Understanding Heat Produced by a Conductor: Joule's Law

This problem involves calculating the heat produced in a conductor when the electric current flowing through it changes. The amount of heat produced by a conductor carrying current is described by Joule's Law of Heating.

Joule's Law of Heating

Joule's Law states that the heat (\(H\)) produced in a resistor when a current (\(I\)) flows through it for a time (\(t\)) is directly proportional to the square of the current, the resistance (\(R\)) of the conductor, and the time for which the current flows. Mathematically, this is expressed as:

\(H = I^2 R t\)

In this problem, the conductor and the time duration are not mentioned to be changing. This means the resistance (\(R\)) of the conductor and the time (\(t\)) for which the current flows remain constant.

Therefore, the heat produced is directly proportional to the square of the current:

\(H \propto I^2\) (when R and t are constant)

Analyzing the Given Information

We are given the initial conditions:

• Initial current (\(I_1\)) = 1 A
• Initial heat produced (\(H_1\)) = 2000 J

We are asked to find the heat produced (\(H_2\)) when the current is doubled.

• New current (\(I_2\)) = 2 \(\times\) Initial current (\(I_1\)) = 2 \(\times\) 1 A = 2 A

Step-by-Step Calculation

Let's use Joule's Law for both cases:

For the initial case (current \(I_1\)):

\(H_1 = I_1^2 R t\)

We know \(H_1 = 2000\) J and \(I_1 = 1\) A. So,

\(2000 \text{ J} = (1 \text{ A})^2 R t\)

\(2000 = 1 \times R t\)

\(2000 = R t \quad \text{(Equation 1)}\)

For the new case (current \(I_2\)):

\(H_2 = I_2^2 R t\)

We know \(I_2 = 2 I_1 = 2 \times 1 = 2\) A. So,

\(H_2 = (2 \text{ A})^2 R t\)

\(H_2 = 4 R t \quad \text{(Equation 2)}\)

Now, we can substitute the value of \(R t\) from Equation 1 into Equation 2:

\(H_2 = 4 \times (R t)\)

\(H_2 = 4 \times 2000 \text{ J}\)

\(H_2 = 8000 \text{ J}\)

Alternatively, since \(H \propto I^2\) when \(R\) and \(t\) are constant, we can write the ratio:

\(\frac{H_2}{H_1} = \frac{I_2^2 R t}{I_1^2 R t}\)

\(\frac{H_2}{H_1} = \frac{I_2^2}{I_1^2} = \left(\frac{I_2}{I_1}\right)^2\)

We know \(I_2 = 2 I_1\), so \(\frac{I_2}{I_1} = 2\). Substituting this into the ratio equation:

\(\frac{H_2}{H_1} = (2)^2 = 4\)

\(H_2 = 4 \times H_1\)

\(H_2 = 4 \times 2000 \text{ J}\)

\(H_2 = 8000 \text{ J}\)

When the current through the conductor is doubled, the heat produced becomes four times the original heat.

Conclusion

The amount of heat produced when the current through the conductor is doubled will be 8000 J.

Revision Table: Key Concepts

Additional Information: Factors Affecting Heat Production

The heat produced in a conductor depends on several factors:

• Current (\(I\)): As shown by Joule's Law, the heat produced is proportional to the square of the current. Higher current leads to significantly more heat.
• Resistance (\(R\)): Heat produced is directly proportional to the resistance of the conductor. Materials with higher resistance produce more heat for the same current and time. This is why heating elements in appliances are made of high-resistance materials like nichrome.
• Time (\(t\)): The heat produced is directly proportional to the duration for which the current flows. The longer the current flows, the more heat is produced.

Understanding these factors is crucial for designing electrical circuits and appliances, especially concerning heat dissipation and safety.