Question

The rms value of an ac current of \(50 \mathrm{~Hz}\) is 10 amp. The time taken by the alternating current in reaching from zero to maximum value and the peak value of current will be, (a) \(2 \times 10^{-2}\) sec and \(14.14 \mathrm{amp}\) (b) \(1 \times 10^{-2}\) sec and \(7.07 \mathrm{amp}\) (c) \(5 \times 10^{-3}\) sec and \(7.07\) amp (d) \(5 \times 10^{-3} \mathrm{sec}\) and \(14.14 \mathrm{amp}\)

Short answer

The peak value of current is \(14.14 \mathrm{amp}\) and the time taken to reach the maximum value is \(5 \times 10^{-3} \mathrm{sec}\). Therefore, the correct answer is (d) \(5 \times 10^{-3} \mathrm{sec}\) and \(14.14 \mathrm{amp}\).

Step-by-step solution

Recall the rms and peak value relationship

We will need to analyze the relationship between the rms and the peak value of the AC current. The relationship is given by \(I_{rms} = \frac{I_{peak}}{\sqrt{2}}\). We will use this equation to find the peak current value given the rms value.

Calculate the peak value

We are given the rms value, \(I_{rms} = 10 \mathrm{~amp}\). We can now calculate the peak value of current using the equation from Step 1: \(I_{peak} = I_{rms} \times \sqrt{2} = 10 \times \sqrt{2} = 14.14 \mathrm{~amp}\).

Recall the formula for time period of AC current

To find the time taken for the AC current to reach its maximum value, we will first have to calculate the time period of the given AC current. The formula to calculate the time period of an AC current is \(T = \frac{1}{f}\), where \(T\) is the time period and \(f\) is the frequency in Hertz.

Calculate the time period

We are given the frequency, \(f = 50 \mathrm{~Hz}\). We can now calculate the time period, \(T\), using the formula from Step 3: \(T = \frac{1}{f} = \frac{1}{50} = 0.02 \mathrm{~sec}\).

Calculate the time taken to reach the maximum value

We know the time taken for one full cycle of the AC current is \(T = 0.02 \mathrm{~sec}\). Since the AC current reaches its peak value exactly one-quarter of the way through its cycle, we can calculate the time taken to reach the maximum value by dividing the time period by 4: \(Time~Taken = \frac{T}{4} = \frac{0.02}{4} = 5 \times 10^{-3} \mathrm{~sec}\).

Choose the correct answer from the options

With the calculated values of \(I_{peak} = 14.14 \mathrm{~amp}\) and the time taken to reach maximum value as \(5 \times 10^{-3} \mathrm{~sec}\), we can choose the correct answer from the given options: (d) \(5 \times 10^{-3} \mathrm{sec}\) and \(14.14 \mathrm{amp}\) is the correct answer.