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A wire is bent into three circular segm...

A wire is bent into three circular segment of radius `r = 10 cm` as shown in. Each segment is a quadrant of a circle `ab` lying in the `x-y` plane, `bc` lying in the `y-z` plane and `ca` lying in the `z-x` plane.
(a) if a magnetic field `B` points in the positive `x` direction, what is the magnitude of the emf developed in the wire when `B` increases at the rate of `3 mTs^(-1)`?
(b) What is the direction of teh current in the segment `bc`.

Text Solution

Verified by Experts

The correct Answer is:
(a) `(3pi)/(4)xx 10^(-5)V`; (b) from c to b
(a) The given loop can be broken into three areas, each quarter circle, lying in `xy, yz and zx` plane. Since magnetic field is along `x` axis, so the flux through the areas in `xyand zx` plane will be zero. Flux will be only through the area in `yz` planes.
So net flux: `phi = (pir^(2))/(4)B = (pi)/(4)(0.10)^(2)B`
Induced emf : `e = (dphi)/(dt) = (pi)/(4)(0.10)^(2)(dB)/(dt)`
`rarr` `e = (pi)/(400) xx 3 xx 10^(-3) = (3pi)/(4) xx 10^(-5) v`.
(b) Due to this induced emf, current in the loop should flow from `c to b` as per the lenz's law, so that induced current opposes the increasing flux.
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