JEE Main · 2021 · 27 Aug 2021 · Shift 2 · Alternating Current

An ac circuit has an inductor and a resistor of resistance

R

in series, such that

X_{L} = 3 R

. Now, a capacitor is added in series such that

X_{C} = 2 R

. The ratio of new power factor with the old power factor of the circuit is

5 : x

The value of

x

is ____.

Answer: 1

Solution

Initial power factor (LR circuit):

cos ϕ_{1} = \frac{R}{R ^{2} + X _{L}^{2}} = \frac{R}{R ^{2} + 9 R ^{2}} = \frac{1}{10}

. After adding capacitor (LCR circuit):

cos ϕ_{2} = \frac{R}{R ^{2} + ( X _{L} - X _{C} ) ^{2}} = \frac{R}{R ^{2} + ( 3 R - 2 R ) ^{2}} = \frac{R}{R ^{2} + R ^{2}} = \frac{1}{2}

. Ratio:

\frac{c o s ϕ _{2}}{c o s ϕ _{1}} = \frac{1/ 2}{1/ 10} = \frac{10}{2} = 5

. Given ratio is

5 : x

, so

x = 1

JEE Main · 2021 · 27 Aug 2021 · Shift 2 · Numerical answer