Faraday's Law

The main source of energy of the world is the change in magnetic flux. This article demonstrates how a change in magnetic flux generates energy, that is emf called induced emf and how Faraday's law relates to this induced emf. You must know what magnetic flux is before reading this article. Some applications of Faraday's law include GFCI (Ground Fault Circuit Interrupter) and production of sound from electric guitars.

The main idea behind Faraday's law is the change in magnetic flux, that is the rate of change of magnetic flux. And the heart of this law is the rate of change of magnetic flux induces emf in an electric circuit. Note that magnetic flux is not magnetic field; the flux includes area as well. Now we take a look at how the magnetic flux can be changed and hence emf can be induced in an electric circuit.

If the flux changes in any possible way, the emf is induced in the circuit. Here it is illustrated with simple examples. In Figure 1, you can see a permanent magnet moving towards the coil. In this case the magnetic field is changing(increasing), and there is an induced emf indicated by the deflection in galvanometer shown.

Figure 1 Faraday's law; magnet is moving towards the coil.
Figure 2 Faraday's law; magnet is moving away from the coil.

In Figure 2, the magnet is moved away from the coil and again the magnetic field is changing (decreasing) and the galvanometer shows deflection in opposite direction now. In Figure 3, the magnet is stationary and there is magnetic field but it is not changing and therefore there is no deflection in the galvanometer.

Figure 1 Faraday's law; magnet is stationary.

The direction of galvanometer, that is which way the galvanometer needle is deflected or the direction of induced emf is given by the Lenz's law discussed in the next article. For now we simply focus on the galvanometer needle is deflected in opposite directions or the emf induced has opposite directions on increasing and decreasing the magnetic flux.

Note that calling "direction of emf" does not mean it is a vector quantity. You may know emf quiet well, and if not review the article on emf. Again note that we always call "direction of current" but that does not mean, it is a vector quantity. To reminder you, if current is positive, emf is positive, and if current is negative, emf is negative, that is emf and current are in the same direction.

We know that the expression of magnetic flux is $\Phi_B = BA\cos \theta$. And in the above figures, we showed the situation of only $B$ is changing, but the area and the angle $\theta$ (angle between magnetic filed and area vector) can also change. If there is any change in $B$, $A$ and $\theta$, that is if there is the change in magnetic flux, there is the induced emf.

The Faraday's law is stated as

FARADAY'S LAW: The induced emf in an electric circuit is caused by the rate of change of magnetic flux at some part of the circuit.

Mathematically, if $\mathcal{E}$ is the induced emf,

\[\mathcal{E} = -\frac{d\Phi_B}{dt} \tag{1} \label{1}\]

where $\Phi_B$ is the magnetic flux. The subscript indicates it is magnetic flux different from electric flux. The negative sign in the above equation is explained by Lenz law in the next article. But let's talk a little bit about the sign of induced emf here without lenz law.

This involves right hand rule where the thumb points in the direction of area vector as illustrated in Figure 4 and Figure 5 below (the hand is not shown, you can understand that). Note that the area vector can have two possible directions, you need to choose one direction as positive and stick with it. Now first know whether the flux is increasing or not. For example, you can easily know this by whether the magnetic filed is approaching or moving away as in Figures 1 and 2.

Figure 4 The magnetic flux is becoming more positive.
Figure 5 The magnetic flux is becoming more negative.

If the flux is increasing, the rate of change $d\Phi_B/dt$ is positive, the induced emf is in opposite direction to that indicated by the curled fingers, that is negative. If the flux is decreasing, the rate of change of magnetic flux is negative and the induced emf is in the direction of curled fingers, that is positive. Make sure to know whether the flux is becoming more positive, less positive, more negative or less negative to determine the sign of rate of change of flux.

Faraday's Law and Induced Electric Fields

If emf is induced because of the change in magnetic flux, there must be induced electric field. You can see in Figures 5 and 6 that, the induced electric field is tangent at each point on the loop. Note that the flux is changing within the loop in Figure 6, that is the magnetic field is not passing through the conductor and still there is induced emf in the loop. You can surely conclude that the force that causes the charges to flow within the conductor is not the magnetic field.

Figure 4 The change in flux induces emf in the loop.
Figure 5 The magnetic flux is changing within the area bounded by the loop, the loop is outside the changing magnetic field and still there is induced emf in the loop.

You'll see in Maxwell's equations that, a changing magnetic flux induces electric field and a changing electric flux induces magnetic field. Do not confuse that a change in field is the change in flux but the change in flux may not be the change in field. What's happening here is that the magnetic field is changing and that induces electric field which causes the induced emf in the loop.

For an induced electric field $\vec E$, and if $d\vec l$ is the length element of the path, the line integral of the induced electric field $\vec E$ over the closed path is $\oint \vec E \cdot d\vec l$. Therefore, $\mathcal{E} =\oint \vec E \cdot d\vec l$, and we can rewrite the Faraday's law as

\[\oint \vec E \cdot d\vec l = - \frac{d\Phi_B}{dt} \tag{2} \label{2}\]

The induced electric fields are different from electrostatic electric fields. In electrostatic electric fields, the work done over a closed path is always zero. In induced electric fields, the work done over a closed path is not zero, however, the line integral over the closed path of the loop is the emf and hence the rate of change of magnetic flux.

In conservative field, the work done by the field over a closed path is always zero. In electrostatic electric fields we discussed, the work done by the field over a closed path is always zero. But here in Faraday's law, that is not the case. In Faraday's law you'll get a different kind of electric field and the work done by that field over a closed path is not zero but that is equal to the emf induced in the loop due to the change in magnetic flux. Such induced electric fields or nonelectrostatic electric fields are not conservative in nature.

From this point you should understand that two kinds of electric fields can exist, one conservative (electrostatic) and another nonconservative (nonelectrostatic). Do not confuse with the same letter $E$ for both field types, we omitted any subscript here but you can understand.

Electromagnetism
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