outside the fraction or omitting the division would give an incorrect dependence on distance.