Faraday’s Law

An emf and a current are induced in a circuit whenever the magnetic flux through this circuit changes. This phenomenon is known as the electromagnetic induction. A law showing the relationship between the induced emf and the changing magnetic flux in any loop, including closed circuit is called as the Faraday’s law. This law is referred to as the central principle of electromagnetic induction.

Faraday did experiments on moving a magnet to a loop of wire connected on an ammeter and without any source of emf. As the magnet is moved into the loop, the ammeter reading shows that a current is present on the loop, or the term is current is induced in the loop. When the magnet is not moving, even if its inside the loop, there is no current induced on the loop. Lastly, as the magnet is moved away from the loop, the ammeter shows that a current is induced with a value having opposite sign with that of the induced current as the magnet moves toward the loop.

Through his experiments, Faraday then concluded that as the magnetic field through the loop changes, electric current can be induced into the loop. This induced current is present only if the magnet is in motion, that is, when the magnetic field is changing. If the magnet is stationary, meaning the magnetic field is constant or not changing, and the magnetic field reaches a steady value, there will be no current induced into the loop or the current in the loop disappears. Thus, it is normal to say that there is an emf induced in the loop due to the changing magnetic field.

The Faraday’s law of induction is written mathematically as

\(\varepsilon = -{d\Phi_B \over dt}\)     (1)

Equation (1) implies that when magnetic flux through the loop changes with time, an emf is induced on it. \(\Phi_B\) is the magnetic flux of the loop which is equal to \(\int \vec B \cdot d\vec A\). The above equation is true to every loop of wire.

If there are N loops of wire, that is, a coil, with the same area, the magnetic flux at every loop is \(\Phi_B\) then an emf is induced in each loop of the coil. In this case, the loops are in series, thus, the total emf in the coil is the sum of the emfs of each loop. Mathematically, this can be expressed as

\(\varepsilon = -N {d\Phi_B \over dt}\)     (2)

In the case of a loop with an area A in a uniform magnetic field, the magnetic flux through the loop will be equal to \(BA \cos \theta\), in which \(\theta\) is the angle between the magnetic field and the normal to the loop, the induced emf is given by the equation

\(\varepsilon = -{d \over dt} BA \cos \theta\)     (3)

The above equation implies that there are different ways in which the emf can be induced in a circuit. It may be the magnitude of the magnetic field, the area enclosed by the loop, or the angle between the magnetic field and the normal to the loop can change with time or combination of any of these factors.