Problem 53 - University Physics with Modern Physics

Question

\(\mathrm{A}^{4} 540-\mathrm{W}^{\prime \prime}\) electric heater is designed to operate from \(120-\mathrm{V}\) lines. (a) What is its resistance? (b) What current does it draw? (c) If the line line voltage drops to 110 \(\mathrm{V}\) , what power does the heater take? (Assume that the resistance is constant. Actually, it will change because of the change in temperature.) (d) The heater coils are metallic, so that the resistance of the heater decreases with decreasing temperature. If the change of resistance with temperature is taken into account, will the electrical power consumed by the heater be larger or smaller than what you calculated in part (c)? Explain.

Step-by-Step Solution

Answer

(a) 26.67 Ω, (b) 4.5 A, (c) 453.85 W, (d) Larger than 453.85 W.

Step 1: Calculate Resistance using Power Formula

We know that the power \( P \) of an electric heater can be calculated using the formula \( P = V^2 / R \), where \( V \) is the voltage and \( R \) is the resistance. Given \( P = 540 \) watts and \( V = 120 \) volts, we rearrange the formula to find \( R \):\[ R = \frac{V^2}{P} = \frac{120^2}{540} = \frac{14400}{540} = 26.67\, \Omega. \]Thus, the resistance of the heater is \( 26.67 \Omega \).

Step 2: Determine the Current Drawn by the Heater

We use Ohm's Law \( I = V/R \), where \( I \) is the current, \( V \) is the voltage, and \( R \) is the resistance previously calculated. Substitute the values \( V = 120 \) volts and \( R = 26.67 \Omega \):\[ I = \frac{120}{26.67} \approx 4.5\, \text{A}. \]So, the current drawn by the heater is approximately \( 4.5 \, \text{A} \).

Step 3: Calculate Power at Reduced Voltage

With the voltage dropped to \( 110 \) volts and resistance remaining constant \( R = 26.67 \Omega \), calculate the new power \( P' \) using \( P' = V'^2 / R \):\[ P' = \frac{110^2}{26.67} = \frac{12100}{26.67} \approx 453.85\, \text{W}. \]Thus, the power consumed by the heater at 110 volts is approximately \( 453.85 \) watts.

Step 4: Consider Effect of Resistance Changing

If the resistance decreases due to the heater cooling, \( R \) will be less than 26.67 \( \Omega \). From \( P = V^2 / R \), a lower \( R \) results in higher power consumption what was calculated previously. Thus, if temperature effects on resistance are considered, power consumed is likely higher than \( 453.85 \) watts.

Key Concepts

Ohm's Law

Ohm's Law is a fundamental principle in electrical engineering that relates voltage (V), current (I), and resistance (R) in an electric circuit. The law is elegantly simple and states that the current flowing through a conductor between two points is directly proportional to the voltage across the two points. This relationship gives us the formula: \( I = \frac{V}{R} \).
This means that if you know any two of these quantities, you can calculate the third. For instance, in our heater problem, once we determined the resistance, we could calculate the current given the voltage. This relationship helps us understand how varying resistance will affect how much current a device will draw at a specified voltage.

Electric Resistance

Electric resistance is the opposition to the flow of electric current in a circuit. It's comparable to friction for electrical flow. Resistance is measured in ohms (\(\Omega\)), and in simple terms, more resistance in a circuit means less current flows. Devices like heaters and bulbs are designed with a specific resistance to ensure they work correctly under standard voltages.
In our exercise, the heater's resistance was calculated to be \( 26.67 \, \Omega \) using the rearranged power formula, which is \( R = \frac{V^2}{P} \). This calculation is pivotal when determining the device's current requirements and power consumption in different conditions.

Current Calculation

Calculating the current is crucial in electric circuits for ensuring the safe operation of devices. Knowing how many amperes a device draws helps in choosing appropriate power supplies and ensures electrical safety.
In the original exercise, once the resistance \( R \) was known, we utilized Ohm’s Law \( I = \frac{V}{R} \) to calculate the current. With the heater's normal operating voltage at 120 volts and a resistance of 26.67 \(\Omega\), the current was found to be approximately 4.5 amperes. This calculation ensures the heater is supplied with right power without overloading circuits.

Power Calculation

In electrical engineering, power is defined as the rate at which energy is used or consumed in a circuit. It's calculated using the formula \( P = \frac{V^2}{R} \), where \( P \) is power in watts, \( V \) is voltage, and \( R \) is resistance. This formula helps us calculate the power usage of electrical devices at different voltages.
In the exercise, when the voltage dropped to 110 volts, the power consumed was recalculated to approximately 453.85 watts. Power consumption can fluctuate with changes in both voltage and resistance. Understanding this helps in assessing the efficiency and energy consumption of electrical appliances under varying conditions.