Question

A constant force $F$ is being applied on a rod of length $l$ kept at rest on two parallel conducting rails connected at ends by resistance $R$ in uniform magnetic field $B$ as shown.

• A. the power delivered by force will be constant with time.
• B. the power delivered by force will be increasing first and then it will decrease
• C. the rate of power delivered by the external force will be increasing continuously
• D. the rate of power delivered by external force will be decreasing continuously before becoming zero.

Answer 1

The correct option is D the rate of power delivered by external force will be decreasing continuously before becoming zero.

$\text{Emf}$ generated in the rod is $e=BVl$

Current through the resistance is $I=\frac{BVl}{R}$

Magnetic force on rod is $IlB$
Using FBD and balancing forces, we
$F-IBl=ma=m\frac{dV}{dt}$
$F-\frac{B^2{l}^{2}V}{R}=m\frac{dV}{dt}$
on solving above differential equation, we get
$\Rightarrow$ $V=V_m[1-e^{\frac{-t}{\tau }}]$
at $t\to \infty$ $V=V_m$(constant)

$(a,b)$ $P=F.V$
As $V$ is variable, $P$ is variable exponentially
$\to (a,b)$ are incorrect

$(c,d)\frac{dP}{dt}=F\frac{dV}{dt}=\frac{F{V}_m{e}^{\frac{-t}{\tau }}}{\tau }$
So, rate at which power changes is exponentially decreasing with time.
Only $\left(d\right)$ Correct