## Electromagnetic Induction

# Motional Electromotive Force and Eddy Current

If a rod of length **l **moves perpendicular to a magnetic field** B, **with a velocity **v ,**then induced emf produced in it given by

**E = B . v × l = B v l**

If a metallic rod of length l rotates about one of its ends in a plane perpendicular to the magnetic field, then the induced emf produced across its ends is given by

**E = 1/2 B****ωl ^{2}**

If a metallic disc of radius r rotates about its own centre in a plane perpendicular to the magnetic field B, then the induced emf produced between the centre and the edge is given by

**E = 1/2 B****ωr ^{2}**

**Eddy Currents:**

If a piece of metal is placed in a varying magnetic field or rotated with high speed in a uniform magnetic field , then induced current set up in the piece are like whirlpool of air called eddy currents. These are also known as Facault’s current.

**i = - e/R**

### View the Topic in this video From 01:11 To 25;06

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1. If rod is moving by making an angle θ with the direction of magnetic field or length, induced emf *e* = *Bvl* sin θ

2. If conducting rod moves on two parallel conducting rails as in time t distance travelled by conductor = vt

Area generated *A* = *lvt*. Flux linked with this area Φ = *BA* = *Blvt*. Hence induced emf |e| = \frac{d\phi}{dt} = Bvl

3. Induced current : i = \frac{e}{R} = \frac{Bvl}{R}

4. Magnetic force : Conductor *PQ* experiences a magnetic force in opposite direction of it's motion and F_{m} = Bil = B\left[\frac{Bvl}{R}\right]l = \frac{B^{2}vl^{2}}{R}

5. Power dissipated in moving the conductor : For uniform motion of rod PQ, the rate of doing mechanical work by external agent or mech. power delivered by external source is given as P_{mech} = P_{ext} = \frac{dW}{dt} = F_{ext}.v = \frac{B^{2}vl^{2}}{R} \times v = \frac{B^{2}v^{2}l^{2}}{R}

6. Cycle wheel : In a conducting wheel each spoke of length l is rotating with angular velocity ω in a given magnetic field as

e_{net} = \frac{1}{2} B\omega l^{2}; \omega = 2\pi v

7. Induced emf in coil : Induced emf also changes in periodic manner that's why phenomenon called periodic EMI.

e = -\frac{d\phi}{dt} = NBA \ \omega \sin \omega t