The heating of a coil is proportional to the nth power of the time of flow of current through it, where n is equal to:
+ 1
Understanding the Heating Effect of Electric Current
When electric current flows through a conductor, such as a coil, it experiences resistance. This resistance opposes the flow of charge, and in overcoming this resistance, electrical energy is converted into heat energy. This phenomenon is known as the heating effect of electric current or Joule heating.
Joule's Law of Heating
The amount of heat produced in a conductor due to the flow of electric current is described by Joule's Law of Heating. According to Joule's Law, the heat produced is:
- Directly proportional to the square of the current (I) flowing through the conductor.
- Directly proportional to the electrical resistance (R) of the conductor.
- Directly proportional to the time (t) for which the current flows.
Mathematically, Joule's Law is expressed by the formula:
\(\text{H} = \text{I}^2\text{Rt}\)
Where:
- \( \text{H} \) is the heat produced.
- \( \text{I} \) is the current flowing through the coil.
- \( \text{R} \) is the resistance of the coil.
- \( \text{t} \) is the time for which the current flows.
Determining the Power of Time in Heating
The question states that the heating of a coil is proportional to the nth power of the time of flow of current through it. This can be written as:
\(\text{H} \propto \text{t}^\text{n}\)
From Joule's Law, the formula for heat produced is \( \text{H} = \text{I}^2\text{Rt} \). If we consider a specific coil with a fixed resistance (\( \text{R} \)) and a constant current (\( \text{I} \)) flowing through it, the terms \( \text{I}^2 \) and \( \text{R} \) are constant.
Therefore, for constant current and resistance, the heat produced is directly proportional to the time (\( \text{t} \)).
\(\text{H} \propto \text{t}\)
We can also write this as:
\(\text{H} \propto \text{t}^1\)
Comparing the relationship from Joule's Law (\( \text{H} \propto \text{t}^1 \)) with the relationship given in the question (\( \text{H} \propto \text{t}^\text{n} \)), we can determine the value of \( \text{n} \).
\(\text{t}^\text{n} = \text{t}^1\)
For this equality to hold true for any time \( \text{t} \) (except potentially 0), the exponents must be equal.
\(\text{n} = 1\)
Thus, the heating of a coil is proportional to the first power of the time of flow of current through it.
Step-by-Step Analysis
- Identify the physical principle involved: The heating of a coil due to electric current is governed by Joule's Law of Heating.
- Recall or state Joule's Law formula: \( \text{H} = \text{I}^2\text{Rt} \).
- Understand the meaning of proportionality: The symbol \( \propto \) means "is proportional to". If \( \text{H} \propto \text{X} \), it means \( \text{H} = \text{kX} \), where \( \text{k} \) is a constant.
- From \( \text{H} = \text{I}^2\text{Rt} \), identify the dependence of \( \text{H} \) on \( \text{t} \). Assuming \( \text{I} \) and \( \text{R} \) are constant for a given situation, \( \text{H} \) is directly proportional to \( \text{t} \).
- Express this proportionality using the power notation: \( \text{H} \propto \text{t}^1 \).
- Compare this to the statement in the question: \( \text{H} \propto \text{t}^\text{n} \).
- Equate the powers of \( \text{t} \): Since the proportionality holds, \( \text{n} \) must be equal to 1.
Revision Table: Key Concepts of Joule Heating
| Concept | Description | Formula/Relationship |
|---|---|---|
| Joule's Law | Describes heat produced by current in a resistor. | \( \text{H} = \text{I}^2\text{Rt} \) |
| Heat (\( \text{H} \)) | Energy converted from electrical to thermal. | Measured in Joules (J). |
| Current (\( \text{I} \)) | Rate of flow of charge. | Measured in Amperes (A). \( \text{H} \propto \text{I}^2 \) (for constant R, t) |
| Resistance (\( \text{R} \)) | Opposition to current flow. | Measured in Ohms (\( \Omega \)). \( \text{H} \propto \text{R} \) (for constant I, t) |
| Time (\( \text{t} \)) | Duration of current flow. | Measured in seconds (s). \( \text{H} \propto \text{t} \) (for constant I, R) |
Additional Information on Electric Heating
The heating effect of electric current has many practical applications. Examples include electric heaters, toasters, electric irons, kettles, and incandescent light bulbs (where the filament heats up and glows).
It is important to note that Joule's Law \( \text{H} = \text{I}^2\text{Rt} \) applies when the current is constant. If the current varies with time, integration over time would be required to calculate the total heat produced.
The power dissipated as heat in a resistor can also be calculated using related formulas derived from Ohm's Law \( \text{V} = \text{IR} \):
- Power \( \text{P} = \text{IV} \)
- Substitute \( \text{V} = \text{IR} \): \( \text{P} = \text{I}(\text{IR}) = \text{I}^2\text{R} \)
- Substitute \( \text{I} = \text{V/R} \): \( \text{P} = (\text{V/R})\text{V} = \text{V}^2\text{/R} \)
Since Heat \( \text{H} = \text{P} \times \text{t} \), we get \( \text{H} = \text{I}^2\text{Rt} \) and \( \text{H} = (\text{V}^2\text{/R})\text{t} \). All these formulas show a direct linear proportionality between the heat produced and the time for which the current flows, assuming other factors are constant.
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