# Complete expression for current induced in a conductive loop

What is the complete general expression for the electric current $$i$$ induced in a coreless conductive loop of inductance $$L$$ and resistance $$R$$ when it is subjected to a given external magnetic flux, which is varying linearly from zero to some $$\Phi_{EXT}$$ in time interval $$t_1-t_0$$. Assume $$i(t_0)=0$$

According to the Lenz Law, the current induced in the loop generates a counter-flux $$\Phi_L$$ that opposes the external flux $$\Phi_{EXT}$$ , which attempts to thread the loop.

The Net flux through the loop is equal to the sum of the external flux attempting to thread the loop and the counter-flux due to the induced current $$\Phi_{NET} = \Phi_{EXT}+\Phi_L$$.

When the resistance of the loop is zero then the opposition of the counter-flux to the external flux is complete ( $$\Phi_L=-\Phi_{EXT}$$ ) and $$\Phi_{NET}$$ remains constant perpetually.

The expression for the induced current must reduce to $$i(t)=0$$ when R is infinite.
Also, this expression must reduce to $$i(t)=\frac{\Phi_L(t)}{L}$$ or to $$i(t)=\frac{-\Phi_{EXT}(t)}{L}$$ when R is zero (see this answer).

Obviously, the expression:

$$i(t)=-\frac{d \Phi_{EXT}}{d t}/R$$

...does not fulfill the latter condition, because it ignores the opposing magnetic flux $$\Phi_L$$ generated by the current induced in the loop.

• So what is the expression for the electric current induced in a conductive loop by a varying external flux that reduces to $\frac{\Phi}{L}$ as R approaches zero? Commented Oct 23, 2020 at 20:36