Question 216: AC fundamentals

Question

To light, a $$50 \mathrm{~W}, 100 \mathrm{~V}$$ lamp is connected, in series with a capacitor of capacitance $$\frac{50}{\pi \sqrt{x}} \mu F$$, with $$200 \mathrm{~V}, 50 \mathrm{~Hz} \,\mathrm{AC}$$ source. The value of $$x$$ will be ___________.

Accepted Answer

$${X_C} = {1 \over {wc}} = {{\pi \sqrt x } \over {2\pi  	imes 50 	imes 50}} 	imes {10^6}$$
$$v_R^2 + v_C^2 = {(200)^2}$$
$$v_C^2 = {200^2} - {100^2}$$
$${v_C} = 100\sqrt 3 \,V$$
$${v_R} = 100\,V$$
$$P = {{{V^2}} \over R}$$
$$R = {{100 	imes 100} \over {50}} = 200\,\Omega $$
$${i_{rm}} = {1 \over 2}\,A$$
$${1 \over 2} 	imes {x_C} = 100\sqrt 3  \Rightarrow {10^{ - 6}} 	imes {{\sqrt x } \over {5000}} 	imes {1 \over 2} = 100\sqrt 3 $$
$${{{{10}^{ - 6}}\sqrt x } \over {10000 	imes 100}} = \sqrt 3 $$
$$\sqrt x  = \sqrt 3 $$
$$x = 3$$

AC fundamentals

To light, a $50 \mathrm{~W}, 100 \mathrm{~V} lamp is connected, in series with a capacitor of capacitance \frac{50}{\pi \sqrt{x}} \mu F, with 200 \mathrm{~V}, 50 \mathrm{~Hz} \,\mathrm{AC} source. The value of x$ will be ___________.

Integer / Numeric Answer Type