Question

A bulb of rating (100W, 200V) is connected across rails by a wire as shown in figure. A rod of mass 1 kg and length l = 1m is placed on the rails and is pulled by a constant force F. If uniform magnetic field B = 2T exists in the space perpendicular to the plane, then

• A. brightness of the bulb increases linearly with time.
• B. maximum force applied for which bulb does not fuse is 1N.
• C. if F = 1N, bulb never gives output power of 100 W.
• D. if F = 0.5 N, bulb can give output power of 50 W.

Answer 1

Answer: (B), (C)

if F = 1N, bulb never gives output power of 100 W.


Induced EMF = $\epsilon$
$F_{net}=F-F_B=F-ilB$
$ma=F-ilB=F-\frac{\epsilon }{R}lB=F-\frac{l^2{B}^{2}v}{R}$
$\frac{dv}{dt}=a=\frac{FR-l^2{B}^{2}v}{mR}$
$\frac{dv}{FR-l^2{B}^{2}v}=\frac{dt}{mR}\Rightarrow$
On solving velocity comes but to be increasing exponentialy with time. The current is proportional to velocity, so brightness increases exponentialy with time.
For $\epsilon =200\Rightarrow i=\frac{1}{2}A$
By $F_B=ilB=\frac{1}{2}\times 1\times 2=1N$
Therefore, $F_{max}=1N$

If $F=F_{max}=1N,i=\frac{1}{2}A$ at $t=\infty$, so we never get 100 w output power in a finite time.

If $F=0.5N,i=\frac{1}{4}$ &
Power $=i^{2}R=25$ Watt