Question

A $750Hz,20V(rms)$ source is connected to a resistance of $100\Omega$, an inductance of $0.1803H$ and a capacitance of $10\mu F$ all in series. The time in which the resistance (heat capacity $2J/{}^{\circ }C$ ) will get heated by $10^{\circ }C$. (assume no loss of heat to the surroundings) is close to :

• A. $418s$
• B. $245s$
• C. $348s$
• D. $365s$

Answer 1

Answer: (C)

$f=750Hz,V_{rms}=20V$
$R=100\Omega ,L=0.1803H$
$C=10\mu F,S=2J/{}^{\circ }C$
$Z=\sqrt{R^2+{\left(X_L-X_C\right)}^2}$
$=\sqrt{R^2+(\omega L-1/\omega C{)}^2}$
$=\sqrt{R^2+{\left(2\pi fL-\frac{1}{2\pi fC}\right)}^2}$
Putting values
$|Z|=834\Omega$
In AC power $P=V_{rms}{i}_{rms}\cos \varphi$
Cos $\varphi =\frac{R}{|Z|}{i}_{rms}=\frac{V_{rms}}{|Z|}$
$=\frac{V_{rms}^{2}R}{(|Z|{)}^2}$
$={\left(\frac{20}{834}\right)}^2\times 100=0.0575J/s$
$H=Pt=S\Delta \theta$
$t=\frac{2(10)}{0.0575}=348$ sec