Question

In an AC circuit an alternating voltage, $E=200\sqrt{2}\sin 100t\,\,volts$ is connected to a capacitor of capacity $1\,\mu F$. The rms value of the current in the circuit is:
$\begin{align}
  & A.\,\,20\,mA \\
 & B.\,\,10\,mA \\
 & C.\,\,100\,mA \\
 & D.\,\,200\,mA \\
\end{align}$

Answer 1

Hint: To get the rms value of the current in the circuit, we should first know the rms value of the voltage in the circuit as well the Impedance, so that using this value we can find the rms value of the current in the circuit. Since, it is an AC circuit so the alternating voltage formula and value that is given can help to find the values that we need.

Formula used:
$\begin{align}
  & E={{E}_{0}}\sin \omega t \\
 & {{E}_{rms}}=\dfrac{{{E}_{0}}}{\sqrt{2}} \\
 & {{X}_{C}}=\dfrac{1}{\omega C} \\
 & {{I}_{rms}}=\dfrac{{{E}_{rms}}}{{{X}_{C}}} \\
\end{align}$

Complete Step-by-Step solution:
From the question, we know that:
$C=1\,\mu F=1\times {{10}^{-6}}\,F$, and
$E=200\sqrt{2}\sin 100t\,\,volts$
According to the alternating voltage formula, we know that:
$E={{E}_{0}}\sin \omega t$
When we compare both the equation of alternating voltage that is mentioned above, we can deduce that:
${{E}_{0}}=200\sqrt{2}\,volts$ and $\omega =100\,{rad}/{sec}\;$
Now,
The rms value of the voltage in the circuit is:
$\begin{align}
  & {{E}_{rms}}=\dfrac{{{E}_{0}}}{\sqrt{2}} \\
 & \Rightarrow {{E}_{rms}}=\dfrac{200\sqrt{2}}{\sqrt{2}} \\
 & \Rightarrow {{E}_{rms}}=200\,volts \\
\end{align}$
Also,
The capacitive reactance in the circuit is:
$\begin{align}
  & {{X}_{C}}=\dfrac{1}{\omega C} \\
 & \Rightarrow {{X}_{C}}=\dfrac{1}{100\times {{10}^{-6}}}\Omega \\
 & \Rightarrow {{X}_{C}}={{10}^{4}}\Omega \\
\end{align}$
Hence,
The rms value of the current in the circuit can be found using the below formula:
${{I}_{rms}}=\dfrac{{{E}_{rms}}}{{{X}_{C}}}$
After substituting the value’s in the above equation, we get:
$\begin{align}
  & {{I}_{rms}}=\dfrac{200}{{{10}^{4}}}\,A \\
 & \therefore {{I}_{rms}}=20\,mA \\
\end{align}$
Therefore, the correct answer is Option (A).

Additional Information:
The root mean square (abbreviated RMS or rms) is a statistical measure of the magnitude of a varying quantity. We use the root mean square to express the average current or voltage in an AC system. The RMS current and voltage (for sinusoidal systems) are the peak current and voltage over the square root of two.

Note:
This is a little mathematical question, as many formulas are being used here, but if you see properly all the formulas are related to each other in some or the other way. It becomes easy to get the solution for this question, if all the formulas are known already and the concepts are clear. Once all the values are found, then getting the answer is easy.