question_answer

An arc lamp requires a direct current of 10 A at 80 V to function. If it is connected to a 220 V (r.m.s.), 50 Hz ac supply, the series inductor needed for it to work is close to

A) 80 H      

B) 0.08 H

C) 0.044 H        

D) 0.065 H

Correct Answer: D

Solution :

[d]: For a dc source \[I=10A,V=80V\]             Resistance of the arc lamp,\[R=\frac{V}{I}=\frac{80}{10}=8\Omega \] For an ac source, \[{{\varepsilon }_{rms}}=220V\] \[\upsilon =50Hz\] \[\omega =2\pi \times 50=100\pi \,\text{rad}\,{{\text{s}}^{-1}}\]       Arc lamp will glow if I = 10 A,     \[\therefore \]\[I=\frac{{{\varepsilon }_{rms}}}{\sqrt{{{R}^{2}}+{{\omega }^{2}}{{L}^{2}}}}\]or\[{{R}^{2}}+{{\omega }^{2}}{{L}^{2}}={{\left( \frac{{{\varepsilon }_{rms}}}{I} \right)}^{2}}\] or\[{{8}^{2}}+{{(100\pi )}^{2}}{{L}^{2}}={{\left( \frac{220}{10} \right)}^{2}}\]or\[{{L}^{2}}=\frac{{{22}^{2}}-{{8}^{2}}}{{{(100\pi )}^{2}}}\] \[\therefore \]\[L=\frac{\sqrt{30\times 14}}{100\pi }=0.065H\]