Calculate the peak current in a 2.5-k Ω resistor connected to a 220-V rms ac source.

Verified step by step guidance

1. Step 1: Understand the given values and the formula to use. The problem provides the resistance \( R = 2.5 \; \text{k}\Omega = 2500 \; \Omega \) and the root mean square (rms) voltage \( V_{\text{rms}} = 220 \; \text{V} \). The goal is to calculate the peak current \( I_{\text{peak}} \).
2. Step 2: Recall the relationship between rms voltage and peak voltage. The peak voltage \( V_{\text{peak}} \) is related to the rms voltage by the equation: \( V_{\text{peak}} = \sqrt{2} \cdot V_{\text{rms}} \).
3. Step 3: Use Ohm's Law to relate the peak current to the peak voltage and resistance. Ohm's Law states \( I = \frac{V}{R} \). For peak current, this becomes \( I_{\text{peak}} = \frac{V_{\text{peak}}}{R} \).
4. Step 4: Substitute \( V_{\text{peak}} = \sqrt{2} \cdot V_{\text{rms}} \) into the equation for \( I_{\text{peak}} \). This gives \( I_{\text{peak}} = \frac{\sqrt{2} \cdot V_{\text{rms}}}{R} \).
5. Step 5: Plug in the given values \( V_{\text{rms}} = 220 \; \text{V} \) and \( R = 2500 \; \Omega \) into the equation \( I_{\text{peak}} = \frac{\sqrt{2} \cdot 220}{2500} \). Simplify the expression to find the peak current.

Key Concepts

Ohm's Law

Ohm's Law states that the current (I) flowing through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the resistance (R) of the conductor. It is mathematically expressed as I = V/R. This fundamental principle is essential for calculating current in electrical circuits.

RMS Voltage

RMS (Root Mean Square) voltage is a way of expressing the effective value of an alternating current (AC) voltage. It represents the equivalent direct current (DC) voltage that would deliver the same power to a resistor. For a sinusoidal AC source, the RMS voltage is approximately 0.707 times the peak voltage, making it crucial for calculations involving AC circuits.

Peak Current

Peak current refers to the maximum instantaneous current that flows through a circuit during one cycle of an AC waveform. It can be calculated from the RMS current using the relationship I_peak = I_rms × √2. Understanding peak current is important for analyzing the performance and safety of electrical components in AC circuits.