# Can Lenz's rule really be used to determine the direction of induced current?

Faraday's Law states that the induced emf in a closed circuit is the negative of the rate of change of magnetic flux through the surface enclosed by the loop. To obtain the direction of the induced current, we may use Lenz Law, which states that emf will be induced so as to oppose the change that caused it, namely the change in magnetic flux. But I have a doubt regarding this.

The flux that appears in Faraday's law is not just the flux due to the external magnetic field, but rather the net flux due to both the external field and that produced by the current itself. So, the "change" that causes the induced current is due to the induced current as well. That said, how do we know which direction the emf will be induced in?

What if the flux due to the current has a greater magnitude that the flux due to the external field? Does that mean that the emf would now be induced to counteract this flux? I am confused.

• The induced current is born to oppose its sourc, why will it add to the flux of the external magnetic field? – Lelouch Jul 10 '16 at 7:20
• On the contrary it will decrease the flux due to the external field – Lelouch Jul 10 '16 at 7:21
• Maxwells equation refers to the total B field, which includes that of the induced current as well, but for practical purposes we neglect it due to its small contribution. – Lelouch Jul 10 '16 at 7:57
• Yes, but how can we be sure that the flux change ($\Delta \Phi_i,$ say), due to the induced current is less than the 'original' flux change ($\Delta \Phi_o,$ say)? That's the kernel of the question, surely. – Philip Wood Nov 23 '18 at 11:20

Suppose that, at time t = 0, a steadily rising, externally-generated magnetic flux, given by$$\Phi_{ext}=\lambda t$$begins to pass through the coil, which we assume to be in a closed circuit of total resistance $$R$$.

The induced emf, $$\mathscr{E},$$ will be given by$$\mathscr{E}=-\frac{d}{dt}(\Phi_{ext}+\Phi_i),$$in which $$\Phi_i$$ is the flux due to the current, $$\mathscr{E}/R$$. In fact, we can write$$\Phi_i=\frac{L}{R}\mathscr{E}$$in which L is the coil's self inductance (to be assumed constant).

If we substitute from the first and last of these equations into the middle one, separate variables and integrate between $$t=0$$ (when $$\Phi_{i}=0)$$ and $$t=t$$ we find $$\Phi_{i}=-\lambda \frac{L}{R} (1-e^{-\frac{R}{L}t}).$$ Differentiating, $$\frac{d\Phi_{i}}{dt}=-\lambda\ e^{-\frac{R}{L}t}$$that is $$\frac{d\Phi_{i}}{dt}=-\frac{d\Phi_{ext}}{dt}\ e^{-\frac{R}{L}t}$$

We've shown that the rate of change of the induced flux is opposite in direction to that of the external flux, but that (because the exponential factor drops from 1 to 0 as time goes on) its magnitude is never greater than that of the external flux. Therefore we've saved Lenz's law (even if we imagine that the law is referring to the opposing of change in external flux, rather than change in total flux).

Notes

1. At time $$t=0$$ it would seem that the rates of change are equal and opposite.

2. This approach seems to me somewhat clumsy. I'd guess that there's a much slicker method.

• This looks like the right approach to me, but I'm not sure how you got to your fourth equation. Can you fill in a step or two there? – Michael Seifert Nov 23 '18 at 14:48
• Also, I agree there should be a "slicker method", probably based on the underlying differential equation$$\frac{d\Phi_i}{dt} + \frac{R}{L} \Phi_i = - \frac{d \Phi_{ext}}{dt}$$and an argument about the signs of $\dot{\Phi}_i$ in the solutions of said equations. – Michael Seifert Nov 23 '18 at 14:50
• @Michael Seifert (a) I first solved the DE to get $\Phi_i$ as a function of $t$. Then I differentiated it. If I have a moment or two I'll put in some more steps. (b) My thoughts entirely, though I wondered if we had to assume that the external flux changed linearly with time. I can't believe that this assumption is necessary, but having played for a while I couldn't come up with anything quite as convincing as what I posted. – Philip Wood Nov 23 '18 at 15:05

Yes it can, in all cases with a 100% certainty. As to why, the best way to understand would be to imagine, have a physical model, conception in the mind about how electrons behave and what makes them to do so.