Question

Explain Biot-Savart law and find the unit of \( \mu_0 \) with the help of the Biot-Savart equation.

Hint:

The Biot-Savart law helps us calculate the magnetic field generated by a current element in a conductor.

Answer 1

Step 1: Biot-Savart Law.
The Biot-Savart law gives the magnetic field \( \vec{B} \) produced at a point by a small current element. The law is given by: \[ d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2} \] where: - \( d\vec{B} \) is the infinitesimal magnetic field, - \( \mu_0 \) is the permeability of free space, - \( I \) is the current, - \( d\vec{l} \) is the infinitesimal length element of the wire, - \( \hat{r} \) is the unit vector from the current element to the point of observation, - \( r \) is the distance from the current element to the point of observation.
Step 2: Unit of \( \mu_0 \).
Rearranging the Biot-Savart law to isolate \( \mu_0 \): \[ \mu_0 = \frac{4\pi r^2 d\vec{B}}{I d\vec{l} \times \hat{r}} \] From this, we can see that the units of \( \mu_0 \) are derived as follows: \[ [\mu_0] = \frac{\text{T m}^2}{\text{A}} \] Thus, the unit of \( \mu_0 \) is \( \frac{\text{T m}^2}{\text{A}} \), where \( \text{T} \) is the tesla, \( \text{m} \) is meters, and \( \text{A} \) is amperes.
Step 3: Conclusion.
The unit of \( \mu_0 \) is \( \frac{\text{T m}^2}{\text{A}} \).