Question

Write Bio-Savart expansion in vector form.
OR
Why there is no end point of magnetic field lines?

Hint:

Magnetic field lines are closed loops; they never terminate in space. This is in contrast to electric field lines, which originate from positive charges and terminate at negative charges.

Answer 1

a. Bio-Savart Law (Vector Form):
The Bio-Savart law gives the magnetic field \( \vec{B} \) at a point in space due to a small current element. In vector form, it is expressed as:
\[ \vec{B} = \frac{\mu_0}{4\pi} \int \frac{I \, d\vec{l} \times \hat{r}}{r^2} \]
Where:
- \( \mu_0 \) is the permeability of free space,
- \( I \) is the current flowing through the wire,
- \( d\vec{l} \) is the infinitesimal vector element of the wire in the direction of the current,
- \( \hat{r} \) is the unit vector pointing from the current element to the point where the magnetic field is being calculated,
- \( r \) is the distance between the current element and the point of observation.
This law is used to calculate the magnetic field produced by a current-carrying conductor.
b. Why there is no end point of magnetic field lines?
Magnetic field lines do not have an end point because the magnetic field produced by a magnetic dipole or a current-carrying conductor forms a closed loop. This means that the lines start from the north pole (or current source) and loop around to the south pole (or return to the conductor), but they do not terminate in empty space. They always form continuous, closed loops.