Material and Temperature Dependence of Resistance

Introduction

Key Concepts

1. Electrical Resistance

Electrical resistance ($R$) is a measure of the opposition that a material offers to the flow of electric current. It is quantified by the equation: $$ R = \rho \frac{L}{A} $$ where:

• ρ (rho) is the resistivity of the material.
• L is the length of the conductor.
• A is the cross-sectional area.

2. Resistivity and Its Dependence on Material

Resistivity ($\rho$) is an intrinsic property of materials that quantifies how strongly a material opposes the flow of electric current. Different materials have different resistivities, which determine their suitability for various applications.

• Conductors: Metals like copper and aluminum have low resistivity, making them excellent conductors used in electrical wiring.
• Insulators: Materials such as rubber and glass have high resistivity, preventing current flow and used for protective coatings.
• Semi-conductors: Materials like silicon have resistivity values between conductors and insulators, crucial for electronic components.

The resistivity of a material is influenced by its atomic structure and the number of free charge carriers available for conduction.

3. Temperature Dependence of Resistance

Temperature significantly affects the resistance of materials, primarily metals and semiconductors, but in opposite ways.

• Metals: In metallic conductors, as temperature increases, resistance increases. This is because higher temperatures cause increased lattice vibrations, scattering free electrons and hindering their flow.
• Semiconductors: For semiconductors, resistance decreases with an increase in temperature. Higher temperatures provide more energy to electrons, increasing the number of charge carriers.

The relationship between resistance and temperature for metals is typically linear and can be expressed as: $$ R_T = R_0 \left[1 + \alpha (T - T_0)\right] $$ where:

• R_T is the resistance at temperature T.
• R₀ is the original resistance at reference temperature T₀.
• α is the temperature coefficient of resistance.

4. Temperature Coefficient of Resistance

The temperature coefficient of resistance ($\alpha$) quantifies how much a material's resistance changes with temperature. It is defined as: $$ \alpha = \frac{1}{R} \left( \frac{\partial R}{\partial T} \right)_{T_0} $$ A positive $\alpha$ indicates that resistance increases with temperature (common in metals), while a negative $\alpha$ signifies a decrease (observed in semiconductors and some alloys).

5. Practical Implications

Understanding the material and temperature dependence of resistance has several practical applications:

• Temperature Sensors: Devices like thermistors exploit the temperature dependence of resistance to measure temperature changes.
• Electrical Wiring: Selecting materials with appropriate resistivity ensures efficient power transmission and minimizes energy losses.
• Electronic Components: Semiconductors are foundational in creating components like diodes and transistors, which are sensitive to temperature variations.

6. Graphical Representation

Plotting resistance versus temperature provides visual insights into how different materials respond to temperature changes. Typically:

• Metals: The graph shows a positive slope, indicating increasing resistance with temperature.
• Semiconductors: The graph exhibits a negative slope, showing decreasing resistance as temperature rises.

7. Calculations Involving Resistance and Temperature

Consider a copper wire with a resistance ($R_0$) of 10 $\Omega$ at 20°C. If the temperature coefficient ($\alpha$) of copper is $0.00393$ per °C, the resistance at 100°C ($R_T$) can be calculated as: $$ R_T = R_0 \left[1 + \alpha (T - T_0)\right] $$ $$ R_T = 10 \Omega \left[1 + 0.00393 \times (100 - 20)\right] $$ $$ R_T = 10 \Omega \left[1 + 0.00393 \times 80\right] $$ $$ R_T = 10 \Omega \left[1 + 0.3144\right] $$ $$ R_T = 10 \Omega \times 1.3144 = 13.144 \Omega $$

8. Impact on Ohm's Law

Ohm's Law states that $V = IR$, where $V$ is voltage, $I$ is current, and $R$ is resistance. Since resistance can vary with temperature and material, it's essential to consider these factors when applying Ohm's Law in practical scenarios. For instance, in environments where temperature fluctuates, the resistance and thus the current may change, affecting the performance of electrical devices.

9. Alloy Design for Temperature Stability

To mitigate the effects of temperature on resistance, alloys are often designed to have minimal temperature coefficients. For example, constantan, an alloy of copper and nickel, has a low $\alpha$, making its resistance relatively stable over a range of temperatures. Such materials are ideal for precision instruments and applications where consistent resistance is critical.

10. Superconductivity

At extremely low temperatures, certain materials exhibit superconductivity, a state where their electrical resistance drops to zero. This phenomenon defies the typical temperature dependence of resistance and has profound implications for high-efficiency power transmission and advanced technological applications. However, superconductivity occurs below a critical temperature specific to each material, limiting its practical use.

Comparison Table

Summary and Key Takeaways