The figure given below shows an LCR series circuit with two switches and . When switch is closed keeping open, the phase difference

Question

The figure given below shows an LCR series circuit with two switches $S_{1}$ and $S_{2}$ . When switch $S_{1}$ is closed keeping $S_{2}$ open, the phase difference $(ϕ)$ between the current and source voltage is $30 °$ and phase difference is $60 °$ when $S_{2}$ is closed keeping $S_{1}$ open. The value of $(3 L_{1} - L_{2})$ is ______ H. [2026]

Smart Achievers · Accepted Answer

(2)

Case - I

$ω = 300 rad/sec$

$S_{2}$

$\tan 30 ° = \frac{| X_{L_{1}} - X_{C} |}{R}$

$\frac{1}{\sqrt{3}} = \frac{| X_{L_{1}} - \frac{100}{3} |}{R}$ $. . . (i)$

Case - II

$\tan 60 ° = \frac{| X_{L_{2}} - X_{C} |}{R}$

$\sqrt{3} = \frac{| X_{L_{2}} - \frac{100}{3} |}{R}$ $. . . (i i)$

$\frac{1}{\sqrt{3} \times \sqrt{3}} = \frac{X_{L_{1}} - \frac{100}{3}}{X_{L_{2}} - \frac{100}{3}}$

$3 X_{L_{1}} - 100 = X_{L_{2}} - \frac{100}{3}$

$3 X_{L_{1}} - X_{L_{2}} = 100 - \frac{100}{3} = \frac{200}{3}$

$3 L_{1} - L_{2} = \frac{200}{3 ω} = \frac{200}{3 \times 300} = \frac{2}{9}$

Solution

1 $\frac{9}{2}$

2 $\frac{2}{9}$

3 $\frac{1}{3}$

4 $3$

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Solution

1 92

2 29

3 13

4 3

Ans. (2) Case - I ω=300 rad/sec XC=1ωC=1300×100×10-6=10.03=1003 Ω tan30°=|XL1-XC|R 13=|XL1-1003|R ...(i) Case - II tan60°=|XL2-XC|R 3=|XL2-1003|R ...(ii) 13×3=XL1-1003XL2-1003 3XL1-100=XL2-1003 3XL1-XL2=100-1003=2003 3L1-L2=2003ω=2003×300=29

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1 $\frac{9}{2}$

2 $\frac{2}{9}$

3 $\frac{1}{3}$

4 $3$