Understanding Temperature Dependence of Resistance

The electrical resistance of a conductor changes with temperature. For most metallic conductors, resistance increases as temperature increases. This relationship can be described using the temperature coefficient of resistance.

Analyzing the Given Problem

We are given the following information:

• Initial temperature, \(T_0 = 47^\circ C\).
• Temperature coefficient of resistance, \(\alpha = 2 \times 10^{-4} K^{-1}\).
• The final resistance, \(R_T\), is 30% more than the initial resistance, \(R_{T_0}\).

We need to find the final temperature, \(T\), at which this resistance increase occurs.

Formula for Temperature Dependence of Resistance

The resistance of a conductor at a temperature \(T\) can be related to its resistance at a reference temperature \(T_0\) using the formula:

\(R_T = R_{T_0} [1 + \alpha (T - T_0)]\)

Where:

• \(R_T\) is the resistance at temperature \(T\).
• \(R_{T_0}\) is the resistance at the reference temperature \(T_0\).
• \(\alpha\) is the temperature coefficient of resistance.
• \(T - T_0\) is the change in temperature.

Step-by-Step Calculation

Convert Initial Temperature to Kelvin

The temperature coefficient is given in \(K^{-1}\), so we should work with temperatures in Kelvin. We convert the initial temperature \(T_0\) from Celsius to Kelvin by adding 273:

\(T_0 = 47^\circ C + 273 = 320 K\)

Express the Final Resistance

The problem states that the final resistance \(R_T\) is 30% more than the initial resistance \(R_{T_0}\). This can be written as:

\(R_T = R_{T_0} + 0.30 R_{T_0}\)

\(R_T = 1.30 R_{T_0}\)

Substitute Values into the Formula

Now, substitute this relationship for \(R_T\) into the temperature dependence formula:

\(1.30 R_{T_0} = R_{T_0} [1 + \alpha (T - T_0)]\)

Simplify the Equation

We can divide both sides of the equation by \(R_{T_0}\) (assuming \(R_{T_0}\) is not zero, which is true for a conductor):

\(1.30 = 1 + \alpha (T - T_0)\)

Isolate the Temperature Difference Term

Subtract 1 from both sides:

\(1.30 - 1 = \alpha (T - T_0)\)

\(0.30 = \alpha (T - T_0)\)

Substitute the Value of Alpha

Substitute the given value of \(\alpha = 2 \times 10^{-4} K^{-1}\):

\(0.30 = (2 \times 10^{-4} K^{-1}) (T - 320 K)\)

Solve for the Temperature Difference (T - T₀)

Divide 0.30 by \(\alpha\):

\(T - 320 K = \frac{0.30}{2 \times 10^{-4} K^{-1}}\)

\(T - 320 K = \frac{0.30}{0.0002} K\)

\(T - 320 K = \frac{3000}{2} K\)

\(T - 320 K = 1500 K\)

Solve for the Final Temperature (T)

Add 320 K to both sides:

\(T = 1500 K + 320 K\)

\(T = 1820 K\)

Thus, the temperature at which the resistance of the conductor becomes 30% more than its resistance at 47°C is 1820 K.

Summary of Results

Revision Table: Conductor Resistance and Temperature

Here's a quick summary of the relationship between temperature and conductor resistance:

Additional Information: Factors Affecting Electrical Resistance

Beyond temperature, the electrical resistance of a conductor depends on several physical characteristics:

• Material: Different materials have different inherent resistivities. Conductors like copper and aluminum have low resistivity, while insulators like rubber and glass have very high resistivity.
• Length (\(L\)): Resistance is directly proportional to the length of the conductor. A longer wire offers more resistance to current flow.
• Cross-sectional Area (\(A\)): Resistance is inversely proportional to the cross-sectional area. A thicker wire offers less resistance.
• Temperature (\(T\)): As discussed, temperature affects resistance, primarily for conductors. For semiconductors and insulators, the relationship can be different.

The resistance \(R\) of a uniform conductor can be expressed using resistivity (\(\rho\)) as:

\(R = \rho \frac{L}{A}\)

Resistivity \(\rho\) is the material property that changes with temperature, and \(\alpha\) describes how \(\rho\) changes with temperature, which in turn affects \(R\).