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Chapter 12: Problem 17

An electric heater of resistance \(8 \Omega\) draws 15 A from the service mains 2 hours. Calculate the rate at which heat is developed in the heater.

Short Answer

Expert verified
The rate of heat development in the heater is 1800 W.

Step by step solution

01

Identify Given Values

The resistance of the heater, \(R = 8 \Omega\), and the current, \(I = 15 \text{A} \), are provided. The duration isn't needed for calculating the rate.
02

Use the Formula for Power

The rate at which heat is developed is equivalent to the power. Use the formula for power: \[ P = I^2 \times R \]
03

Calculate the Power

Substitute the given values into the formula: \[ P = (15 \text{A})^2 \times 8 \Omega = 225 \text{A}^2 \times 8 \Omega = 1800 \text{W} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Resistance
Resistance measures how much a material opposes the flow of electric current. It’s denoted by the symbol \(\text{R}\) and is measured in ohms, symbolized as \( \Omega \). There are several factors that affect resistance, including the material, length, and cross-sectional area of the conductor.
For example, if a wire has high resistance, it will not allow current to pass through easily, making it less efficient for conducting electricity.
In the given exercise, the resistance \( \text{R}\) of the electric heater is \(8\Omega\). Understanding resistance is crucial for determining how electrical devices work and how much energy they will consume.
Current
Current is the flow of electric charge in a circuit. It’s represented by the symbol \( \text{I} \) and measured in amperes (A).
Current flow is driven by the potential difference (voltage) and is opposed by the resistance of the circuit. The more significant the current, the higher the amount of charge flowing past a point per second.
In our exercise, the current \( \text{I} \) through the heater is \ 15 \text{A} \, which means there is a substantial amount of charge moving through the resistance.
Power Formula
Power measures the rate at which energy is consumed or converted. In an electrical circuit, power \(\text{P}\) can be calculated using the formula:

\[ P = I^2 \times R \]

Here, \(I\) is the current and \(R\) is the resistance. This formula helps determine the energy consumed per second.
Using our exercise: \[ P = (15 \text{A})^2 \times 8 \Omega = 225 \text{A}^2 \times 8 \Omega = 1800 \text{W} \]
So, the power (rate of heat development) in the heater is \ 1800 W \, indicating how much electrical energy gets converted to heat each second.
Ohm's Law
Ohm's Law is fundamental to understanding electrical circuits. It states that the current \( \text{I} \) through a conductor is directly proportional to the voltage \( \text{V} \) across it and inversely proportional to the resistance \( \text{R} \). The law is given by the equation:

\[ V = I \times R \]

This means if you increase the voltage, the current increases if the resistance remains constant. Conversely, if the resistance rises, the current decreases for a given voltage.
Although our problem focuses on power calculation, understanding Ohm's Law helps in grasping how voltage, current, and resistance interact in any circuit.

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Most popular questions from this chapter

Which uses more energy, a \(250 \mathrm{~W}\) TV set in \(1 \mathrm{hr}\), or a \(1200 \mathrm{~W}\) toaster in 10 minutes?

Show how you would connect three resistors, each of resistance \(6 \Omega\), so that the combination has a resistance of (i) \(9 \Omega\), (ii) \(4 \Omega\).

Two conducting wires of the same material and of equal lengths and equal diameters are first connected in series and then parallel in a circuit across the same potential difference. The ratio of heat produced in series and parallel combinations would be- (a) \(1: 2\) (b) \(2: 1\) (c) \(1: 4\) (d) \(4: 1\)

A piece of wire of resistance \(R\) is cut into five equal parts. These parts are then connected in parallel. If the equivalent resistance of this combination is \(R^{\prime}\), then the ratio \(R / R^{\prime}\) is - (a) \(1 / 25\) (b) \(1 / 5\) (c) 5 (d) 25

A hot plate of an electric oven connected to a \(220 \mathrm{~V}\) line has two resistance coils \(\mathrm{A}\) and \(\mathrm{B}\), each of \(24 \Omega\) resistance, which may be used separately, in series, or in parallel. What are the currents in the three cases?
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