Question

a) A personal computer with a monitor and keyboard requires \(60 \mathrm{W}\) at \(110 \mathrm{V}(\mathrm{rms})\). Calculate the rms value of the current carried by its power cord. b) A laser printer for the personal computer in (a) is rated at \(80 \mathrm{W}\) at \(110 \mathrm{V}(\mathrm{rms})\). If this printer is plugged into the same wall outlet as the computer, what is the rms value of the current drawn from the outlet?

Short answer

a) 0.545 A; b) 1.272 A total current drawn.

Step-by-step solution

Understanding the Formula for Current

The formula for the root mean square (rms) current is given by Ohm's Law: \[ I_{rms} = \frac{P}{V_{rms}} \]where \( P \) is the power in watts and \( V_{rms} \) is the rms voltage. Let's use this formula to find the rms current for both devices.

Calculating RMS Current for the Computer

Given that the power \( P \) for the computer is \( 60 \mathrm{W} \) and the \( V_{rms} \) is \( 110 \mathrm{V} \), substitute into the formula:\[ I_{rms} = \frac{60 \mathrm{W}}{110 \mathrm{V}} \]This simplifies to:\[ I_{rms} = 0.545 \mathrm{A} \]This is the rms current for the computer.

Calculating RMS Current for the Printer

For the laser printer, given \( P = 80 \mathrm{W} \) and \( V_{rms} = 110 \mathrm{V} \), substitute into the formula:\[ I_{rms} = \frac{80 \mathrm{W}}{110 \mathrm{V}} \]This simplifies to:\[ I_{rms} = 0.727 \mathrm{A} \]This is the rms current for the printer.

Total RMS Current Drawn from Outlet

When both the computer and the printer are plugged into the same outlet, the currents add up. So,\[ I_{total} = I_{computer} + I_{printer} \]Substitute the currents calculated:\[ I_{total} = 0.545 \mathrm{A} + 0.727 \mathrm{A} = 1.272 \mathrm{A} \]Thus, the total rms current drawn from the outlet is \( 1.272 \mathrm{A} \).

Key concepts

Ohm's Law

Ohm’s Law is a foundational principle in understanding electricity. It describes the relationship between voltage, current, and resistance in an electrical circuit. Formulated by Georg Simon Ohm, the law is expressed as: \[ V = IR \] where:

• \( V \) is the voltage across the circuit, measured in volts (V).
• \( I \) is the current flowing through the circuit, measured in amperes (A).
• \( R \) is the resistance within the circuit, measured in ohms (Ω).

In the context of calculating RMS current, as in the problem above, Ohm's Law can be adapted using power to find current. This adaptation simplifies to \( I_{rms} = \frac{P}{V_{rms}} \), showing that current is equal to power divided by voltage. This calculation is key to understanding how much electrical current is flowing in devices connected to an outlet.

Electrical Power

Electrical power is the rate at which electrical energy is transferred by an electric circuit. The formula to calculate power in watts (W) is given by:\[ P = VI \] This equation highlights that power is the product of voltage and current.
In the exercises provided:

• The computer operates at \( 60 \, \mathrm{W}\).
• The laser printer operates at \( 80 \, \mathrm{W}\).

These power ratings indicate how much energy each device uses per second. By understanding the power consumption, you can better manage the electrical load in your home and prevent circuit overloads. Managing power is also crucial for longer-term efficiency and cost management.

RMS Voltage

RMS voltage is a crucial concept in AC circuit analysis. The acronym RMS stands for "Root Mean Square," and it provides a more accurate measure of the average voltage in an AC system.
AC voltage constantly varies, but RMS voltage allows us to treat it as if it were a steady DC voltage delivering the same power. In effect, RMS voltage is about 0.707 times the peak voltage of an AC wave. This conversion simplifies the process of calculating power and current in AC circuits.
In the problem scenario, the RMS voltage is given as 110 V. This value is used to determine how much current each device draws when operating normally. Using RMS values simplifies the computation and interpretation of AC circuits, allowing for straightforward calculations much like those with DC.

Basic Circuit Analysis

Basic circuit analysis involves understanding and calculating different electrical parameters in a circuit, such as voltage, current, and resistance. Using fundamental laws like Ohm's Law and principles of power, one can analyze and predict how electric circuits behave under various conditions.
In the textbook exercise, basic circuit analysis was used to determine the current drawn by the computer and printer from a single outlet. This involves understanding the total RMS current by summing individual currents:\[ I_{total} = I_{computer} + I_{printer} \]This simple addition helps in predicting whether the outlet capacity is adequate and ensures safe operation of both devices simultaneously without exceeding the circuit's current thresholds. This basic understanding is crucial for daily activities, preventing issues like tripping breakers or damaging appliances due to overloading.