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- Things to remember

The direction of induced e.m.f and hence current can be determined by one of the following two methods:

1. Lenz’s law which has already discussed

2. Fleming’s Right Hand Rule

Stretch the forefinger, middle finger and the thumb of the right hand mutually perpendicular to each other. If the force finger points in the direction of the magnetic field, the thumb gives the direction of the motion of the conductor then the middle finger gives the direction of the induced current.

From Faraday’s law, induced e.m.f is given by

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

It shows that e.m.f depends on the number of turns N and rate of change of magnetic flux\(\phi \). Since magnetic flux, \(\phi = BA\: \cos \: \theta \).

So magnetic flux can be changed by

- Changing the intensity of magnetic field (B).
- Changing the orientation \((\theta)\) of the coil with respect to the magnetic field.
- Changing the area of the conducting circuit.

Consider a straight conductor PQ of length l moving at right angles to a uniform magnetic field B with a velocity v. suppose the conductor moves through a small distance x in time t. then area swept out by the conductor is given by

\begin{align*} \text {Area swept} &=l \times x \\ \therefore \text {Magnetic flux cut,} \: \phi &= B \times \text {Area swept} \\ &= b\times l \times x \\ &= Blx \\\end{align*}

From Faraday’s law of electromagnetic inducton, the magnitude of induced e.m.f in the conductor is given by

\begin{align*} \\ e &= N \frac {d\phi }{dt} = N \frac {d}{dt} (Bl\: x) \\ &= N\: Bl \frac {dx}{dt} \\ \therefore e &= Bl\:v \:\:\: \left [\therefore N = 1, \: \text {and} \: \frac {dx}{dt} = v \right ] \\ \end{align*}

**Special case:**

If the conductor moves at an angle \(\theta \) to which thw conductor moves across the field is \(v\: \sin\: \theta\).

$$\therefore \;\;\; \epsilon = Bv\:l\sin \theta $$

The direction of the induced e.m.f. can be determined by Fleming's right hand rule .

Consider a rectangular coil being rotated with constant angular velocity \(\omega \) in the uniform magnetic field about an axis perpendicular to the field.

Let

\begin{align*} N &= \text {number of turns of the coil} \\ A &= \text {area of each turn} \\ B &= \text {strength of magnetic field.} \\ \end{align*}

Suppose initially i.e. at t = 0, the plane of the coil is perpendicular to the direction of the magnetic field. Let the coil rotates in anticlockwise direction through an angle \(\theta = \omega t\) in time t. At this instant, the perpendicular to the plane of the coil makes an angle \(\theta \) with the direction of he field. Therefore, at this instant, the magnetic flux \(\phi \) through each turn of the coil is given by

\begin{align*} \phi &= AB\: \cos\: \theta \\ &= AB\cos\: \omega t \\\end{align*}

Using Faraday's laws of electromagnetic induction, induced e.m.f in the coil is given by

\begin{align*} E &= -N \frac {d\phi }{dt} \\ &= -N \frac {d}{dt} (AB\cos\: \omega t), \\ \text {where N is number of turns of the coil.} \\ &= -N\: AB \frac {d}{dt}(\cos\: \omega t) \\&= -N\: AB(-sin\: \omega t)\: \omega \\ \epsilon &= N\: AB\omega\: \sin\: \omega t \dots (i) \\ \end{align*} The magnitude of induced e.m.f. will be maximum (\(\epsilon _0\))

\begin{align*}\\ \text {when} \: \sin\: \omega t = 1 \\ \text {i.e.} \: \epsilon_0 &= N\: AB\omega \\ \text {So equation} \: (i)\: \text {becomes} \\ \therefore \epsilon &= \epsilon _0\: \sin\: \omega t \dots (ii) \end{align*}

Thus a coil rotating with a constant angular velocity in a uniform magnetic field produces a sinusoidally alternating e.m.f. If a resistor of resistance R is connected across the coil, the resultant current will be sinusoidal and is given by

\begin{align*} I &= \frac {\epsilon}{R} = \frac {\epsilon _0 \sin \: \omega t}{R} \\ \therefore I &= I_0\sin\: \omega t \\ \text {where}\: \frac {\epsilon _0}{R} = I_0 \: \text {is maximum value} \\ \end{align*}

Such a rotating coil in a uniform magnetic field is the basic operating principle of an a.c. generator.

Reference

Manu Kumar Khatry, Manoj Kumar Thapa, et al.*Principle of Physics*. Kathmandu: Ayam publication PVT LTD, 2010.

S.K. Gautam, J.M. Pradhan. *A text Book of Physics*. Kathmandu: Surya Publication, 2003.

According to Faraday's law , e.m.f depends on the number of turns N and rate of change of magnetic flux.

If the force finger points in the direction of the magnetic field, the thumb gives the direction of the motion of the conductor then the middle finger gives the direction of the induced current.

From Faraday’s law, induced e.m.f is given by \( E = -N \frac {d\phi }{dt} \).

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