Current Electricity | CUET | PHYSICS| JEE | APNA CAREER

1. Introduction to Current Electricity

Current Electricity deals with the flow of electric charges through conductors under the influence of an electric field. Unlike electrostatics, where charges are at rest, this chapter focuses on steady (constant) flow of charges, known as direct current (DC). The study of current electricity forms the foundation for understanding electrical circuits, household wiring, electronic devices, and power transmission systems.

Electric current is essential in daily life — from lighting bulbs and charging mobile phones to operating motors and computers. In this chapter, we explore the microscopic origin of current, macroscopic laws governing it (such as Ohm’s law), circuit analysis techniques, and energy transformations in electrical systems.

A steady current requires a closed circuit and a source of energy (such as a cell or battery) that maintains a potential difference, continuously doing work to move charges against the resistance offered by the conductor.

2. Electric Current

Definition: Electric current is the rate of flow of electric charge through any cross-section of a conductor. Mathematically, if a charge ΔQ passes through a cross-section in time Δt, the current I is given by:

$$I = \frac{\Delta Q}{\Delta t}$$

For continuous flow, in the limit as Δt approaches zero:

$$I = \frac{dQ}{dt}$$

• SI Unit: Ampere (A), where 1 A = 1 coulomb per second (1 C/s).
• Nature: Current is a scalar quantity. However, it has a direction — the conventional current flows from higher potential to lower potential (positive to negative terminal), which is opposite to the actual flow of electrons (negative charges).
• Types of Current:
  • Direct Current (DC): Flows in one direction only (e.g., from cells, batteries).
  • Alternating Current (AC): Periodically reverses direction (studied in the next chapter).

Example: If 5 coulombs of charge flow through a wire in 2 seconds, the current is I = 5/2 = 2.5 A.

To maintain a steady current, the circuit must be closed, and an external agency (such as a battery) must continuously supply energy to move charges from lower to higher potential energy regions.

3. Current Density

Current density (J) at any point in a conductor is the current per unit area, when the area is perpendicular to the direction of current flow:

$$\mathbf{J} = \frac{I}{A}$$

where A is the cross-sectional area.

• Nature: Vector quantity, directed along the flow of positive charge.
• In non-uniform conductors, current density may vary across the cross-section.
• Relation to microscopic parameters will be derived later.

Current density helps in understanding localized flow, especially in thick or irregularly shaped conductors.

4. Flow of Electric Charge in Metallic Conductors (Microscopic View)

In metals, free electrons (conduction electrons) are responsible for current. These electrons are not completely free; they collide frequently with lattice ions.

When no electric field is applied, electrons move randomly with thermal velocities, resulting in zero net current.

When an electric field E is applied (due to potential difference), electrons experience a force F = -eE (negative because electrons are negatively charged). They accelerate but soon collide and lose momentum, leading to a steady average velocity called drift velocity.

5. Drift Velocity and Its Derivation

Drift velocity (v_d) is the small average velocity acquired by free electrons in the direction opposite to the electric field.

Derivation: Consider an electron of mass m and charge -e. Under electric field E, acceleration a = eE/m (magnitude).

Between collisions, the average time is the relaxation time τ (mean time between two successive collisions).

The drift velocity gained in time τ is:

$$v_d = a \tau = \frac{eE}{m} \tau$$

Thus,

$$v_d = \frac{e \tau}{m} E$$

(The direction is opposite to E for electrons.)

Typical value of v_d is very small (~10^{-4} m/s), even though thermal speed is ~10^6 m/s. This explains why current appears instantly in a circuit — the electric field propagates at nearly the speed of light, setting all electrons into drift almost simultaneously.

6. Relation Between Current and Drift Velocity

Consider a conductor of cross-sectional area A and length l. Let n be the number density of free electrons (number per unit volume).

In time dt, the electrons drift a distance v_d dt. The volume containing electrons that cross a section in dt is A × v_d dt.

Total charge crossing the section: ΔQ = n × (A v_d dt) × e

Therefore, current:

$$I = \frac{\Delta Q}{dt} = n e A v_d$$

Substituting v_d:

$$I = n e A \left( \frac{e \tau}{m} E \right) = \frac{n e^2 A \tau}{m} E$$

This is a fundamental microscopic expression for current.

Mobility (μ): Defined as drift velocity per unit electric field:

$$\mu = \frac{v_d}{E} = \frac{e \tau}{m}$$

Mobility is higher for materials with longer relaxation time or lighter charge carriers. Conductivity is related to mobility as σ = n e μ.

7. Ohm’s Law

Statement: At constant temperature and physical conditions, the current flowing through a conductor is directly proportional to the potential difference applied across its ends.

Mathematically:

$$V \propto I \quad \Rightarrow \quad V = I R$$

or

$$I = \frac{V}{R}$$

where R is the constant of proportionality, called electrical resistance.

• SI Unit of Resistance: Ohm (Ω), where 1 Ω = 1 V/A.
• Ohmic Conductors: Materials that obey Ohm’s law (e.g., most metals at moderate temperatures). Their V-I graph is a straight line through origin.
• Non-Ohmic Conductors: Do not obey Ohm’s law (e.g., semiconductors, diodes, filament lamps). Their V-I graph is non-linear.

Limitations of Ohm’s Law:

• Valid only when temperature and other physical conditions remain constant.
• Not applicable to semiconductors or at very high currents where heating effects are significant.

8. Electrical Resistance and Resistivity

Resistance (R) depends on the geometry and material of the conductor:

$$R = \rho \frac{l}{A}$$

where:

• l = length of conductor
• A = cross-sectional area
• ρ = resistivity (specific resistance) of the material.
• Resistivity (ρ): Intrinsic property of the material. It is the resistance of a unit length and unit cross-sectional area conductor.
  • SI Unit: Ω·m
  • Good conductors (metals): ρ ~ 10^{-8} to 10^{-6} Ω·m
  • Insulators: ρ ~ 10^{12} to 10^{17} Ω·m or higher
  • Semiconductors: Intermediate values, highly temperature-dependent.

Conductivity (σ): Reciprocal of resistivity, σ = 1/ρ. Unit: Siemens per meter (S/m) or mho/m.

Derivation of R = ρ l / A: From earlier, I = (n e² A τ / m) E

But E = V/l (uniform field), so:

$$I = \frac{n e^2 A \tau}{m} \frac{V}{l}$$

$$V = I \left( \frac{m l}{n e^2 \tau A} \right)$$

Comparing with V = I R, we get:

$$R = \frac{m l}{n e^2 \tau A} \quad \Rightarrow \quad \rho = \frac{m}{n e^2 \tau}$$

This shows resistivity depends on material properties (n, τ) but not on dimensions.

9. Variation of Resistivity with Temperature

For most metals, resistivity increases with temperature:

$$\rho = \rho_0 (1 + \alpha \Delta T)$$

where:

• ρ₀ = resistivity at reference temperature (usually 0°C or 20°C)
• α = temperature coefficient of resistivity (positive for metals, ~0.004 per °C for copper)
• ΔT = rise in temperature

Reason: Higher temperature increases lattice vibrations, leading to more frequent collisions and reduced relaxation time τ.

For semiconductors and insulators, α is negative — resistivity decreases with temperature because more charge carriers are excited.

Resistance variation follows similar relation:

$$R = R_0 (1 + \alpha \Delta T)$$

10. Combination of Resistors

Resistors can be combined in series or parallel to achieve desired equivalent resistance.

Series Combination:

• Same current I flows through each resistor.
• Total voltage V = V₁ + V₂ + V₃ + …
• Equivalent resistance:
  $$R_s = R_1 + R_2 + R_3 + \dots$$
• Voltage divides in proportion to resistances (voltage divider rule).

Parallel Combination:

• Same voltage V across each resistor.
• Total current I = I₁ + I₂ + I₃ + …
• Equivalent resistance:
  $$\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots$$
• Current divides in inverse proportion to resistances (current divider rule).

Advantages:

• Series: Increases resistance, used for higher voltage division.
• Parallel: Decreases resistance, provides multiple paths, used in household wiring.

11. Electromotive Force (EMF) and Cells

EMF (ε): The work done by the source per unit charge in moving charge from negative to positive terminal inside the cell. It is the maximum potential difference when no current flows (open circuit).

• SI Unit: Volt (V)
• EMF is not a force; it is energy per unit charge.

A real cell has internal resistance (r) due to the electrolyte and electrodes.

Terminal Voltage (V):

• When supplying current (discharging): V = ε – I r
• When being charged: V = ε + I r

Cells in Series:

• Total EMF = ε₁ + ε₂ + … (if aiding)
• Total internal resistance = r₁ + r₂ + …

Cells in Parallel (identical cells):

• Total EMF = ε
• Total internal resistance = r / n (for n cells)

12. Kirchhoff’s Laws

These laws are based on conservation principles and are powerful tools for solving complex circuits.

Kirchhoff’s Current Law (KCL) or Junction Rule: The algebraic sum of currents meeting at a junction is zero.

$$\sum I = 0$$

( Currents entering = currents leaving ). This follows from conservation of charge — no charge accumulates at a junction in steady state.

Kirchhoff’s Voltage Law (KVL) or Loop Rule: The algebraic sum of potential differences (including EMFs and voltage drops) around any closed loop is zero.

$$\sum V = 0$$

This follows from conservation of energy — net work done in a closed loop is zero in electrostatic field (conservative).

Application Steps:

1. Assign currents with assumed directions.
2. Apply KCL at junctions.
3. Apply KVL in loops, considering sign conventions (rise or drop in potential).
4. Solve the simultaneous equations.

13. Wheatstone Bridge

A circuit used to measure unknown resistance accurately.

It consists of four resistances P, Q, R, S arranged in a diamond shape, with a galvanometer between one pair of opposite junctions and battery across the other.

Balanced Condition: When no current flows through the galvanometer:

$$\frac{P}{Q} = \frac{R}{S}$$

Applications: Meter bridge (practical form using a uniform wire), strain gauges, and precise resistance measurement.

14. Meter Bridge (Slide Wire Bridge)

A simplified Wheatstone bridge using a 1-meter uniform wire. Unknown resistance X is compared with known resistance R.

Balancing length l from one end gives:

$$\frac{X}{R} = \frac{l}{100 - l}$$

End corrections and sensitivity considerations are important in experiments.

15. Potentiometer

A device for accurate measurement of potential difference, comparison of EMFs, and internal resistance.

Principle: When a steady current flows through a long uniform wire, the potential gradient (potential drop per unit length) is constant.

Comparison of Two EMFs:

$$\frac{\varepsilon_1}{\varepsilon_2} = \frac{l_1}{l_2}$$

(where l₁ and l₂ are balancing lengths).

Measurement of Internal Resistance: Using a known resistance box and finding balancing lengths with and without shunt.

Advantages: Does not draw current from the cell at balance point, hence highly accurate.

16. Electrical Energy and Power

When current flows through a resistor, electrical energy is converted into heat (Joule heating).

Power (P): Rate of energy dissipation.

$$P = V I = I^2 R = \frac{V^2}{R}$$

Energy (H or W) dissipated in time t:

$$H = V I t = I^2 R t = \frac{V^2 t}{R}$$

This is Joule’s Law of Heating.

Practical Units: Kilowatt-hour (kWh) = 3.6 × 10^6 J (commercial unit of electrical energy).

Applications: Electric heaters, fuses (designed to melt at high current due to I²R heating), incandescent lamps.

17. Important Formula Summary

• I = Q/t = n e A v_d
• v_d = (e E τ)/m = μ E
• R = ρ l / A
• ρ = m / (n e² τ)
• V = I R (Ohm’s law)
• R = R₀ (1 + α ΔT)
• Series: R_s = Σ R_i
• Parallel: 1/R_p = Σ (1/R_i)
• Terminal voltage: V = ε – I r
• Power: P = I²R = V I
• Kirchhoff’s laws as stated above
• Wheatstone: P/Q = R/S
• Potentiometer: ε₁/ε₂ = l₁/l₂

18. Numerical Problem-Solving Approach

For effective problem-solving:

• Draw clear circuit diagrams with labels.
• Assign currents and apply Kirchhoff’s laws systematically.
• For series-parallel, reduce step by step.
• Check units and significant figures.
• For temperature effects, apply the linear approximation carefully.

Example Numerical (illustrative): A wire of resistance 10 Ω is stretched to double its length. What is the new resistance? (Assume volume constant).

Solution: Volume constant ⇒ A ∝ 1/l. New length = 2l, new area = A/2. R' = ρ (2l) / (A/2) = 4 (ρ l / A) = 4 × 10 = 40 Ω.

Many such problems appear in examinations testing conceptual understanding combined with calculation.

19. Real-World Applications and Advanced Insights

• Household Wiring: Parallel connection ensures same voltage (220 V) across appliances; series would divide voltage undesirably.
• Fuses and Circuit Breakers: Protect against overload using Joule heating.
• Batteries in Devices: Internal resistance affects efficiency; lower r gives higher terminal voltage.
• Superconductors: Zero resistivity below critical temperature — revolutionary for power transmission (no losses).
• Semiconductor Devices: Temperature dependence enables thermistors and sensors.
• Biological Systems: Nerve impulses involve ion currents similar to drift.

In competitive exams, questions often combine multiple concepts — e.g., potentiometer with temperature variation or complex circuits with cells of different EMFs.

20. Common Misconceptions and Tips for Mastery

• Current does not “get used up” in a circuit; charge is conserved.
• Electrons move slowly, but energy propagates quickly.
• Resistance is not only due to collisions but also material property.
• EMF is not the same as terminal voltage when current flows.
• Study NCERT examples thoroughly; practice numericals daily.

Revision Strategy:

1. Memorize definitions and formulas with units.
2. Understand derivations (especially drift velocity and resistivity).
3. Master circuit solving with Kirchhoff’s laws.
4. Solve previous year questions and standard numericals.
5. Visualize with diagrams — field lines, electron drift, circuit symbols.