Electric Resistance, Resistivity, and Conductivity

Vocabulary flashcards covering Ohm's Law, Resistance, Resistivity, Conductivity, reshaped wire relations, and graphical slopes.

Resistance

The opposition to the flow of current through a conductor, expressed as $R = \frac{V}{I}$ with the unit of ohm ($\Omega = \text{volt}/\text{Amp}$).

Factors Affecting Resistance ($R$)

1) Kind of material, 2) Temperature ($R \propto \text{Temp}$), 3) Length ($R \propto L$), and 4) Area ($R \propto \frac{1}{A}$).

Superconductor

A conductor at absolute zero ($0\,\text{K} = -273^\circ\text{C}$) where atomic vibrations and electron collisions cease, causing electrical resistance $R$ to vanish completely.

Specific Resistance (Resistivity)

The resistance of a conductor of unit area and unit length, represented as $\rho_e = \frac{R A}{L}$ with units of $\Omega \cdot \text{m}$ or $\frac{\text{Volt} \cdot \text{m}}{\text{Ampere}}$.

Factors Affecting Resistivity ($\rho_e$)

Kind of material and temperature only; resistivity is independent of length ($L$) and area ($A$).

Conductivity ($\sigma$)

The inverse (reciprocal) of resistivity, given by $\sigma = \frac{1}{\rho_e} = \frac{L}{R A}$ with units of $\frac{1}{\Omega \cdot \text{m}}$, $\Omega^{-1} \cdot \text{m}^{-1}$, or $\frac{\text{Amp}}{\text{Volt} \cdot \text{m}}$.

Reshaped Wire Formulas (Volume Constant)

When a wire is reshaped (stretched, pressed, pulled, drawn, or folded), volume remains constant ($\text{Volume} = L \times A$), leading to $\frac{R_1}{R_2} = \frac{L_1^2}{L_2^2} = \frac{A_2^2}{A_1^2} = \frac{r_2^4}{r_1^4}$.

Percentage of Change Formula

The percentage change of a quantity $X$ calculated as \% \Delta X = \left(\frac{X_{\text{new}} - X_{\text{old}}}{X_{\old}}\right) \times 100.

Wire Radius Doubled and Length Halved

For a wire where radius is doubled ($r_2 = 2$) and length is halved ($L_2 = 1$ relative to $L_1 = 2$), the new resistance becomes $R_2 = \frac{1}{8} R_1$ (decreases 8 times).

Wire Pressed with Radius Tripled

For a reshaped wire pressed such that its radius is tripled ($r_2 = 3 r_1$), the ratio $\frac{R_1}{R_2} = \frac{r_2^4}{r_1^4} = \frac{81}{1}$ yields $R_2 = \frac{1}{81} R_1$ (decreases 81 times).

Wire Stretched with Length Increased by 50%

For a wire stretched until its length increases by $50\%$ ($L_2 = 1.5 L_1$), the resistance relation $\frac{R_1}{R_2} = \frac{L_1^2}{L_2^2} = \frac{1^2}{1.5^2} = \frac{4}{9}$ yields $R_2 = \frac{9}{4} R_1$ (increases $\frac{9}{4}$ times).

Graph of Resistivity vs Reciprocal Conductivity

A linear graph plotting $\rho_e$ against $\frac{1}{\sigma}$ with slope equal to $\rho_e \cdot \sigma = 1$, yielding an angle $\theta = 45^\circ$ because $\tan(\theta) = 1$.