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A square loop of side 12 cm with its sid...

A square loop of side `12 cm` with its sides parallel to `x` and `y`- axes is moved with a velocity `8 cm//s` along positive `x`-direction in an environment containing magnetic field along `+ve` z-direction. The field has a gradient of `10^(-3) "tesla"//"em"` along `-ve` x-direction (increasing along `-ve` x-axis) and also decreases with time at the rate of `10^(-3) "tesla"//s`. The emf induced in the loop is

Text Solution

Verified by Experts

The magnetic field in loop varies with position .x. of loop and also with time ismultaneously.
The rate of change of flux due to variation of .b.
with time is `(dphi)/(dt) =Axx(dB)/(dt)`
The rate of change of flux due to variation B with position .x. is
`(dphi)/(dt)=Axx(dB)/(dt)=A(dB)/(dx)xx(dx)/(dt)=A(dB)/(dx)xxv`
Since both cause decrease in flux, the two effects will add up
`therefore` The net emf induced
`e=(dphi)/(dt) =A(dB)/(dt) +A(dB)/(dx)xxv=A[(dB)/(dt)+v.(dB)/(dx)]`
`=144 xx10^(-4)[10^(-3)+8xx10^(-3)]`
`144 xx 9xx10^(-7)=129.6xx10^(-6) V`
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