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# Faraday’s Laws and Lenz’s Law of Electromagnetic Induction

## Faraday’s Laws of Electromagnetic Induction

*Michael Faraday* (an English scientist) performed a series of experiments to demonstrate the phenomenon of electromagnetic induction and he summed up his conclusions into two laws, known as *Faraday's laws of electromagnetic induction.*

## First Law of Electromagnetic Induction

The first law states that *"when a magnetic flux linking a conductor or coil changes, an EMF is induced in the conductor or coil"*. Therefore, the first law tells about the condition under which the emf is induced in a conductor or coil.

## Second Law of Electromagnetic Induction

The second law states that *"The magnitude of the induced emf in the conductor or coil is directly proportional to the rate of change of magnetic flux linkages"*. Therefore, the second law gives the magnitude of induced emf in the conductor or coil.

**Explanation** − Consider a coil has N turns and the magnetic flux linking the coil changes from $\varphi_{1}$ to $\varphi_{2}$ in t seconds. Therefore,

$$\mathrm{Initial\: magnetic\: flux \:linkage,ψ_{1} = N\varphi_{1}}$$

$$\mathrm{Final \:magnetic\: flux \:linkage, ψ_{2} = N\varphi_{2}}$$

$$\mathrm{\therefore Induced \:EMF,e\propto\frac{ψ_2-ψ_1}{t}\propto\frac{N\varphi_2-N\varphi_1}{t}}$$

$$\mathrm{\Longrightarrow e=k(\frac{N\varphi_2-N\varphi_1}{t})}$$

Where k is a constant of proportionality and in SI units its value being 1. Therefore,

$$\mathrm{\Longrightarrow e=(\frac{N\varphi_2-N\varphi_1}{t})}$$

In differential form,

$$\mathrm{e=N\frac{d\varphi}{dt}}$$

The above equation represents Faraday’s second law mathematically.

## Lenz’s Law

A *German scientist, Emil Lenz*, gave a rule to determine the direction of induced EMF by the electromagnetic induction in a conductor or coil, which is stated as follows −

*"The current due to induced emf will flow in such a direction so as to oppose the cause that produces it i.e. the current due to induced emf will set up a magnetic flux to oppose the change in magnetic flux which produced it".*

*Mathematically*, the Lenz’s law is given by adding a minus in the R.H.S. of Faraday’s second law i.e.

$$\mathrm{e=-N\frac{d\varphi}{dt}}$$

Here, the minus sing shows that the induced emf opposes the changing magnetic field that induces it. Therefore, the Lenz’s law directly follows from the *law of conservation of energy* i.e. in order to induce emf in a coil or conductor, some energy must be expanded in the opposition.

## Numerical Example

A magnetic flux of 25 mWb is linked with coil of 600 turns. If this magnetic flux reversed in a time of 3 ms, calculate the induced emf in the coil.

### Solution

$$\mathrm{Change\: in \:magnetic\: flux,d\varphi=\varphi_{2} - \varphi_{1}=25-(-25)=50nmb}$$

$$\mathrm{Time\: taken\: for\: the\: change,dt=3ms}$$

$$\mathrm{\therefore Induced\: EMF,e = N\frac{d\varphi}{dt}=600\times(\frac{50\times10^{-3}{3}}\times10^{-3})=10000=10kV}$$

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