Current Electricity

What this chapter is about

This chapter moves from electrostatics (charges at rest) to charges in motion — the electric current. You study what happens when a steady potential difference is maintained across a conductor, how free electrons drift under an electric field, and why different materials resist the flow of current to different extents. The ideas here connect the microscopic picture (electrons colliding with ions) to the macroscopic quantities you measure with ammeters and voltmeters.

You learn Ohm's law, resistivity, the temperature dependence of resistance, and how resistances combine in series and parallel. The chapter also covers electromotive force (emf), internal resistance of cells, and Kirchhoff's rules for analysing circuits with more than one loop. Finally, you meet the Wheatstone bridge and the potentiometer — instruments that exploit null methods to measure resistance and compare emfs with high precision.

After working through this chapter, you should be able to analyse any DC circuit, predict currents and potential differences, and design simple measurement setups using a metre bridge or potentiometer.

Key ideas

• Electric current I is the net charge flowing through a cross-section per unit time: I = Q/t. The SI unit is the ampere (A), where 1 A = 1 C/s.
• In a metallic conductor, current arises from the drift of free electrons; drift velocity vd is related to current by I = n A e vd, where n is the number density of free electrons, A is cross-sectional area, and e is the electron charge.
• Ohm's law states that, at constant temperature, the potential difference V across a conductor is directly proportional to the current I through it: V = I R. The constant R is called resistance (unit: ohm, Ω).
• Resistivity ρ of a material is defined by R = ρ L/A, where L is length and A is cross-sectional area. Resistivity depends on the material and temperature, not on the conductor's dimensions.
• For metals, resistivity increases roughly linearly with temperature; for semiconductors and insulators, resistivity decreases as temperature rises.
• Resistors in series add directly (Req = R₁ + R₂ + …); resistors in parallel combine as 1/Req = 1/R₁ + 1/R₂ + ….
• The emf (ε) of a cell is the work done per unit charge by non-electric forces inside the cell; the terminal voltage V = ε − I r, where r is internal resistance.
• Kirchhoff's junction rule (conservation of charge): the algebraic sum of currents at any junction is zero. Kirchhoff's loop rule (conservation of energy): the algebraic sum of potential changes around any closed loop is zero.

Formulas and facts to remember

1. Current: I = Q/t (charge per unit time).
2. Drift velocity relation: I = n A e vd.
3. Ohm's law: V = I R.
4. Resistance from resistivity: R = ρ L/A.
5. Temperature dependence of resistivity: ρ(T) = ρ₀ [1 + α (T − T₀)], where α is the temperature coefficient.
6. Series combination: Req = R₁ + R₂ + …
7. Parallel combination: 1/Req = 1/R₁ + 1/R₂ + …
8. Terminal voltage of a cell: V = ε − I r.
9. Wheatstone bridge balance condition: P/Q = R/S (no current through the galvanometer).
10. Potentiometer principle: potential drop along a uniform wire is proportional to length; used to compare emfs or measure internal resistance.

Worked examples

Example 1: Finding drift velocity

A copper wire of cross-sectional area 1.0 × 10⁻⁶ m² carries a steady current of 2.0 A. The free-electron density in copper is 8.5 × 10²⁸ m⁻³. Calculate the drift velocity of the electrons.

Solution

Use I = n A e vd.

Rearranging: vd = I / (n A e).

Substituting values: vd = 2.0 / (8.5 × 10²⁸ × 1.0 × 10⁻⁶ × 1.6 × 10⁻¹⁹) vd = 2.0 / (1.36 × 10⁴) vd ≈ 1.5 × 10⁻⁴ m/s.

The drift velocity is about 0.15 mm/s — remarkably slow, yet the electric signal travels almost instantaneously because the electric field is established throughout the wire at nearly the speed of light.

Example 2: Equivalent resistance and current

Three resistors of 6 Ω, 3 Ω and 2 Ω are connected in parallel across a 12 V battery of negligible internal resistance. Find the total current drawn from the battery.

Solution

For parallel combination: 1/Req = 1/6 + 1/3 + 1/2 = 1/6 + 2/6 + 3/6 = 6/6 = 1.

So Req = 1 Ω.

Total current I = V/Req = 12/1 = 12 A.

Example 3: Potentiometer – comparing two cells

A potentiometer wire of length 100 cm has uniform cross-section. When a standard cell of emf 1.02 V is connected, the null point is at 51.0 cm. When an unknown cell replaces the standard cell, the null point shifts to 68.0 cm. Find the emf of the unknown cell.

Solution

For a potentiometer, emf is proportional to balancing length: ε₁/ε₂ = l₁/l₂.

Let ε₁ = 1.02 V (standard), l₁ = 51.0 cm, l₂ = 68.0 cm.

ε₂ = ε₁ × (l₂/l₁) = 1.02 × (68.0/51.0) = 1.02 × 1.333 ≈ 1.36 V.

The emf of the unknown cell is approximately 1.36 V.

Common mistakes

• Confusing emf with terminal voltage → Remember that terminal voltage equals emf only when no current flows; otherwise V = ε − I r.
• Adding resistances using the parallel formula when they are actually in series → Trace the current path: if the same current passes through both, they are in series.
• Taking drift velocity to be very high because current appears instantaneous → Drift velocity is tiny; what propagates fast is the electric field, not the electrons themselves.
• Forgetting sign conventions in Kirchhoff's loop rule → Assign a direction to the loop; drop potential across a resistor if traversing in the direction of current, rise if opposite. For a cell, rise if going from negative to positive terminal.
• Using the Wheatstone bridge formula when the bridge is not balanced → The relation P/Q = R/S holds only when the galvanometer shows zero deflection.

Quick revision

1. Current is rate of flow of charge; drift velocity is very small (order 10⁻⁴ m/s in copper at typical currents).
2. Ohm's law: V = I R; valid at constant temperature for ohmic conductors.
3. R = ρ L/A — longer wires have more resistance; thicker wires have less.
4. Series resistances add; for parallel, reciprocals add.
5. Kirchhoff's rules let you solve any circuit: junction rule conserves charge, loop rule conserves energy.
6. Potentiometer gives a null method for comparing emfs without drawing current from the cell under test.