Question

A parallel plate capacitor having plates of radius $6\text{ cm}$ has capacitance $100\text{ pF}$. It is connected to $230\text{ V}$ a.c. supply with angular frequency $300\text{ rad/s}$. The r.m.s. value of current is

• A. $6.9 \times 10^{-6}\text{ A}$
• B. $2.3 \times 10^{-5}\text{ A}$
• C. $6.9 \times 10^{-5}\text{ A}$
• D. $6.9 \times 10^{-7}\text{ A}$

Hint:

Notice that the question provides the physical radius of the plates ($6\text{ cm}$). This is extra information intended to complicate the question! Since the total capacitance value ($100\text{ pF}$) is already explicitly given, you can ignore the radius entirely and compute the answer directly using $I = V\omega C$.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
An alternating current supply is connected across a parallel plate capacitor. We are given the structural radius of the plates, the capacitance $C = 100\text{ pF}$, the operating effective voltage $V_{\text{rms}} = 230\text{ V}$, and the angular frequency $\omega = 300\text{ rad/s}$. We need to compute the root-mean-square (r.m.s.) value of the alternating current, $I_{\text{rms}}$.

Step 2: Key Formula or Approach:
The r.m.s. current in a purely capacitive AC circuit is governed by Ohm's law with capacitive reactance $X_C$: $$I_{\text{rms}} = \frac{V_{\text{rms}}}{X_C}$$ Since the capacitive reactance is defined as $X_C = \frac{1}{\omega C}$, we can combine these equations into a single expression: $$I_{\text{rms}} = V_{\text{rms}} \times \omega C$$

Step 3: Detailed Explanation:
Let's list and convert the parameters provided into standard SI units: $$\text{Capacitance, } C = 100\text{ pF} = 100 \times 10^{-12}\text{ F} = 10^{-10}\text{ F}$$ $$\text{Angular frequency, } \omega = 300\text{ rad/s}$$ $$\text{R.M.S. Voltage, } V_{\text{rms}} = 230\text{ V}$$ Substitute these values directly into our current equation: $$I_{\text{rms}} = 230 \times 300 \times 10^{-10}$$ Multiply the numerical constants: $$230 \times 300 = 69,000 = 6.9 \times 10^4$$ Now, combine the power of ten terms: $$I_{\text{rms}} = 6.9 \times 10^4 \times 10^{-10} = 6.9 \times 10^{-6}\text{ A}$$ This matches option (A).

Step 4: Final Answer:
The r.m.s. value of the current is $6.9 \times 10^{-6}\text{ A}$, which corresponds to option (A).