Question

What are the maximum values of (a) current and (b) voltage when an incandescent 60 -W light bulb (at \(110 \mathrm{~V})\) is connected to a wall plug labeled \(110 \mathrm{~V} ?\)

Short answer

Answer: The maximum current for the light bulb is calculated using the formula \(I_\text{max} = \frac{110 \mathrm{~V}}{R}\), where R is the resistance of the light bulb. The maximum voltage for the light bulb is given and is 110 V.

Step-by-step solution

Calculate the resistance of the light bulb

First, we need to find the resistance of the light bulb. We can do this using the power formula: \(P = V^2/R\), where P is the power in watts, V is the voltage in volts, and R is the resistance in ohms. Rearranging the formula for R gives: \(R = \frac{V^2}{P}\). Given the power of the bulb (P) to be 60 W and the voltage (V) to be 110 V, we can plug in these values and calculate the resistance of the light bulb: \(R = \frac{(110 \mathrm{~V})^2}{60 \mathrm{~W}}\)

Calculate the maximum current

Now that we have the resistance, we can use Ohm's law to calculate the maximum current. Ohm's law states: \(I = \frac{V}{R}\), where I is the current in amperes, V is the voltage in volts, and R is the resistance in ohms. Using the voltage given for the wall plug, 110 V, and the calculated resistance, we can find the maximum current: \(I_\text{max} = \frac{110 \mathrm{~V}}{R}\)

Calculate the maximum voltage

The maximum voltage is already given by the problem as 110 V. This is because the voltage across the light bulb is determined by the voltage of the wall plug to which it is connected. So, \(V_\text{max} = 110 \mathrm{~V}\)

Final answers

Calculating the values from the previous steps, we find the maximum current and voltage for the light bulb: (a) Maximum Current: \(I_\text{max} = \frac{110 \mathrm{~V}}{R}\) (b) Maximum Voltage: \(V_\text{max} = 110 \mathrm{~V}\)

Key concepts

Electric Power

Electric power refers to the rate at which electrical energy is transferred by an electric circuit. It's expressed in watts (W), named after James Watt. In simple terms, it's calculated as the product of voltage and current. Knowing how electric power works is essential when using household appliances, like light bulbs.

In the context of the incandescent 60-W light bulb, the electric power given is 60 watts. This means the bulb uses 60 joules of electrical energy every second it operates. When you connect this bulb to a 110-volt wall outlet, you know the power (P) and the voltage (V). The equation for electric power is: \[ P = V \times I \] Where

• \( P \) is the power in watts,
• \( V \) is the voltage in volts, and
• \( I \) is the current in amperes.

This relationship highlights how voltage and current combine to expend electrical energy as power. For everyday devices, understanding this can help comprehend how much electricity they consume and how efficient they are.

Resistance Calculation

Resistance is a measure of how much an object opposes the flow of electric current. This property is expressed in ohms (\( \Omega \)). For our incandescent light bulb problem, knowing the resistance helps us further understand how the bulb operates under a specified voltage.

The formula connecting power (P), voltage (V), and resistance (R) is: \[ R = \frac{V^2}{P} \] Using this formula, we can determine the resistance of the bulb. Given that the power is 60 W and the voltage is 110 V, you calculate:\[ R = \frac{(110 \ \text{V})^2}{60 \ \text{W}} \] This step is crucial because it leads to the next calculation of current using Ohm’s law. Calculating resistance helps design circuits capable of supporting the electrical load, ensuring that components operate safely and efficiently.

Current and Voltage Analysis

Current is the flow of electric charge in a circuit, measured in amperes (A). In this scenario with the light bulb, determining the maximum current involves using the resistance previously calculated and the given voltage through Ohm's Law:

\[ I = \frac{V}{R} \] Where

• \( I \) is the current in amperes,
• \( V \) is the voltage in volts, and
• \( R \) is the resistance in ohms.

This allows us to find the maximum current the light bulb can safely handle when connected to the 110-volt source. The maximum voltage is always determined by the supply, in this case, 110 V, since that's the specification of the outlet providing the power.

Understanding current and voltage relationships through these formulas not only helps in solving textbook problems but also in practical applications like determining electrical loads in household installations.