Question

Explain Biot-Savart's law for a small current-carrying conductor.

Hint:

Biot-Savart's law is fundamental in calculating the magnetic field produced by a current-carrying conductor, and the field depends on the current, the shape of the conductor, and the position where the field is being calculated.

Answer 1

Step 1: Biot-Savart's Law.
Biot-Savart's law gives the magnetic field \( d\vec{B} \) produced at a point due to a small element of current-carrying conductor. According to this law, the magnetic field is directly proportional to the current \( I \), the length of the conductor \( dl \), and the sine of the angle \( \theta \) between the current element and the position vector \( \vec{r} \) from the element to the point. The law is mathematically expressed as: \[ d\vec{B} = \frac{\mu_0}{4\pi} \frac{I \, dl \, \sin \theta}{r^2} \hat{r} \] where:
- \( d\vec{B} \) is the magnetic field produced by the small element,
- \( \mu_0 \) is the permeability of free space,
- \( I \) is the current in the conductor,
- \( dl \) is the length of the current element,
- \( r \) is the distance from the current element to the point where the magnetic field is being calculated,
- \( \theta \) is the angle between the current element and the position vector,
- \( \hat{r} \) is the unit vector along the position vector.
Step 2: Understanding the law.
Biot-Savart's law allows us to calculate the magnetic field due to a current-carrying conductor by integrating over the length of the conductor. The magnetic field produced at any point depends on the geometry of the current path and the position of the point.
Step 3: Conclusion.
Thus, Biot-Savart's law provides a way to calculate the magnetic field produced by a current-carrying conductor by considering the contribution of each infinitesimal segment of the conductor.