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A copper wire ab of length l, resistance...

A copper wire `ab` of length `l`, resistance `r` and mass `m` starts sliding at `t=0` down a smooth, vertical, thick pair of connected condcuting rails as shown in figure.A uniform magnetic field `B` exists in the space in a direction perpendicular to the plane of the rails which options are correct.

A

The magnitude and direction of the induced current in the wire when speed of the wire `v` is `(vBl)/r`, `a` to b

B

The downward acceleration of the wire at this instant `g-(B^(2)l^(2))/(mr)v`.

C

The velocity of the wire as a function of time `v_(m)(1-e^(-"gt"//v_(m)))`(where `v_(m)=(mgr)/(B^(2)l^(2)))`

D

The displacement of the wire as a function of time `v_(m)t-v_(m)^(2)/"g" (1-e^(-"gt"//v_(m)))`(where `v_(m)=(mgr)/(B^(2)l^(2)))`

Text Solution

Verified by Experts

The correct Answer is:
A, B, C, D
(A)`i=(BvL)/r(a "to"b)` , (B)`mg-(B^(2)l^(2)v)/r=m(dv)/(dt) therefore "acc"^(n)=(dv)/(dt)=g-(B^(2)l^(2)v)/(mr)`
`mg-Bil=(Bl(Bv_(m)l))/r,v_(m)=(mgr)/(B^(2)l^(2))`
( C)` underset(0)overset(v)int(bv)/(g-(B^(2)l^(2)v)/(mr))=underset(0)overset(t)int(bt) therefore v=v_(m)(1-e^((-"gt")/v_(m)))` (D) `(ds)/(dt)=v underset(0)overset(s)intds=underset(0)overset(t)intv_(m)(1-e^((-"gt")/v_(m)))dt`
`s=v_(m)t-(1-e^((-"gt")/v_(m))) rArr ` (g) Heat produced per sec=`i^(2)r=((B^(2)V_(m)^(2)l^(2))/r^(2))r=mgv_(m)`
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