# Ampere's (Circuital) Law

Ampere's circuital law states that the line integral of the magnetic field around a closed loop C is equal to $\mu_0$ times the total electric current passing through the loop i.e.,
\begin{align}
\oint_{C}\vec{B}\cdot\mathrm{d}\vec{l}=\mu_0 I_{enc}
\end{align}

## Applications of Ampere's Circuital Law

Ampere's circuital law is very useful in problems having symmetry. It can be used to find magnetic field of an infinitely long straight conductor, a long solenoid and a toroid.

## Problems from IIT JEE

**Problem (IIT JEE 1993): **
A current $I$ flows along the length of an infinitely long, straight, thin-walled pipe. Then the magnetic field

- at all points inside the pipe is the same, but not zero.
- at any point inside the pipe is zero.
- is zero only on the axis of the pipe.
- is different at different points inside the pipe.

**Solution:**
By symmetry, the magnetic field inside the pipe is circumferential. Take a circular loop of radius $r$ with centre along the axis of the pipe. By symmetry, the magnitude of magnetic field is same throughout the loop.

By Ampere's law, $\oint \vec{B}\cdot\mathrm{d}\vec{l}=B(2\pi r)=\mu_0I_\text{enc}$. Since $I_\text{enc}=0$, we get $B=0$ for all $r$. Thus, $B=0$ inside the pipe. The readers are encouraged to explain why there can not be a uniform field along the axis, similar to field of a solenoid.

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