Question

A rod of mass `m` and resistance `r` is placed on fixed, resistanceless, smooth conducting rails (closed by a resistance `R`) and it is projected with an initial velocity `u`.Find its velocity as a function of time.

Answer 1

Let at an instant the velocity of the rod be v. The emf induced in the rod will be vBl. The electrically equivalent circuit is shown in the following diagram.

`:.` Current in the circuit `i=(Blv)/(R+r)`
At time t
Magnetic force acting on the rod is F=ilB, opposite to the motion of the rod.

`ilB=-m(dV)/(dt)`.....(1)
`i=(Blv)/(R+r)`....(2)
Now solving these two equation
`(B^(2)l^(2)V)/(R+r)=-m*(dV)/(dt)`
`-(B^(2)l^(2))/((R+r)m)*dt=(dV)/(V)`
let `(B^(2)l^(2))/((R+r)m)=k`
`-K.dt=(dV)/(V)`
`int_(u)^(v)(dV)/(V)=int_(0)^(t)-K.dt`

In `((v)/(u))=-Kt`
`V=ue^(-Kt)`