Introduction And Ohm’s Law

PhysicsCurrent ElectricityFor NEET aspirants

Electric current is the rate of flow of electric charge through a conductor, defined as $I = d Q / d t$ and measured in amperes (A). In metallic conductors, current arises from the drift of free electrons under an applied electric field. Ohm's law states that the potential difference across a conductor is directly proportional to the current through it at constant temperature, giving $V = I R$ . This concept forms the foundation of DC circuit analysis in both the JEE (Physics - Current Electricity) and NEET (Physics - Current Electricity) syllabi, covering current density, drift velocity, mobility, resistivity, temperature dependence, and V-I characteristics.

Key Formulas - Quick Reference

Electric current: $I = \frac{d Q}{d t} = n e A v_{d}$
Current density: $J = \frac{I}{A} = n e v_{d}$
Drift velocity: $v_{d} = \frac{e E τ}{m} = \frac{e V τ}{m L}$
Mobility: $μ = \frac{v _{d}}{E} = \frac{e τ}{m}$
Ohm's law (macroscopic): $V = I R$
Ohm's law (microscopic): $J = σ E$
Resistance: $R = ρ \frac{L}{A}$
Conductivity: $σ = \frac{1}{ρ} = \frac{n e ^{2} τ}{m} = n e μ$
Temperature dependence: $ρ_{T} = ρ_{0} [1 + α (T - T_{0})]$
EMF and terminal voltage: $V = ε - I r$ (discharging), $V = ε + I r$ (charging)

1. Electric Current

Flow of electric charge constitutes electric current. For a conductor, if charge $δ Q$ flows through a cross-section of area $A$ in time $δ t$ , the average current is:

I_{avg} = \frac{δ Q}{δ t}, I_{inst} = \frac{d Q}{d t}

SI unit: ampere (A) = coulomb/second. Current is a scalar, but conventional current is taken in the direction of positive charge flow (opposite to electron flow in metals).

2. Mechanism of Current Flow in Metallic Conductors

Free electrons in a metal move randomly in all directions with thermal velocities of the order of $1 0^{5}$ m/s. The average velocity is zero, so no net current flows in the absence of an external field.

When a potential difference is applied, an electric field $E$ is set up inside the conductor:

Applied electric field $\to$ Force on electrons $\to$ Drift of electrons $\to$ Current

Figure 1: Free electrons in a conductor drift in one direction under the applied field, while conventional current

I

is defined in the opposite direction.

Drift Velocity ( $v_{d}$ )

Drift velocity is the average velocity acquired by free electrons in the direction opposite to the applied field. Under the field $E$ , an electron experiences force $F = e E$ and acceleration $a = e E / m$ . Between two successive collisions (mean free time $τ$ ), it gains velocity, and after each collision, the direction randomises. The steady drift velocity is:

v_{d} = \frac{e E τ}{m}

where $τ$ is the relaxation time (average time between two successive collisions of an electron with lattice ions).

Figure 2: A single electron follows a zigzag path due to random thermal collisions with lattice ions; the small net drift velocity

v_{d}

(arrow, top) points opposite to the applied field.

Total charge crossing a cross-section in one second gives the current:

I = n e A v_{d}

Here $n$ is the number density of free electrons, $e$ is electron charge, $A$ is cross-sectional area, and $A v_{d}$ is the volume swept per second by charge carriers.

Current Density ( $J$ )

Current density is a vector quantity representing current per unit cross-sectional area:

J = \frac{I}{A} \overset{n}{^} = n e v_{d}

SI unit: A/m $^{2}$ . For a non-uniform cross-section, $I = \int J \cdot d A$ .

Mobility ( $μ$ )

Mobility is defined as the magnitude of drift velocity per unit electric field:

μ = \frac{∣ v _{d} ∣}{E} = \frac{e τ}{m}

SI unit: m $^{2}$ /(V $\cdot$ s). Mobility measures how readily charge carriers respond to an applied field. Electrons in copper have $μ \approx 4.3 \times 1 0^{- 3}$ m $^{2}$ /(V $\cdot$ s).

Solved Example 1

Q: A steady current passes through a cylindrical conductor. Is there an electric field inside the conductor?

Solution

Yes. In electrostatics (no current), the electric field inside a conductor is zero because the surface is equipotential. However, when a potential difference is applied and a steady current flows, the situation is no longer static - a non-zero electric field $E = V / L$ exists inside the conductor to sustain the drift of electrons against collisions.

3. Ohm's Law

Statement: The potential difference across a conductor is directly proportional to the current flowing through it at a given temperature.

V \propto I \Rightarrow V = I R

The proportionality constant $R$ is the resistance of the conductor. SI unit: ohm ( $Ω$ ) = volt/ampere.

Microscopic Form of Ohm's Law

Combining $I = n e A v_{d}$ , $v_{d} = e E τ / m$ , and $V = E L$ :

I = n e A \cdot \frac{e E τ}{m}

= \frac{n e ^{2} τ}{m} \cdot A E

= \frac{n e ^{2} τ}{m} \cdot \frac{A V}{L}

Comparing with $V = I R$ :

R = \frac{m}{n e ^{2} τ} \cdot \frac{L}{A}

\Rightarrow ρ = \frac{m}{n e ^{2} τ}, σ = \frac{n e ^{2} τ}{m} = n e μ

The microscopic (or vector) form of Ohm's law is:

J = σ E

This form is more fundamental - it holds at every point inside the conductor, whereas $V = I R$ is the integrated form for a finite conductor.

4. Resistivity ( $ρ$ ) and Conductivity ( $σ$ )

The resistance of a conductor depends on its geometry and material:

R = ρ \frac{L}{A}

where $ρ$ is the resistivity of the material. SI unit: $Ω \cdot$ m.

Reciprocal of resistivity is conductivity:

σ = \frac{1}{ρ}, SI unit: (Ω \cdot m)^{- 1} or S/m (siemens per metre)

Reciprocal of resistance is conductance ( $G = 1/ R$ ), SI unit: siemens (S) or mho.

Range of Resistivity

Material type	Resistivity $ρ$ ( $Ω \cdot$ m)	Temperature coefficient $α$
Conductors (metals)	$\sim 1 0^{- 8}$	Positive
Semiconductors (Si, Ge)	$1 0^{- 4}$ to $1 0^{4}$	Negative
Insulators	$\sim 1 0^{17}$	Negative (very small)
Superconductors ( $T < T_{c}$ )	$0$ (exactly)	Not applicable

5. Temperature Dependence of Resistivity

For metals, resistivity increases with temperature because increased lattice vibrations reduce the relaxation time $τ$ . For small temperature variations:

ρ_{T} = ρ_{0} [1 + α (T - T_{0})]

where $α$ is the temperature coefficient of resistivity (unit: K $^{- 1}$ or $^{\circ}$ C $^{- 1}$ ).

For pure metals: $α > 0$ (resistance rises with $T$ ).
For semiconductors and insulators: $α < 0$ (more charge carriers get thermally excited at higher $T$ ).
For superconductors: resistivity is exactly zero below the critical temperature $T_{c}$ .
For alloys like manganin, constantan: $α \approx 0$ (used as standard resistors).

6. V-I Characteristics: Ohmic vs Non-ohmic Conductors

A conductor is ohmic if it obeys Ohm's law ( $V \propto I$ ) - its V-I graph is a straight line through the origin, and $R$ is constant (independent of $V$ and $I$ ).

A conductor is non-ohmic if the V-I graph is not a straight line. Examples:

Semiconductor diode: conducts only in forward bias; V-I curve is asymmetric and nonlinear.
Filament bulb (tungsten): V-I curve bends because resistance rises with temperature as the filament heats up.
Thyristor: exhibits negative differential resistance in part of its curve.
Electrolytes: deviate at high fields.

Figure 3: V-I characteristics of an ohmic conductor (linear), a semiconductor diode (asymmetric, nonlinear), and a tungsten filament bulb (curve bends due to heating).

7. EMF vs Terminal Voltage

Every real cell has an internal resistance $r$ . The EMF ( $ε$ ) of a cell is the potential difference across its terminals when no current flows (open circuit).

The terminal voltage ( $V$ ) is the potential difference across the terminals when current $I$ flows through the cell.

Discharging (cell delivers current to external circuit): $V = ε - I r$ . Terminal voltage is less than EMF.
Charging (external source pushes current backward through cell): $V = ε + I r$ . Terminal voltage is greater than EMF.
Open circuit ( $I = 0$ ): $V = ε$ .
Short circuit ( $V = 0$ ): $I_{m a x} = ε / r$ .

Solved Example 2

Q: A copper wire has $n = 8.5 \times 1 0^{28}$ free electrons/m $^{3}$ and cross-section $A = 2 \times 1 0^{- 6}$ m $^{2}$ . If a current of 3 A flows through it, find the drift velocity of electrons.

Solution

Using $I = n e A v_{d}$ :

v_{d} = \frac{I}{n e A} = \frac{3}{( 8.5 \times 1 0 ^{28} ) ( 1.6 \times 1 0 ^{- 19} ) ( 2 \times 1 0 ^{- 6} )}

v_{d} \approx 1.1 \times 1 0^{- 4} m/s = 0.11 mm/s

Note how small the drift velocity is (~0.1 mm/s) compared to thermal velocity ( $1 0^{5}$ m/s), even though the current appears substantial.

Key Insight

Despite drift velocities being tiny, currents flow "instantaneously" because the electric field propagates through the conductor at nearly the speed of light. All free electrons throughout the conductor start drifting almost simultaneously the moment the field is applied.

Frequently Asked Questions

What is the difference between drift velocity and thermal velocity? (JEE / NEET)

Thermal velocity is the random motion of free electrons due to temperature, of the order of

1 0^{5}

m/s, with zero average (no net current). Drift velocity is the average velocity acquired in the direction opposite to the applied field, of the order of

1 0^{- 4}

m/s. Drift velocity is much smaller but produces the net current.

Why does resistance of a metal increase with temperature but decrease for a semiconductor? (JEE Main / NEET)

In metals, the number of free electrons is roughly constant with temperature, but increased lattice vibrations reduce the relaxation time

τ

, so

ρ = m / (n e^{2} τ)

increases. In semiconductors, higher temperature releases more electrons from the valence band into the conduction band, so

n

rises exponentially, dominating over the drop in

τ

- net resistivity falls.

Is Ohm's law a universal law of physics? (JEE / NEET)

No. Ohm's law is not universal - it is an empirical relation obeyed only by ohmic conductors (most metals) at constant temperature. Semiconductor diodes, filament bulbs, electrolytes, and gases at high fields all violate

V \propto I

. However, the microscopic form

J = σ E

holds locally in a wider range of materials.

When is terminal voltage of a battery greater than its EMF? (JEE Main / NEET)

Terminal voltage exceeds EMF only when the battery is being charged - an external source pushes current backwards through the cell, and

V = ε + I r

. In normal discharge, terminal voltage is less than EMF by

I r

. On open circuit (no current), terminal voltage equals EMF.

Previous year questions on Introduction And Ohm’s Law

22 questions from past papers, each with a step-by-step solution.

Show all 22 questions

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