Ohm’s Law establishes the fundamental linear relationship between the electric current flowing through a conductor and the potential difference applied across its ends. Formulated by German physicist Georg Simon Ohm, it serves as the cornerstone for analyzing electrical circuits.
Statement of the Law
The electric current (I) flowing through a metallic conductor is directly proportional to the potential difference (V) across its ends, provided all physical conditions—such as temperature, mechanical strain, dimensions, and material composition—remain strictly constant. Mathematically: V∝I V=IR Where R is a constant of proportionality known as the Electrical Resistance of the conductor.
V-I Characteristic Curves
- Ohmic Conductors: Materials that strictly obey Ohm’s law show a linear, straight-line relationship when voltage is plotted against current (V−I graph). The constant slope of this line represents the resistance (R). Examples include pure metals like copper, aluminum, and silver under normal operating conditions.
- Non-Ohmic Conductors: Materials where the ratio of V to I is not constant, resulting in a non-linear V−I curve. Examples include semiconductors, diodes, transistors, vacuum tubes, and electrolytes.
Electrical Resistance (R)
Resistance is the intrinsic property of a material to oppose or impede the orderly flow of electric charge carriers (electrons) through it.
Cause of Resistance
At the microscopic level, free electrons in a conductor accelerate under the influence of an external electric field. As they drift toward the positive terminal, they continuously collide with the constituent fixed positive ions and atoms of the material’s crystal lattice. These frequent collisions dissipate kinetic energy as heat and hinder the smooth movement of electrons, giving rise to electrical resistance.
Units and Dimensions
- SI Unit: The SI unit of resistance is the Ohm, denoted by the Greek letter omega (Ω).
- Definition of 1 Ohm: A conductor has a resistance of 1 Ω if a potential difference of 1 Volt across its ends causes a current of 1 Ampere to flow through it (1 Ω=1 V/A).
- Dimensional Formula: [M1L2T−3A−2]
Factors Governing Electrical Resistance
The total resistance of a uniform conductor depends on both its structural geometry and the intrinsic properties of the material.
1. Length of the Conductor (l)
The resistance of a conductor is directly proportional to its length (R∝l). A longer conductor increases the distance electrons must travel, leading to a higher number of lattice collisions.
2. Area of Cross-Section (A)
The resistance is inversely proportional to the cross-sectional area of the conductor (R∝A1). A thicker wire offers a wider path for electron flow, reducing the frequency of collisions per unit volume. Doubling the radius of a wire reduces its resistance to one-fourth.
3. Nature of the Material (Resistivity)
Combining the geometric dependencies yields the structural formula for resistance: R=ρAl Where ρ (rho) is the constant of proportionality termed Electrical Resistivity or Specific Resistance.
4. Temperature of the Medium
Temperature significantly alters atomic arrangements and lattice vibrations, thereby modifying resistance:
- Metals (Conductors): Resistance increases with rising temperature. Higher thermal energy causes lattice ions to vibrate with greater amplitude, increasing the probability of electron collisions and reducing the relaxation time (τ).
- Semiconductors and Insulators: Resistance decreases with rising temperature. Thermal energy breaks covalent bonds, liberating more free electrons and holes for conduction, which outweighs the effect of lattice vibrations.
- Alloys: Materials like Manganin, Constantan, and Nichrome exhibit a negligible, near-zero temperature coefficient of resistance, making them ideal for standard resistors.
Electrical Resistivity vs. Electrical Conductivity
Electrical Resistivity (ρ)
Resistivity is the measure of a material’s intrinsic resistance to current flow per unit dimensions. It equals the resistance of a conductor having a unit length (1 m) and a unit cross-sectional area (1 m2).
- SI Unit: Ohm-meter (Ω⋅m).
- UPSC Prelims Core Fact: Resistivity is an intrinsic material property. It depends exclusively on the nature of the material and its temperature. It is entirely independent of the shape, length, or cross-sectional area of the object.
Electrical Conductivity (σ)
Conductivity is the reciprocal of electrical resistivity, representing the ease with which electric charge flows through a substance. σ=ρ1
- SI Unit: Siemens per meter (S/m) or Ω−1⋅m−1 (also written as mho/m).
Classification of Materials based on Resistivity Values
| Material Type | Resistivity Range (Ω⋅m) | Conduction Characteristics | Common Examples |
|---|---|---|---|
| Good Conductors | 10−8 to 10−6 | Abundant free electrons; very low opposition to current. | Silver (best conductor), Copper, Aluminum. |
| Alloys | 10−6 to 10−4 | Relatively high resistivity; does not oxidize (burn) easily at high temperatures. | Nichrome, Manganin, Constantan. |
| Semiconductors | 10−5 to 105 | Conditional conductivity; intermediate free charge carrier density. | Silicon, Germanium. |
| Insulators | 1012 to 1019 | Tightly bound electrons; practically zero free charge carriers. | Glass, Teflon, Hard Rubber, Dry Wood. |
Microscopic Derivation of Ohm’s Law
Ohm’s law can be substantiated from fundamental electron transport parameters. The current flowing through a conductor is related to the electron drift velocity (vd) by: I=nAevd Where n is the free electron density, A is the cross-sectional area, and e is the elementary electronic charge. The drift velocity can be expressed in terms of the applied electric field (E=lV) and relaxation time (τ): vd=mleVτ Substituting vd back into the current equation: I=(mlne2Aτ)V⟹V=(ne2τm)Al⋅I Comparing this with V=IR, we extract the precise expressions for Resistance and Resistivity: R=(ne2τm)Al ρ=ne2τm Where m is the mass of an electron and τ is the average relaxation time between successive collisions.
Resistor Combinations in Circuits
In practical applications, individual resistive elements are grouped together in specific configurations to manage voltage drops and net current flow.
Series Combination
- Configuration: Resistors are connected end-to-end sequentially so that the entire current flows through each resistor in turn.
- Current Rules: The current (I) remains identical through all resistors.
- Voltage Rules: The total applied voltage splits across the individual resistors proportional to their resistance values (V=V1+V2+V3).
- Equivalent Resistance (Rs): The total effective resistance is the sum of individual resistances: Rs=R1+R2+R3+⋯+Rn
- Key Inference: The equivalent resistance in a series circuit is always greater than the highest individual resistance in the combination.
Parallel Combination
- Configuration: Resistors are connected across common junctions so that the circuit branches out into multiple paths.
- Current Rules: The total input current splits among the parallel branches based on their resistances (I=I1+I2+I3).
- Voltage Rules: The potential difference (V) across each parallel branch remains exactly the same.
- Equivalent Resistance (Rp): The reciprocal of the equivalent resistance is equal to the sum of the reciprocals of individual resistances: Rp1=R11+R21+R31+⋯+Rn1
- Key Inference: The equivalent resistance in a parallel circuit is always smaller than the smallest individual resistance in the combination. Domestic household wiring is always configured in parallel to ensure every appliance receives the same operational voltage and functions independently.