What is the rms current through a resistor of `50 Omega` when an alternating voltage given by `V 140 sin(314)t` is connected to it?

Options

A. `1.5A`
B. `1.98A`
C. `2.5A`
D. `2.98A`

Explanation

To find the RMS current through a resistor of 50 ohms when connected to an alternating voltage given by \( V = 140 \sin(314t) \), we can follow these steps:

### Step 1: Identify the RMS Voltage
The given voltage is in the form \( V(t) = V_m \sin(\omega t) \), where \( V_m \) is the maximum voltage (amplitude) and \( \omega \) is the angular frequency. Here, \( V_m = 140 \) volts.

The RMS (Root Mean Square) voltage \( V_{rms} \) for a sinusoidal voltage is given by the formula:
\[ V_{rms} = \frac{V_m}{\sqrt{2}} \]

### Step 2: Calculate the RMS Voltage
Substituting the value of \( V_m \):
\[ V_{rms} = \frac{140}{\sqrt{2}} = \frac{140}{1.414} \approx 98.99 \text{ volts} \]

### Step 3: Use Ohm's Law to Find RMS Current
Ohm's Law states that the current \( I \) through a resistor is given by:
\[ I = \frac{V}{R} \]
where \( V \) is the voltage across the resistor and \( R \) is the resistance.

### Step 4: Substitute Values to Find RMS Current
Now substituting the values we have:
\[ I_{rms} = \frac{V_{rms}}{R} = \frac{98.99}{50} \]
Calculating this gives:
\[ I_{rms} \approx 1.9798 \text{ A} \]

### Step 5: Round the Answer
Rounding to two decimal places, we get:
\[ I_{rms} \approx 1.98 \text{ A} \]

Thus, the RMS current through the resistor is approximately **1.98 A**.

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