A conducting movable rod `AB` lies across the frictionless parallel conducting rails in a uniform magnetic field `vecB` whose magnitude at `t=0` is `B_(0)`. The rod `AB` is given velocity `v` right ward and it continues to move with same velocity through out the motion [The acceleration due to gravity is along negative `z-`axis, i.e. `vecg=10m//s^(2)(-hatk)`, mass of rod `=m`.] Which of the following graphs will be the best representation of magnitude of magnetic field versus time.

Options

A.
B.
C.
D.

Explanation

Since speed is constant, it means there is no induced `emf phi=l(b+vt)B`
`e=lBv+l(b+vt)(dB)/(dt)=0`
`rArr (b+vt)(dB)/(dt)=-Bv`
`rArr int_(B_(0))^(B)(dB)/(B)=-vint_(0)^(t)(dt)/(b+vt)`
`rArrlnB//B_(0)=-v[(1)/(v)ln((b+vt)/(b))]=ln((b)/(b+vt))`
`rArr B=(B_(0)b)/(b+vt)`
Second Method :
So flux does not change with time, so
`phi(1)=phi(t+dt)`
`rArrBl(b+vt)=(B+dB)l[b+v(t+dt)]`
`rArr (b+vt)dB+Bvdt=0`
`rArr (dB)/(B)=-V(dt)/(b+vt)rArrB=(B_(0)b)/(b+vt)`