First slide

AC Circuits

Question

In $L C$ circuit the inductance $L = 40 mH$ and capacitance $C = 100 μ F$ . If a voltage $V (t) = 10 \sin (314 t)$ is applied to the circuit, the current in the circuit is given as

Moderate

A

$5.2 \cos 314 t$
B

$0.52 \cos 314 t$
C

$10 \cos 314 t$
D

$0.52 \sin 314 t$

Solution

$\begin{array}{l} X_{L} = w L = 314 \times 40 \times 10^{- 3} = 12560 \times 10^{- 3} = 12.56 Ω \\ X_{C} = \frac{1}{w c} = \frac{1}{314 \times 100 \times 10^{- 6}} = \frac{1000}{314} = 31.84 Ω \end{array}$
Phaser diagram
$\begin{array}{l} V_{\max} = i_{\max} (X_{C} - X_{L}) \\ \Rightarrow 10 = i_{\max} (31.84 - 12.56) \\ \Rightarrow i_{\max} = \frac{10}{19.28} = 0.52 \\ S i n c e, X_{C} > X_{L} C u r r e n t l e a d s t h e v o l t a g e b y \frac{π}{2} \end{array}$
$i = i_{\max} \sin (w t + \frac{π}{2}) = 0.52 \sin (314 t + \frac{π}{2}) = 0.52 \cos (314 t)$

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