A. Ohmic Loss in a Conductor:
When an electric current flows through a conductor, such as a wire, it encounters resistance within the conductor. This resistance is primarily due to the collisions of electrons with the lattice of atoms or ions in the conductor's material. Ohmic loss, also known as Joule heating or resistive heating, refers to the power that is dissipated or lost in the form of heat as a result of these collisions and the associated resistance.
The power dissipated as ohmic loss in a conductor can be calculated using the formula:
P = I^2R
Where:
P = Power dissipated as heat (in watts, W)
I = Current flowing through the conductor (in amperes, A)
R = Resistance of the conductor (in ohms, Ω)
The power comes from the energy supplied by the voltage source that drives the current through the conductor. As the electrons move through the conductor and encounter resistance, they lose some of their kinetic energy, which is converted into heat energy. This heat energy is the source of the power dissipated as ohmic loss.
B. Relations between Temperature Coefficients and Coefficient of Linear Expansion:
The temperature coefficient of resistance (α_R), temperature coefficient of resistivity (α_ρ), and coefficient of linear expansion (α) of a conductor are related as follows:
α_R = α_ρ - 2α
Here's the derivation:
Resistance (R) of a conductor depends on its resistivity (ρ) and its dimensions (length, L, and cross-sectional area, A) according to the formula:
R = ρ * (L / A)
The resistivity (ρ) of a material changes with temperature, and this change is described by the temperature coefficient of resistivity (α_ρ):
Δρ = α_ρ * ρ0 * ΔT
Where Δρ is the change in resistivity, α_ρ is the temperature coefficient of resistivity, ρ0 is the resistivity at a reference temperature, and ΔT is the change in temperature.
The resistance (R) also changes with temperature, and this change is described by the temperature coefficient of resistance (α_R):
ΔR = α_R * R0 * ΔT
Where ΔR is the change in resistance, α_R is the temperature coefficient of resistance, R0 is the resistance at a reference temperature, and ΔT is the change in temperature.
The resistance (R) of a conductor is directly related to its dimensions (L and A) through its linear dimensions:
R = ρ0 * (L / A)
The linear dimensions (L and A) of a conductor change with temperature, and this change is described by the coefficient of linear expansion (α):
ΔL = α * L0 * ΔT
ΔA = α * A0 * ΔT
Where ΔL is the change in length, ΔA is the change in cross-sectional area, α is the coefficient of linear expansion, L0 and A0 are the dimensions at a reference temperature, and ΔT is the change in temperature.
Combining equations (4) and (5) yields:
ΔR = α * (ρ0 / A0) * R0 * ΔT
Equating this expression with equation (2) gives:
α_R * R0 * ΔT = α * (ρ0 / A0) * R0 * ΔT
Simplifying and rearranging the terms:
α_R = α_ρ - 2α
C. Limitations of Ohm's Law:
Ohm's law, which relates voltage (V), current (I), and resistance (R) through the equation V = IR, has certain limitations:
Linearity: Ohm's law assumes that the relationship between voltage and current is linear, meaning that it holds true for linear resistive components like resistors. In reality, some components, such as diodes and transistors, do not obey Ohm's law because their current-voltage characteristics are nonlinear.
Temperature Dependence: Ohm's law assumes that the resistance of a conductor remains constant with temperature. However, most materials exhibit changes in resistance with temperature due to the temperature coefficient of resistance (α_R). As the temperature changes, the resistance may deviate from the predicted linear behavior.
Limited Applicability: Ohm's law is primarily applicable to passive electrical components like resistors. It may not apply to complex circuits with varying components, reactive elements like capacitors and inductors, or at very high frequencies.
Non-Ohmic Materials: Materials like semiconductors, superconductors, and nonlinear devices do not follow Ohm's law, making it inadequate for describing their behavior.
Quantum Effects: At the nanoscale, quantum mechanical effects become significant, and Ohm's law may not accurately describe the behavior of electronic devices operating at those scales.
In summary, Ohm's law is a useful simplification for many electrical circuits, but it has limitations when dealing with nonlinear components, temperature variations, or extremely small scales where quantum effects come into play. Engineers and physicists use more advanced circuit analysis techniques to account for these limitations in practical applications.
