# Ampere's circuital law vs. Biot-Savart's law

I know that many questions have been asked regarding this, but browsing over the questions, none of them seem to clarify my doubts.

By applying Ampere's circuital law to a current-carrying circular coil, choosing a circle of radius $a$ which is smaller than the radius of the coil, and integrating over it, I found that magnetic field inside a current carrying coil is zero since there is no current enclosed within the circle. So the rhs in the equation $\oint B\cdot dl = \mu_0 I$ becomes zero which effectively means the magnetic field is zero. However, we already know that by using Biot-Savart's law we can easily derive that magnetic field at the center of a current-carrying circular coil is $B = \frac{\mu_0 I}{2r}$.

How are these two contradictory? If not, have I applied the two laws correctly? How can we derive the magnetic field inside a current carrying circular coil using Ampere's circuital law?

• Just because an integral involving $B$ is zero does not imply $B$ is zero. – knzhou Jul 30 '16 at 0:30
• then it should imply that the dot product is zero.but it doesnot matter whatever the curve be the integral is always zero.it therefore should imply that B is zero. – Pink Jul 30 '16 at 0:33
• This is wrong. For example, think back to classical mechanics. The integral of any conservative vector field is zero. That doesn't mean all conservative vector fields are zero. – knzhou Jul 30 '16 at 0:35
• then what about a long torroidal solenoid.in this case only,we should not come to the conclusion that B is zero inside the torroid not enclosing any current. – Pink Jul 30 '16 at 0:58
• it isn't. it's just pretty small. – knzhou Jul 30 '16 at 0:59