Electrical Resistance — Revision Notes

⚡ 30-Second Revision

Definition: — Opposition to current flow.
Ohm's Law: — $V = IR \implies R = V/I$.
SI Unit: — Ohm ($\Omega$).
Factors: — $R \propto L$, $R \propto 1/A$, $R \propto \rho$, $R \propto T$ (for metals).
Formula: — $R = \rho \frac{L}{A}$.
Resistivity ($\rho$): — Intrinsic material property. Unit: $\Omega \cdot \text{m}$.
Conductivity ($\sigma$): — $\sigma = 1/\rho$. Unit: S/m.
Temperature Dependence: — $R_T = R_0 [1 + \alpha(T - T_0)]$.
Stretching Wire: — If length increases by factor $n$, resistance increases by $n^2$ (volume constant).

2-Minute Revision

Electrical resistance is a material's opposition to electric current, measured in Ohms ($\Omega$). It's defined by Ohm's Law as the ratio of voltage ($V$) to current ($I$), so $R = V/I$. Resistance depends on four main factors: the material's inherent resistivity ($\rho$), its length ($L$), its cross-sectional area ($A$), and its temperature ($T$).

The relationship is given by $R = \rho L/A$. Resistivity is an intrinsic property of the material (unit: $\Omega \cdot \text{m}$), while resistance is specific to a particular object. For most metals, resistance increases with temperature due to increased atomic vibrations, following $R_T = R_0 [1 + \alpha(T - T_0)]$.

Remember that when a wire is stretched, its volume remains constant, so if length increases by a factor 'n', its area decreases by 'n', leading to a resistance increase by $n^2$. Distinguish between Ohmic (linear V-I) and non-Ohmic (non-linear V-I) materials.

This concept is fundamental for circuit analysis and power calculations.

5-Minute Revision

Electrical resistance is the measure of how much a material opposes the flow of electric current. It's a critical concept in physics, especially for NEET. At its core, resistance arises from the collisions between moving electrons and the atoms within a conductor, converting electrical energy into heat.

Ohm's Law provides the fundamental definition: $R = V/I$, where $V$ is the potential difference across the conductor and $I$ is the current flowing through it. The SI unit for resistance is the Ohm ($\Omega$).

Resistance is not a fixed value for all conductors; it depends on several factors:

1
Length ($L$): — Directly proportional ($R \propto L$). Longer wires offer more resistance.
2
Cross-sectional Area ($A$): — Inversely proportional ($R \propto 1/A$). Thicker wires offer less resistance.
3
Nature of Material (Resistivity, $\rho$): — This is an intrinsic property. Good conductors have low resistivity, insulators have high. The formula combining these is $R = \rho L/A$. Resistivity's unit is Ohm-meter ($\Omega \cdot \text{m}$). Conductivity ($\sigma$) is the reciprocal of resistivity, $\sigma = 1/\rho$.
4
Temperature ($T$): — For most metals, resistance increases with temperature. This is because increased thermal vibrations of atoms impede electron flow more. The relationship is $R_T = R_0 [1 + \alpha(T - T_0)]$, where $\alpha$ is the temperature coefficient of resistivity.

Key Scenarios to Remember:

Stretching a wire: — If a wire is stretched to $n$ times its original length, its volume remains constant ($AL = \text{constant}$). This means its area becomes $A/n$. The new resistance will be $R' = \rho (nL)/(A/n) = n^2 (\rho L/A) = n^2 R$. This $n^2$ factor is a common trap.
Ohmic vs. Non-Ohmic: — Ohmic materials (like most metals at constant temperature) obey Ohm's Law, showing a linear V-I graph. Non-Ohmic materials (like semiconductor diodes) do not, and their V-I graph is non-linear.

Example: A wire of resistance $5,Omega$ is stretched such that its length becomes three times its original length. What is its new resistance? Solution: Original resistance $R = 5,Omega$. Length is stretched by $n=3$. New resistance $R' = n^2 R = (3)^2 \times 5,Omega = 9 \times 5,Omega = 45,Omega$.

Prelims Revision Notes

1
Definition & Ohm's Law: — Electrical resistance ($R$) is the opposition to current flow. $R = V/I$. SI unit: Ohm ($\Omega$). $1 \Omega = 1,\text{V/A}$.
2
Factors Affecting Resistance:

* Directly proportional to length ($L$): $R \propto L$. * Inversely proportional to cross-sectional area ($A$): $R \propto 1/A$. * Depends on the nature of the material (resistivity, $\rho$). * Depends on temperature ($T$).

1
Resistivity ($\rho$): — Intrinsic property of a material. $R = \rho L/A \implies \rho = RA/L$. SI unit: Ohm-meter ($\Omega \cdot \text{m}$). Good conductors have low $\rho$, insulators have high $\rho$.
2
Conductivity ($\sigma$): — Reciprocal of resistivity. $\sigma = 1/\rho$. SI unit: Siemens per meter (S/m) or $(\Omega \cdot \text{m})^{-1}$.
3
Temperature Dependence: — For metals, resistance increases with temperature. $R_T = R_0 [1 + \alpha(T - T_0)]$, where $\alpha$ is the temperature coefficient of resistivity (unit: $^circ\text{C}^{-1}$ or $\text{K}^{-1}$). For semiconductors, resistance generally decreases with temperature.
4
Effect of Stretching/Reshaping: — When a wire is stretched, its volume ($V = AL$) remains constant. If length increases by a factor $n$, area decreases by $n$. New resistance $R' = n^2 R$. If radius/diameter changes by a factor $n$, then area changes by $n^2$, and resistance changes by $1/n^4$ (if length is kept constant, which is not the case in stretching).
5
Ohmic vs. Non-Ohmic:

* Ohmic: Materials obeying Ohm's Law (constant $R$, linear $V-I$ graph). E.g., metals at constant temperature. * Non-Ohmic: Materials not obeying Ohm's Law (variable $R$, non-linear $V-I$ graph). E.g., semiconductor diodes, thermistors.

1
Dimensions of Resistance: — $[R] = [ML^2T^{-3}A^{-2}]$.
2
Dimensions of Resistivity: — $[\rho] = [ML^3T^{-3}A^{-2}]$.
3
Color Coding: — (Briefly recall if needed, but less common for NEET direct questions).

Vyyuha Quick Recall

To remember factors affecting resistance: 'Really Long And Thin Materials Resist Too'

Resistance is proportional to Length.
Area (cross-sectional) is inversely proportional.
Temperature affects it.
Material (resistivity) is key.
Resistivity is an Intrinsic property (I for Intrinsic).