Faradays Law of Induction

Faraday's Law of Induction describes how an electric current produces a magnetic field and, conversely, how a changing magnetic field generates an electric current in a conductor. English physicist Michael Faraday gets the credit for discovering magnetic induction in 1830; however, an American physicist, Joseph Henry, independently made the same discovery about the same time, according to the University of Texas.
Demonstrating Faraday's law
Fig. 1: Demonstrating Faraday's law
It is impossible to overstate the significance of Faraday's discovery. Magnetic induction makes possible the electric motors, generators, and transformers that form the foundation of modern technology.
Faraday's law states: The electromotive force around a closed path is equal to the time rate of change of the magnetic flux enclosed by the path.
If a coil of N turns is placed in the region of a changing flux, as in [Fig. 1], a voltage will be induced across the coil as determined by Faraday's law:
$$\bbox[5px,border:1px solid grey]{\mathcal{E} = N{ d\phi_B \over dt}} \tag{1}$$
where $\mathcal {E}$ is the electromotive force (EMF), N represents the number of turns of the coil, $\phi_B$ is the magnetic flux and $d \phi/dt$ is the instantaneous change in flux (in webers) linking the coil. The term linking refers to the flux within the turns of wire. The term changing simply indicates that either the strength of the field linking the coil changes in magnitude or the coil is moved through the field in such a way that the number of flux lines through the coil changes with time.
If the flux linking the coil ceases to change, such as when the coil simply sits still in a magnetic field of fixed strength, $d \phi/dt=0$, and the induced voltage $e = N(d \phi/dt) = N(0) = 0$.
The direction of the electromotive force is given by Lenz's law. The laws of induction of electric currents in mathematical form was established by Franz Ernst Neumann in 1845. Faraday's law contains information about the relationships between both the magnitudes and the directions of its variables. However, the relationships between the directions are not explicit; they are hidden in the mathematical formula.
Fig. 2:Left hand rule(magnetic induction)
Left-hand and right-hand rule for a magnetic field due to a current in a straight wire.
Fig. 3: Left-hand and right-hand rule for a magnetic field due to a current in a straight wire.

Left Hand Rule for Faraday's Law

It is possible to find out the direction of the electromotive force (EMF) directly from Faraday's law, without invoking Lenz's law. A left hand rule helps doing that, as follows;
  • Align the curved fingers of the left hand with the loop (yellow line).
  • Stretch your thumb. The stretched thumb indicates the direction of n (brown), the normal to the area enclosed by the loop.
  • Find the sign of the change in flux. Determine the initial and final fluxes (whose difference is $\Delta \phi_B$) with respect to the normal n, as indicated by the stretched thumb.
  • If the change in flux, $\Delta \phi_B$, is positive, the curved fingers show the direction of the electromotive force (yellow arrowheads).
  • If $\Delta \phi_B$ is negative, the direction of the electromotive force is opposite to the direction of the curved fingers (opposite to the yellow arrowheads).
For a tightly wound coil of wire, composed of N identical turns, each with the same $\phi_B$, Faraday's law of induction states that,
$$\bbox[5px,border:1px solid grey]{\mathcal{E} = -N {d\phi_B \over dt}} \tag{2}$$
where N is the number of turns of wire and $\phi_B$ is the magnetic flux through a single loop.
In order to understand Faraday's Law of Induction, it is important to have a basic understanding of magnetic fields. A magnetic field is often depicted as lines of magnetic flux. In the case of a bar magnet, the flux lines exit from the north pole and curve around to reenter at the south pole. In this model, the number of flux lines passing through a given surface in space represents the flux density, or the strength of the field.

How magnetic induction works?

If we run an electric current through a wire, it will produce a magnetic field around the wire. The direction of this magnetic field can be determined by the right-hand rule. If you extend your thumb and curl the fingers of your right hand, your thumb points in the positive direction of the current, and your fingers curl in the north direction of the magnetic field.
If you bend the wire into a loop, the magnetic field lines will bend with it, forming a toroid, or doughnut shape.
A current-carrying circular loop
Fig. 4: A current-carrying circular loop
If we run a current through a wire loop in a magnetic field, the interaction of these magnetic fields will exert a twisting force, or torque, on the loop causing it to rotate. However, it will only rotate so far until the magnetic fields are aligned. If we want the loop to continue rotating, we have to reverse the direction of the current, which will reverse the direction of the magnetic field from the loop. The loop will then rotate 180 degrees until its field is aligned in the other direction. This is the basis for the electric motor.
Conversely, if we rotate a wire loop in a magnetic field, the field will induce an electric current in the wire. The direction of the current will reverse every half turn, producing an alternating current. This is the basis for the electric generator. It should be noted here that it is not the motion of the wire but rather the opening and closing of the loop with respect to the direction of the field that induces the current. When the loop is face-on to the field, the maximum amount of flux passes through the loop. However, when the loop is turned edge-on to the field, no flux lines pass through the loop. It is this change in the amount of flux passing through the loop that induces the current.

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