Step 1: Understanding the Concept:
Alternating Current (AC) is a type of electrical current in which the direction of the flow of electrons switches back and forth at regular intervals or cycles.
Unlike Direct Current (DC), which flows in a single constant direction, AC current follows a sinusoidal waveform in most power distribution systems.
Because the current value is constantly changing, we need specific ways to measure its "effective" strength.
The Peak Current (\(I_{0}\)) represents the maximum value the current reaches during one cycle, which is the amplitude of the sine wave.
However, most electrical appliances are rated using the Root Mean Square (RMS) value (\(I_{rms}\)).
The RMS value is the equivalent DC current that would produce the same amount of heat energy in a resistor as the AC current over one full cycle.
Mathematically, it is found by taking the square root of the mean of the squares of the instantaneous values of the current over a period.
For a standard sinusoidal wave, the RMS value is always less than the peak value because the current spends most of its time below the peak level.
Step 2: Key Formula or Approach:
The mathematical relationship between the peak current and the RMS current for a sinusoidal wave is derived from calculus.
The fundamental formula is:
\[ I_{rms} = \frac{I_{0}}{\sqrt{2}} \]
Since the question asks us to find the peak current given the RMS value, we rearrange the formula as:
\[ I_{0} = I_{rms} \times \sqrt{2} \]
Here, \(\sqrt{2}\) is a mathematical constant approximately equal to 1.414.
Step 3: Detailed Explanation:
The problem provides us with the following data:
Given RMS current, \(I_{rms} = 5 \text{ A} \).
To find the maximum current (peak current), we perform the following calculation:
\[ I_{0} = 5 \times \sqrt{2} \]
\[ I_{0} = 5 \times 1.4142 \]
By multiplying these two values:
\[ I_{0} = 7.071 \text{ A} \]
Rounding this to two decimal places, we obtain 7.07 A.
This means that although the "effective" current doing work in the circuit is 5 A, the actual electron flow reaches a high point of 7.07 A twice in every cycle (once in the positive direction and once in the negative direction).
Understanding this difference is critical for electrical safety and component selection.
For instance, insulation in wires must be able to withstand the peak voltage and peak current, even if the RMS values are lower.
Circuit breakers and fuses are also designed with these variations in mind.
In this specific problem, our calculated value of 7.07 A matches exactly with the provided option (C).
Step 4: Final Answer:
The peak current is calculated to be 7.07 A.
Therefore, the correct option is (C).