Question 722: Electrostatics & Capacitance

Question

Two capacitors with capacitance values C
1
= 2000 $$ \pm $$ 10 pF and C
2
= 3000 $$ \pm $$ 15 pF are connected in series. The voltage applied across this combination is V = 5.00 $$ \pm $$ 0.02 V. The percentage error in the calculation of the energy stored in this combination of capacitors is ________.

Accepted Answer

Given that, $C_1=(2000 \pm 10) \mathrm{pF} ; \quad C_2=(3000 \pm 15) \mathrm{pF}$ and $V=(5.00 \pm 8.02) \mathrm{V}$.
Combine capacitance of series combination
$$
\begin{aligned}
\frac{1}{C} & =\frac{1}{C_1}+\frac{1}{C_2} ~~~~....(1)\\
\Rightarrow & \frac{1}{C}=\frac{1}{2000}+\frac{1}{3000}=\frac{(6+4)}{12000} \\
\frac{1}{C} & =\frac{10}{12000} \Rightarrow C=1200 \mathrm{pF}
\end{aligned}
$$
Differentiating Eq. (1), we get
$$
\begin{aligned}
-\frac{1}{C^2} \cdot d C & =-\frac{1}{C_1^2} d C_1+\left(-\frac{1}{C_2^2}ight) \cdot d C_2 \\
\Rightarrow \frac{d C}{C^2} & =\left(\frac{d C_1}{C_1^2}+\frac{d C_2}{C_2^2}ight) \\
\Rightarrow \frac{d C}{C} & =\left[\frac{d C_1}{C_1^2}+\frac{d C_2}{C_2^2}ight] \cdot C \\
\Rightarrow \frac{d C}{C} & =\left[\frac{10}{(2000)^2}+\frac{15}{(3000)^2}ight] 	imes 1200 \\
\Rightarrow \frac{d C}{C} & =\left[\frac{10}{4} 	imes 10^{-6}+\frac{15}{9} 	imes 10^{-6}ight] 	imes 1200 \\
& =\left[\frac{10}{4}+\frac{15}{9}ight] 	imes 12 	imes 10^{-4}
\end{aligned}
$$
$$
\Rightarrow \frac{d C}{C}=[2.5+1.66] 	imes 12 	imes 10^{-4}=[4.17 	imes 12] 	imes 10^{-4}
$$
And
$$
\frac{d V}{V}=\frac{0.02}{5}
$$
Energy stored in the capacitor is given by
$$
U=\frac{1}{2} C V^2
$$ÃÂ ÃÂ ÃÂ ÃÂ .....(2)
Differentiating Eq. (2), we get
$$
\begin{aligned}
\frac{d U}{U} 	imes 100 & =\frac{d C}{C} 	imes 100+2 \frac{d V}{V} 	imes 100 \\
& =\left[50.04 	imes 10^{-4} 	imes 100+\frac{2 	imes 0.02}{5} 	imes 100ight] \\
\left(\frac{d U}{U}ight)^{100} & =[0.5004+0.8] \% \Rightarrow 1.30 \%
\end{aligned}
$$

Electrostatics & Capacitance

Two capacitors with capacitance values C 1 = 2000 $ \pm 10 pF and C 2 = 3000 \pm 15 pF are connected in series. The voltage applied across this combination is V = 5.00 \pm $ 0.02 V. The percentage error in the calculation of the energy stored in this combination of capacitors is ________.

Integer / Numeric Answer Type