Question

\(220 \mathrm{~V}, 50 \mathrm{~Hz}\), ac is applied to a resistor. The instantaneous value of voltage is (a) \(220 \sqrt{2} \sin 100 \pi \mathrm{t}\) (b) \(220 \sin 100 \pi \mathrm{t}\) (c) \(220 \sqrt{2} \sin 50 \pi \mathrm{t}\) (d) \(220 \sin 50 \pi \mathrm{t}\)

Short answer

The correct instantaneous value of voltage is (a) \(220 \sqrt{2} \sin 100 \pi \mathrm{t}\).

Step-by-step solution

Calculate Maximum Voltage (V_m)

To find the maximum voltage, we need to use the relation between RMS voltage and maximum voltage. The relation is as follows: \[V_{rms} = \frac{V_m}{\sqrt{2}}\] Now, we can calculate the maximum voltage \((V_m)\) : \[V_m = V_{rms} \times \sqrt{2}\] \[V_m = 220\sqrt{2} \mathrm{~V}\]

Calculate the Angular Frequency (ω)

To find the angular frequency, we need to use the relation between frequency and angular frequency. The relation is as follows: \[\omega = 2\pi f\] Where \(f\) is the frequency. Now, we can calculate the angular frequency \((\omega)\): \[\omega = 2\pi (50) = 100\pi\]

Form the Instantaneous Voltage Equation

Now that we have calculated the maximum voltage \((V_m)\) and angular frequency \((\omega)\), we can write the equation of the instantaneous voltage: \[V(t) = V_m \sin(\omega t)\] \[V(t) = 220\sqrt{2} \sin(100\pi t)\] This matches with option (a). Therefore, the correct answer is: (a) \(220 \sqrt{2} \sin 100 \pi \mathrm{t}\)