Answer: The magnetic induction \(dB\) at a point due to a small current element \(I\,dl\) is directly proportional to the current \(I\), the length \(dl\) and the sine of the angle \(\theta\) between the element and the line joining it to the point, and inversely proportional to the square of the distance \(r\): \(dB = \dfrac{\mu_0}{4\pi}\dfrac{I\,dl\sin\theta}{r^2}\).
[Diagram: a conductor carrying current I; a small element dl at O; point P at distance r from the element; angle θ between dl and r.]
- \(dB \propto I\), \(dB \propto dl\), \(dB \propto \sin\theta\), \(dB \propto \dfrac{1}{r^2}\).
- Combining, \(dB = \dfrac{\mu_0}{4\pi}\dfrac{I\,dl\sin\theta}{r^2}\), where \(\mu_0 = 4\pi\times10^{-7}\) Wb/A m is the permeability of free space.
- Vector form: \(d\vec{B} = \dfrac{\mu_0}{4\pi}\dfrac{I\,d\vec{l}\times\vec{r}}{r^3}\). The direction of \(d\vec{B}\) is perpendicular to the plane containing \(d\vec{l}\) and \(\vec{r}\) (given by the right-hand rule).
- The total field of a conductor is obtained by integrating over all elements: \(\vec{B} = \displaystyle\int d\vec{B}\).
