Question

Write down Biot-Savart law and find the expression for the magnetic field produced by a current-carrying conductor of infinite length, on the basis of it.

Hint:

The Biot-Savart law is useful for calculating the magnetic field produced by a current element. For an infinite straight conductor, the magnetic field decreases inversely with the distance from the conductor.

Answer 1

Step 1: Biot-Savart Law.
The Biot-Savart Law gives the magnetic field \(d\vec{B}\) produced at a point due to a small current element \(I d\vec{l}\) as: \[ d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2}, \] where:
- \(\mu_0\) is the permeability of free space (\(\mu_0 = 4\pi \times 10^{-7}\ \mathrm{T \cdot m/A}\)),
- \(I\) is the current,
- \(d\vec{l}\) is the vector length of the current element,
- \(\hat{r}\) is the unit vector from the current element to the point where the magnetic field is calculated,
- \(r\) is the distance from the current element to the point.

Step 2: Magnetic Field Due to a Long Straight Conductor.
For an infinitely long, straight conductor carrying a current \(I\), we can integrate the Biot-Savart law along the length of the conductor. The magnetic field at a distance \(r\) from the conductor is given by: \[ B = \frac{\mu_0 I}{2\pi r}. \] This expression is derived by integrating the Biot-Savart law for an infinite length conductor. The direction of the magnetic field follows the right-hand rule, meaning the magnetic field circulates around the wire in concentric circles.

Final Answer: The magnetic field produced by an infinitely long current-carrying conductor is given by: \[ B = \frac{\mu_0 I}{2\pi r}, \] where \(r\) is the distance from the wire, and the direction is given by the right-hand rule.