Problem 63 - Fundamentals of Physics

Question

A \(2.0 \mathrm{~kW}\) heater element from a dryer has a length of \(80 \mathrm{~cm}\). If a \(10 \mathrm{~cm}\) section is removed, what power is used by the now shortened element at \(120 \mathrm{~V} ?\)

Step-by-Step Solution

Answer

The power used by the shortened element is approximately 2.286 kW.

Step 1: Calculate Initial Resistance

First, we find the resistance of the original heater. Using the formula for power, \( P = \frac{V^2}{R} \), where \( P = 2000 \mathrm{~W} \) (or \(2.0 \mathrm{~kW}\)) and \( V = 120 \mathrm{~V} \), rearrange to find \( R = \frac{V^2}{P} \). Substituting the given values gives \( R = \frac{(120 \mathrm{~V})^2}{2000 \mathrm{~W}} \). Calculate \( R \).

Step 2: Determine Resistance Per Unit Length

Next, find the resistance per unit length of the original heater. The total length is \( 80 \mathrm{~cm} \), and the resistance is \( R \) from Step 1. Thus, the resistance per unit length \( r = \frac{R}{80 \mathrm{~cm}} \) ohms per centimeter.

Step 3: Find Resistance of Shortened Element

Calculate the new resistance of the shortened heater which is now \( 70 \mathrm{~cm} \) long. Multiply the resistance per unit length by the new length: \( R' = r \times 70 \mathrm{~cm} \).

Step 4: Calculate New Power Consumption

Using the resistance from Step 3 and the voltage \( V = 120 \mathrm{~V} \), calculate the power used by the shortened element. Use \( P' = \frac{V^2}{R'} \) to find the new power \( P' \).

Key Concepts

Resistance Calculation

Resistance is an essential factor in electrical circuits that determines how much an object opposes the flow of electric current. It is measured in ohms (\(\Omega\)). In this exercise, we are dealing with a heater element whose original resistance needs to be calculated. To do so, **Ohm's Law** and related formulas are used. This involves understanding how length impacts resistance.

Consider the original heater element, which was 80 cm long. We found the total resistance using a relation derived from the power formula \( P = \frac{V^2}{R} \), which rearranges to \( R = \frac{V^2}{P} \). For practical calculation, given that power \( P = 2000 \text{ W} \) and voltage \( V = 120 \text{ V} \), the resistance \( R \) could be calculated as follows:

• Substitute the values into the formula: \( R = \frac{(120)^2}{2000} \).
• Compute the initial resistance value.

Once the initial resistance is known, the resistance per unit length is calculated since the length of the wire affects its resistance. The resistance per unit length formula is: \( r = \frac{R}{80} \) Ohms per centimeter.

This way, by knowing the resistance per unit length, we can calculate the resistance for any length derived from the original heater wire.

Ohm's Law

Ohm's Law is a fundamental principle used to calculate the relationship between voltage, current, and resistance in any electrical circuit. It forms the base for understanding how to derive power and resistance, especially in altering conditions such as in this exercise. This law states:

• \( V = IR \)
• Where \( V \) is the voltage in volts, \( I \) is the current in amperes, and \( R \) is the resistance in ohms \( \Omega \)

In this exercise, although we manipulated the power formula from Ohm's Law, it remains an underpinning element to understand the calculations thoroughly. By using known quantities either of power or resistance, we can easily manipulate them to find unknown terms, which can include new resistance values or changes in power output when circuit parameters change as was necessary after altering the heater element length.

Ohm's Law is valuable, as it adapts seamlessly with other formulae to engage complex scenarios where multiple resistances or power factors might be altered, exemplifying its utility in real-world problem-solving.

Electrical Power

Electrical Power is the rate at which electrical energy is transferred by an electric circuit. The power consumption of an electrical device can be deduced using the relationship between voltage and resistance. Here, we use the formula \( P = \frac{V^2}{R} \), where \( P \) is power in watts, \( V \) is voltage in volts, and \( R \) is resistance in ohms.

Upon removing a section of the heating element, its resistance decreases, thus impacting power consumption. Given that the voltage remains the same, the equation \( P' = \frac{V^2}{R'} \) helps in recalculating the power utilization for the shortened element.

• The new resistance \( R' \) (calculated considering the new length using resistance per unit length formula).
• Substitute \( R' \) back into the power equation to find the new power.

When a section of the heater element is removed, it essentially reduces the resistance, leading to potentially higher current draw and consequently more power if the voltage source can supply it. This exercise is a real-world example of how modifications in a circuit can alter its total power consumption.