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.


1 Answer 1


Recall Lenz's Law- The induced electromotive force with different polarities induces a current whose magnetic field opposes the change in magnetic flux through the loop in order to ensure that the original flux is maintained through the loop when current flows in it. That's how the negative sign in the second equation came into being, and I think you have understood that. But look at your first equation! There need not be a negative sign there. Perhaps you thought it would be negative because of Lenz's Law..and that's a mistake. Look at the part of the definition in bold. By assigning a negative sign to the right-hand side of the equation, you are implying that the emf induced(and hence the current) is opposite to the direction of flux itself- and that's a wrong notion. As for the link you provided, go back and check the first answer. There is no negative sign in the equation. Hope I've helped you.

  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$
    – tpg2114
    Commented Oct 23, 2020 at 14:16
  • $\begingroup$ 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? $\endgroup$ Commented Oct 23, 2020 at 20:36

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.