A more general definition of the temperature coefficient of...

A more general definition of the temperature coefficient of resistivity is $$\alpha=\frac{1}{\rho} \frac{d \rho}{d T}$$ where $\rho$ is the resistivity at temperature $T .$ (a) Assuming $\alpha$ is constant, show that $$\rho=\rho_{0} e^{\alpha\left(T-T_{0}\right)}$$ where $\rho_{0}$ is the resistivity at temperature $T_{0} .$ (b) Using the series expansion $e^{x} \approx 1+x$ for $x<<1$ , show that the resistivity is given approximately by the expression $$\rho=\rho_{0}\left[1+\alpha\left(T-T_{0}\right)\right] \text { for } \alpha\left(T-T_{0}\right)<<1$$

A more general definition of the temperature coefficient of resistivity is
$$\alpha=\frac{1}{\rho} \frac{d \rho}{d T}$$
where $\rho$ is the resistivity at temperature $T .$ (a) Assuming $\alpha$ is constant, show that
$$\rho=\rho_{0} e^{\alpha\left(T-T_{0}\right)}$$
where $\rho_{0}$ is the resistivity at temperature $T_{0} .$ (b) Using the series expansion $e^{x} \approx 1+x$ for $x<<1$ , show that the resistivity is given approximately by the expression
$$\rho=\rho_{0}\left[1+\alpha\left(T-T_{0}\right)\right] \text { for } \alpha\left(T-T_{0}\right)<<1$$

Key Concepts

Series Expansion and Approximations

The series expansion, particularly the Taylor series expansion for exponential functions, is used to approximate complex functions under certain conditions. In this context, it allows the exponential expression to be simplified to its linear form for small arguments, which provides a practical approximation for small temperature changes.

Integration and Exponential Functions

Upon separating variables in the differential equation, integration is used to sum the infinitesimal changes in resistivity over changes in temperature. The result of this integration is an exponential function that describes the overall relationship between resistivity and temperature, with the temperature coefficient appearing as an exponent.

Temperature Coefficient of Resistivity

This concept defines the relative change in a material's resistivity per unit change in temperature. It is a measure of how sensitive a material’s resistivity is to temperature variations, which is important in understanding and predicting the behavior of materials under different temperature conditions.

Separable Differential Equations

This mathematical technique involves rewriting a differential equation such that each variable and its differential appear on opposite sides of the equation. For resistivity and temperature, this method allows for the separation of variables, thereby making the equation solvable by integration.

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