Ohm's Law — Revision Notes

⚡ 30-Second Revision

Ohm's Law: — $V = IR$ (Voltage = Current $imes$ Resistance)
Resistance: — $R = \rho \frac{L}{A}$ (Resistivity $imes$ Length / Area)
Resistivity: — $\rho = \frac{m}{ne^2\tau}$ (Microscopic origin)
Conductance: — $G = 1/R$
Conductivity: — $\sigma = 1/\rho$
Temperature Dependence: — $R_T = R_0 [1 + \alpha (T - T_0)]$
Vector Form: — $\vec{J} = \sigma \vec{E}$
Ohmic Conductors: — Linear V-I graph, constant R (e.g., metals at constant T)
Non-Ohmic Conductors: — Non-linear V-I graph, R varies (e.g., diodes, thermistors)
Condition: — Temperature and other physical conditions must be constant for $V=IR$ to hold true.

2-Minute Revision

Ohm's Law, $V=IR$, is a cornerstone of current electricity, stating that current is directly proportional to voltage across a conductor, provided temperature is constant. $R$ is the constant resistance, measured in Ohms ($Omega$). Resistance itself depends on the material's resistivity ($ho$), length ($L$), and cross-sectional area ($A$) as $R = \rho L/A$. Resistivity is an intrinsic property, while resistance is for a specific component.

Crucially, resistance is temperature-dependent. For metals, $R_T = R_0 [1 + alpha (T - T_0)]$, meaning resistance increases with temperature. Materials obeying Ohm's Law are Ohmic (linear V-I graph), while those that don't are non-Ohmic (non-linear V-I graph, e.

g., semiconductors). Remember that if a wire is stretched, its volume remains constant, so changes in length also imply changes in area, affecting resistance significantly (e.g., doubling length quadruples resistance).

The vector form, $\vec{J} = \sigma \vec{E}$, relates current density to the electric field.

5-Minute Revision

Ohm's Law is expressed as $V = IR$, where $V$ is potential difference (Volts), $I$ is current (Amperes), and $R$ is resistance (Ohms). This law holds true for Ohmic conductors (like metals) only when physical conditions, especially temperature, remain constant. The V-I graph for an Ohmic conductor is a straight line passing through the origin, with its slope representing the resistance.

Resistance ($R$) is not just a material property; it also depends on the conductor's geometry. The formula $R = \rho \frac{L}{A}$ shows that resistance is directly proportional to length ($L$) and inversely proportional to the cross-sectional area ($A$).

$\rho$ is the resistivity, an intrinsic material property. For example, if a wire is stretched to double its length, its volume ($AL$) remains constant. This means its area must halve ($A' = A/2$). The new resistance would be $R' = \rho \frac{2L}{A/2} = 4 \rho \frac{L}{A} = 4R$.

This 'stretching' problem is a common NEET trap.

Temperature significantly affects resistance. For metals, resistance increases with temperature, following $R_T = R_0 [1 + \alpha (T - T_0)]$, where $\alpha$ is the temperature coefficient of resistance. For semiconductors, resistance generally decreases with temperature. Be mindful of the reference temperature $T_0$ and the temperature difference $(T - T_0)$.

Non-Ohmic conductors, such as diodes and transistors, do not obey Ohm's Law; their V-I characteristics are non-linear, and their resistance is not constant. The microscopic origin of Ohm's Law relates resistance to the drift velocity of electrons and their collisions with lattice ions, where $\rho = \frac{m}{ne^2\tau}$. The vector form $\vec{J} = \sigma \vec{E}$ provides a more general, localized description of current flow.

Prelims Revision Notes

1
Ohm's Law Statement: — $V = IR$. Current ($I$) is directly proportional to potential difference ($V$) across a conductor, provided temperature and other physical conditions are constant. $R$ is the constant of proportionality, called resistance.
2
Units: — Voltage (V) in Volts, Current (I) in Amperes, Resistance (R) in Ohms ($Omega$).
3
Resistance Formula: — $R = \rho \frac{L}{A}$.

* $\rho$: Resistivity (Ohm-meter, $\Omega \cdot m$), intrinsic property of material. * $L$: Length of conductor (meter, m). * $A$: Cross-sectional area of conductor (square meter, $m^2$).

1
Resistivity (Microscopic): — $\rho = \frac{m}{ne^2\tau}$.

* $m$: mass of electron. * $n$: number density of free electrons. * $e$: charge of electron. * $\tau$: relaxation time (average time between collisions).

1
Conductance (G): — $G = 1/R$, unit Siemens (S) or mho ($\Omega^{-1}$). Measures ease of current flow.
2
Conductivity ($\sigma$): — $\sigma = 1/\rho$, unit Siemens per meter (S/m). Intrinsic material property.
3
Temperature Dependence of Resistance: — $R_T = R_0 [1 + \alpha (T - T_0)]$.

* $R_T$: Resistance at temperature $T$. * $R_0$: Resistance at reference temperature $T_0$. * $\alpha$: Temperature coefficient of resistance (per degree Celsius, $^circ C^{-1}$). Positive for metals (R increases with T), negative for semiconductors (R decreases with T).

1
Ohmic Conductors: — Obey Ohm's Law. Linear V-I graph (straight line through origin). Resistance is constant. Examples: Most metals at constant temperature.
2
Non-Ohmic Conductors: — Do not obey Ohm's Law. Non-linear V-I graph. Resistance varies with V or I. Examples: Diodes, transistors, thermistors, filament lamps.
3
Limitations of Ohm's Law: — Not universal. Fails for non-Ohmic materials, at very high electric fields, or when temperature is not constant.
4
Vector Form of Ohm's Law: — $\vec{J} = \sigma \vec{E}$.

* $\vec{J}$: Current density (Amperes per square meter, $A/m^2$). * $\vec{E}$: Electric field (Volts per meter, V/m). * $\sigma$: Conductivity.

Vyyuha Quick Recall

To remember the Ohm's Law formula and its variations: VIRginia Is Really Very Intelligent. (V=IR, I=V/R, R=V/I). For resistance dependence: Really Long And Round Pipes Are Less Resistant. (R is proportional to L, inversely proportional to A, and depends on $\rho$).