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.

Key concepts

RMS Value

The term "RMS" stands for "Root Mean Square," and it's a way to measure the effective value of an alternating current (AC). RMS value allows us to compare AC with direct current (DC) in terms of its ability to do work or provide power. In the context of AC, RMS value is significant because AC changes its direction and magnitude continuously over time. The formula for calculating the RMS value of current or voltage is derived from statistical methods and helps in understanding the "average" value throughout the AC cycle.- For AC current, the RMS value is calculated as: \[ I_{rms} = \frac{I_{peak}}{\sqrt{2}} \] Here, \(I_{rms}\) is the effective current, and \(I_{peak}\) is the maximum or peak current. The RMS value provides a method to quantify AC by equating it to a DC value that has the same ability to transfer energy.

Peak Value

The peak value of an AC current represents its maximum amplitude during a cycle. It's the highest amount of current that flows at any point in time within one complete cycle of the AC waveform. Understanding the peak value is crucial because it determines the maximum potential that an AC device must handle.The relationship between RMS and peak values is significant for AC calculations. The RMS value is always less than the peak value because the waveform averages out over a cycle.- To calculate the peak current from the RMS value, the formula is: \[ I_{peak} = I_{rms} \times \sqrt{2} \] In practical terms:- If \(I_{rms}\) is known, finding \(I_{peak}\) helps determine the device's capacity and design standards for safety measures.

Frequency

Frequency in AC current refers to how often the current completes a full cycle within one second, measured in hertz (Hz). This concept is vital because it dictates the timing and behavior of the AC waveform and is crucial for ensuring compatibility with electrical devices.- The formula for frequency is: \[ f = \frac{1}{T} \] Where:- \( f \) is the frequency,- \( T \) is the time period, which is the duration of one complete cycle.In the context of the given exercise, a frequency of \(50 \text{ Hz}\) signifies that the AC current completes 50 cycles per second. Frequency is essential in AC systems because it impacts the design and operation of electrical equipment.

Time Period

The time period of an AC current is the time it takes to complete one full cycle of the waveform. It is directly related to frequency, and it plays a crucial role in understanding how AC current operates over time. - The formula to find the time period \(T\) is: \[ T = \frac{1}{f} \] Where:- \( T \) is the time period in seconds,- \( f \) is the frequency in hertz.The concept of time period is vital, as it helps predict the behavior of the AC waveform. For example, knowing the time period allows us to find out how long the waveform takes to peak or return to zero. - In the exercise, with \(f = 50 \text{ Hz}\), the time period \(T\) is \(0.02 \text{ sec}\). This explains how quickly the AC waveform reaches its peak value of current. Understanding the time period helps in effectively designing circuits and planning the synchronization of AC in various applications.