# Why does a changing magnetic field produce a current?

A changing magnetic field induces a current in a conductor. For example, if we move a bar magnet near a conductor loop, a current gets induced in it.

Faraday's law states that

The E.M.F. $\mathcal{E}$ induced in a conducting loop is equal to the rate at which flux $\phi$ through the loop changes with time.

Along with Lenz's law,

$$\mathcal{E} = -\frac{d\phi}{dt}$$

Why is this so? The velocity of the electrons w.r.t. me, the observer is 0, so according to $\vec{F} = q\vec{v} × \vec{B}$, force should be zero in any direction on the electrons in the loop. Then what causes the current to flow and the E.M.F. to be induced? Is the force due to an electric field(the electric field in my reply to Albert in comments) or should I consider the velocity w.r.t. the source of the magnetic field?

Edit : I'm in a frame of reference which is stationary w.r.t. the loop, so from where does the electric force(due to the electric field in my reply to Albert in comments) come from?

• $\mathcal{E}$ is the induced voltage, that is, the change in flux generates an electric field, which is the one who moves the electrons (do not forget the term $qE$ in the Lorentz force) – user126422 Oct 5 '16 at 11:37
• @Albert What caused the electric field? I just learnt a force experienced by a particle near a current carrying conductor is zero and it's explanation by relativity(although I haven't studied relativity but just know that as a consequence of relativity, in a reference frame moving with the drift velocity of the electrons, the negative linear charge density decreases in magnitude compared to the positive charge density of the nuclei and a equal and opposite electric field cancels the magnetic field generated by the motion of the particle in that frame and the current). – Apoorv Potnis Oct 5 '16 at 11:48
• the electric field $E$ comes from Lenz law, you can get E from $\mathcal{E}$ (as $E=-\frac{d \mathcal{E}}{dl}$), where $dl$ is a length element along the wire – user126422 Oct 5 '16 at 12:05
• What is the origin of E.M.F., there should be charges to create an area of higher potential and lower potential and thus an Electric field, shouldn't they? – Apoorv Potnis Oct 5 '16 at 12:09
• It is a fundamental law of nature (later included into maxwell's equations), in this case you do not need charges to create an $E$ – user126422 Oct 5 '16 at 12:20

The third of Maxwell's equations (Faraday's law) says that a changing magnetic field has an E-field curling around it. The closed line integral of this electric field is the EMF that drives the induced current in the conducting wire. At a microscopic level, the curling electric field, which has a significant component parallel to the wire, exerts a force on the charges in the conductor.

If your question is "why are Maxwell's equations the way they are?", I'm afraid that isn't a good question for this site.

• Thanks. My physics textbook didn't state Maxwell's equations at all. I had to look them from another book. – Apoorv Potnis Oct 6 '16 at 3:29

It is indeed a relativistic effect.

In fact you can derive Faraday's law from the Lorentz transform of the electromagnetic field.

A boost (velocity) orthogonal to a magnetic field $\textbf{B}$ transforms in an electric field $\textbf{E}$ that is both orthogonal to the boost (velocity) and the $\textbf{B}$ field.

Only the component of $\textbf{B}$ in the plane of the ring is involved because this component $\textbf{B}_\perp$ is orthogonal to the boost. See the green highlighted term in the Lorentz transform of the electro-magnetic field. The term

• $\textbf{B}_\parallel$ = component of $\textbf{B}$ parallel to the boost.
• $\textbf{B}_\perp$ = component of $\textbf{B}$ orthogonal to the boost.
• $\textbf{B}_\bigotimes$ = same component as $\textbf{B}_\perp$ but 90 degrees rotated with respect to the boost direction.
• At low speeds $\gamma \approx 1$ and $\beta$ is proportional to the velocity.

Now note that only $\textbf{B}_\perp$ contributes to a change in the total flux through the ring while moving the magnet, and that the term $\textbf{B}_\parallel$ does not contribute.

This is how you can derive Faraday's law from the Lorentz transform of the electro-magnetic field.

http://www.physics-quest.org/Book_Chapter_EM_LorentzContr.pdf

I guess you probably already know this, but still let me state it. If you have read Griffith's Electrodynamics or any other book on electromagnetism, one statement is always specified in faraday's law chapter...in different words

NATURE ABHORS A CHANGE IN FLUX

Note that it is the CHANGE in flux, not the flux itself, which the nature dislikes. So according to lenz's law, it(nature...wire) will try to do anything to resist that change in flux. In your case, as the magnetic field increases when the magnet is brought closer to the loop, a clockwise current will flow in the loop as seen from the ammeter's side. This current is due to the electric field set up due to the change in magnetic field. It is given by the closed loop integral of $E.dl$ = -$\frac{\delta B}{\delta t}$A...since the area is unchanging. The electric field is circumferential which moves the charges in the wire....

If the charges are stuck to the wire......guess what happens??.....the wire rotates....which also means current(rotating charges, although stuck)!!!

Take a look at this,... The big , really long, cylinder is the magnet. As it is brought closer, there is an increase in the EMF. but it decays. Again as the magnet's end comes closer to the loop, the EMF spikes in the other direction

• Why does Nature abhor a change in flux? What causes an electric field inside the conductor loop such that the current produced opposes the magnetic field due to magnet? – Apoorv Potnis Oct 5 '16 at 12:11
• The physics ends at how. 'Why' is a metaphysical question. :) – Prasad Mani Oct 5 '16 at 12:12
• So, there's an electric field created just out of nowhere, without charges??? – Apoorv Potnis Oct 5 '16 at 12:14
• Or is there a relativistic explanation? – Apoorv Potnis Oct 5 '16 at 12:15
• One last comment. Physics can probe the nature, questioning it is a whole different thing. Nature behaves someway, we struggle for decades to try and explain it using math and models which can just make predictions. WHY nature behaves a certain way (why does the loop simply not let the flux through it increase and keep everyone happy) cannot be answered by physics – Prasad Mani Oct 5 '16 at 12:44

When the north Pole of magnet is approaching towards the coil ,the coil facing the north Pole behave like a north Pole and both of tries to repel each other and they will try to stop the motion of magnet, so to push the magnet​ against repulsive force work must be done against repulsive force , the work appears in the form​ of electrical energy in anti clockwise directions. ......,......Read this it is complete explaination

As you stated right it has to be a conductor to induce a current from a changing magnetic field. In conductors there are some electrons which are in unbounded states to the nuclei.

It is well known that a magnetic field and an electrons electric field don't interact. So what is interacting? The electron has two more properties, one of them is the electrons magnetic dipole moment.

If a stationary electron is inside a constant magnetic field the electrons magnetic dipole moment gets aligned with the external field and that was is, no current will flow.

If you are "riding" on a changing magnetic field you will see the electron (in the wire) moving (in relation to the changing magnetic field). And there is a phenomenon of deflection of moving charges, called Lorentz force. If you are sitting on the electron you get the same picture, regarding a changing magnetic field the electron seems to move and by this gets deflected. So in both frames the relatively movement between charge and field (no matter does from a third frame the charge moves or the magnetic field), there will be induced a current.

Now the clou. There is a phenomenon of a homopolar generator where both the magnetic field and the disc are rotating together and nevertheless a current is induced:

• Thanks. Finally I understand the case in which I sit on both the electrons of the magnet and the loop taking into consideration the rotatory motion of electrons which produce the magnetic field(Due to your answer). But what if the observer is stationary w.r.t. loop, I could see a magnetic field line piercing the wire but the velocity of the electron in the wire would be zero still. That would contribute to zero magnetic force. – Apoorv Potnis Oct 5 '16 at 17:31
• Are you claiming the E.M.F. only works vecause of the magnetic dipol moment of the charges? – lalala Apr 16 '17 at 10:17
• @lalala You are a very attentive reader. – HolgerFiedler Apr 16 '17 at 11:33
• any references which supports this claim? It seems to be a bit in disagreement what Maxwell told us. – lalala Apr 16 '17 at 11:43
• @lalala A theory has to consist known phenomenons and has not to be in contradiction with this phenomenons. Search inconsistencies in the claims an I'll be very thankful to be cured from doubts. – HolgerFiedler Apr 16 '17 at 11:58