What is the general formula for induced emf of a conducting rod moving on a ? shaped conductor?

A conducting rod of length ‘l’ is moving smoothly on the parallel rails of a π shaped conductor with a constant velocity (v). As an ext-agency is pulling the rod to the right, \(therefore\)these is a rate of change of area \(ltimes v\) of the loop as a result the rate of change of magnetic flux is \(B,l,v\) \(therefore varepsilon =left| frac{dphi }{dt} right|=B,l,v\)

In case of a conducting rod moving on a \(pi -\)shaped conductor, only one side of the loop is moving. But now, we are considering the general case where all sides of the loop are moving in different direction with different velocities.

A part of the loop \(overrightarrow{dl}\) is moving with a velocity ‘v’ as shown in the figure.

In a time dt, \(overrightarrow{dl}\)sweeps out an area shown by parallelogram GFMN shown in the figure.

\(left| overrightarrow{da} right|=Vdtleft( dlsin theta right)\) \(left| frac{overrightarrow{da}}{dt} right|=Vdl,sin theta\) \(frac{overrightarrow{da}}{dt}=overrightarrow{dl}times overrightarrow{V}\) \(overrightarrow{B}.frac{overrightarrow{da}}{dt}=overrightarrow{B}.left( overrightarrow{dl}times overrightarrow{V} right)\) \(=left( overrightarrow{dl}times overrightarrow{V} right).,overrightarrow{B}\) \(=left( overrightarrow{V}times overrightarrow{B} right).,overrightarrow{dl}\)

If all sides of the loop are moving in different direction, then the induced EMF

\(varepsilon =oint{left( overrightarrow{V}times overrightarrow{B} right)}overrightarrow{dl}\)

This is the general formula for induced EMF due to time varying area of the loop.

Special case:

If all parts of the loop are moving with uniform translational velocity of in an uniform magnetic field ‘B’, then the loop observes no change in magnetic flux. Passing through it & hence the induced emf is equal to zero.

It is similar to the rigid loop moving in a uniform magnetic field

\(therefore\)same flux passes through it.

\(varepsilon =oint{overrightarrow{V}times overrightarrow{B}}.,overrightarrow{dl}\) [=overrightarrow{V}times overrightarrow{B}.oint{overrightarrow{dl}}=0]

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