# How does back-EMF oppose the input-current?

A circuit, has current $A$ flowing at a certain $V$.

When there is a change in magnetic-flux, based on Faraday's law of induction & Lenz's law, we know that there is change in Potential Difference now, aside from the source $V$ now we have a induced $-V$ due to the change in magnetic-flux, and it opposes the current, why would it? I understood from lenz's law that it will, but not great detail as to why.

Another thing, if the power-source can be increased, $V$ can potentially increase to oppose the $-V$? And maintain $A$ at the same value it was?

When there is a change in magnetic-flux, based on Faraday's law of induction & Lenz's law, we know that there is change in Potential Difference now, aside from the source V now we have a induced −V due to the change in magnetic-flux, and it opposes the current, why would it? I understood from lenz's law that it will, but not great detail as to why.

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

Note the negative sign. If we integrate both sides by the area through which there is a magnetic field (think of the cross-section of a solenoid, for example), then we get:

$$\int (\nabla \times E) \cdot dA = \oint E\cdot dl = - \frac{\partial \Phi}{\partial t}$$

The middle term follows from applying Stoke's theorem on the first term. The middle term is also the definition of voltage. The final definition is flux ($\Phi$) which is the product of the magnetic field and the cross-sectional area the field passes through, which is how one arrives at the third term.

Another thing, if the power-source can be increased, V can potentially increase to oppose the −V? And maintain A at the same value it was?

The 'opposite V' or back EMF acts to stabilize the current flowing. Suppose you increase V, and hence A, which increases the flux and hence slowly increases the back EMF. Then suppose that you stop increasing V. The back EMF will persist and slowly decay thereafter until the current stops changing (current reaches steady-state). If you were to instead keep increasing V, you'd keep increasing A, keep increasing the flux and keep increasing the back EMF (it would exacerbate the situation).

• Well we know that back-emf will reduce current, instead of stabilizing it, if we want to maintain the same current as before, we increase source-emf to oppose-back emf or "cancel" it, so that current would be unchanged, and stays the same. Jul 23, 2014 at 1:51
• @Key that only makes sense if there's some external flux messing with the inductor in question (e.g., bring a strong magnet near an operating solenoid). Otherwise the applied voltage away from equilibrium is causing the change in current and hence the back EMF. Jul 23, 2014 at 2:14
• There is an external flux, sorry for not stating that earlier, so current can be stabilized by increasing the source-voltage. Jul 23, 2014 at 13:56
• But, if we increase the source-voltage to maintain the same current, wouldn't there resistance increase as well? Jul 30, 2014 at 19:13

I'm don't know if you will find this answer satisfying, but suppose the EMF went the opposite way. Instead of opposing the current, it boosts the current. Then the higher current will produce a higher field and higher EMF which will boost the current, which will produce a higher field and higher EMF ... until the wire melts.

Lenz's Law established stability. The opposite would produce instability. The world as we know it could not exist without the minus sign in Lenz's Law.

This type of argument has its detractors, but perhaps you will get something out of it.

• Valid point, but aside from the wire melting aspect. Lenz's law is the key point of conservation of energy in electodyanmics. Jun 27, 2016 at 12:00

Are you asking why the electrons in wires exhibit this behavior? If so it is due to the fact the path of moving electrons is curved when in a magnetic field and how much it curves increases when the magnetic field increases. Since the electrons in wires are always moving about randomly and never still, their paths' get curved. This net rotation of electrons manifests on the large-scale as a net current change in the wire when the magnetic field changes.

I don't think there is a physical explanation exists as to why the path of electrons curves in a magnetic field(created by a bunch of electrons going in a circle) and why said path has the opposite rotation of it's creator though.