# Induced current in a loop - what about cancellation?

We all know that a time-varying magnetic field through a coil will tend to induce a current in the coil, and the converse is also true.

If you look at the field lines created by current in a coil, they travel in one direction inside the coil, and in the opposite direction outside the coil, like so:

Suppose that we add a time-varying field that is perfectly uniform in magnitude AND direction over the entire area inside and outside the coil. The field inside the coil would tend to induce a current in the coil in one direction, while the field outside the coil would tend to induce a current in the other.

Would the effect be that zero current flows in the coil?

• Can you state your exact question clearly please? Is the first scenario described above the picture related to that described below? I can't make it out. Commented Jun 30, 2017 at 6:00
• Take the coil, without any current in it, and put it inside a magnetic field that is completely uniform in direction, but changing in magnitude - if I were to draw the field lines, they'd all by arrows going upward through the coil and outside the coil. Will a current be induced as the field varies in strength, or will the fields outside the coil tend to cancel the effect? Commented Jul 1, 2017 at 14:43

The field outside the coil do not contribute to the emf generated. It depends only on the rate of change of flux through the path considered. Referto the maxwell equations. It clearly states(in differential form):

$\nabla \times \vec E = -\frac{\partial B}{\partial t}$

Which in integral form can be written as:

$\int_{C} \vec E.dl = -\int \int_{S} \frac{\partial B}{\partial t}.dS$

Here the left hand side represents the contour integral around the circuit/loop/path under consideration. The right side represents a surface integral,over the surface $S$ bounded by the contour $C$. As you can see, the rhs depends only on how the magnetic field behaves inside the loop under consideration.

• Cool, thanks! It's surprising that time-varying current in the coil will produce a field that is circular in nature, with external fields pointing "down" and internal field "up," while induced emf from a time-varying field doesn't depend at all on the external fields. But I guess the math doesn't lie! Commented Jul 3, 2017 at 15:23