Current Electricity

1. Check this before you revise anything

The "Additional Exercises" section has been removed from this chapter, as from all 14 chapters of the current Class 12 Physics book. The questions run contiguously from 3.1 to 3.11 with no gaps, and a full-text search of every chapter finds zero occurrences of the phrase.

That leaves 9 questions for a 26-page chapter. Several substantial topics carry no exercise question at all: the potentiometer, the meter bridge, and the combination of cells in series and parallel are each taught and then never tested here.

The old stub had no solutions file, and duplicated meta-driven sections as headings inside the page body.

Two symbol traps if you work from an extracted PDF. The ohm sign $\Omega$ extracts as a capital "W", so "$5.0\Omega$" appears as "5.0 W". Separately, as in Chapters 1 and 2, the micro symbol extracts as a plain "m". Check any ambiguous value against the printed page.

2. Current and Drift Velocity (Textbook 3.2 to 3.5)

Electric current is the rate of flow of charge:

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

measured in amperes. By convention current is taken in the direction positive charge would move, which is opposite to the actual drift of electrons in a metal.

Drift velocity. Free electrons in a metal move rapidly and randomly, but that motion averages to nothing. An applied field superimposes a slow systematic drift, and it is this drift that constitutes the current:

$I=nAe{v}_d$

where $n$ is the number of free electrons per unit volume.

The number that surprises everyone. Substituting realistic values for copper gives drift speeds of order $10^{-4}$ m/s. An electron takes several hours to travel a few metres along a wire, as Exercise 3.9 works out in detail.

Yet a lamp lights the instant the switch closes. The resolution is that the electron does not have to arrive: the electric field is established along the whole conductor at nearly the speed of light, so every free electron in the circuit — including those already inside the filament — starts drifting almost simultaneously.

3. Ohm's Law and Resistivity (Textbook 3.4 to 3.8)

Ohm's law states that for a conductor at constant temperature:

$V=IR$

Resistance depends on both the material and the geometry:

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

so a longer wire has more resistance and a thicker one less. Resistivity $\rho$ is the material property, independent of shape, and its reciprocal is the conductivity $\sigma$.

The microscopic statement of the same law relates current density to field:

$\vec{j}=\sigma \vec{E}$

Temperature dependence (3.8). For metals, resistance rises with temperature:

$R_2=R_1\left[1+\alpha (T_2-T_1)\right]$

where $R_1$ must be the resistance at the reference temperature $T_1$. Values of $\alpha$ are small, around $10^{-4}$ to $10^{-3}{}^{\circ }$C${}^{-1}$, so large temperature changes are needed to shift the resistance appreciably — which is why Exercise 3.3 gives a rise of a thousand degrees for a 17 per cent change.

When a question supplies currents rather than resistances, as Exercise 3.6 does, convert each to a resistance with $R=V/I$ before applying this relation.

Power. Three equivalent forms follow from $V=IR$:

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

4. Cells, emf and Internal Resistance (Textbook 3.10)

The emf of a cell is the work done per unit charge in driving charge around the complete circuit. Every real cell also has an internal resistance $r$, so the emf is shared between the external and internal paths:

$I=\frac{\epsilon }{R+r}$

Terminal voltage while discharging. Some potential is dropped inside the cell, so what appears across the terminals is less than the emf:

$V=\epsilon -Ir$

Terminal voltage while charging. Here an external supply forces current backwards through the cell, so the internal drop adds instead:

$V=\epsilon +Ir$

This sign reversal is the single most examined subtlety in the chapter, and it is what makes Exercise 3.8 give 11.5 V rather than 4.5 V.

Maximum current. Setting $R=0$, which short-circuits the terminals, gives:

$I_{\max }=\frac{\epsilon }{r}$

For a car battery with $r=0.4\Omega$ this is 30 A — large enough to turn a starter motor, and dangerous enough to explain why shorting the terminals is hazardous.

5. Kirchhoff's Rules and the Wheatstone Bridge (Textbook 3.11 to 3.13)

Series and parallel reduction fails for networks with cross-connections, and Kirchhoff's two rules handle those.

Junction rule. The algebraic sum of currents at any junction is zero — a statement of conservation of charge, since charge cannot pile up at a point.

Loop rule. The algebraic sum of potential changes around any closed loop is zero — a statement of conservation of energy, since a charge carried once round a loop returns to its starting potential.

Choosing current directions. Assume any directions you like and stay consistent. A negative result simply means the true flow is opposite to your assumption, and the magnitude is still correct.

Note that resistors combine oppositely to capacitors:

The Wheatstone bridge is four resistances arranged in a loop with a galvanometer bridging the two midpoints. It is balanced when:

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

At balance no current flows through the galvanometer, the bridge points sit at equal potential, and an unknown resistance can be found from the other three.

Always test the ratio before assuming balance. Exercise 3.7 looks like a bridge but has $\frac{10}{5}=2$ against $\frac{5}{10}=0.5$. These are unequal, so it is not balanced: current does flow through the middle arm, no series-parallel reduction is possible, and the full Kirchhoff treatment is needed.

Solving it gives a total current of $\frac{10}{17}$ A, splitting into $\frac{4}{17}$ A and $\frac{6}{17}$ A, with $\frac{2}{17}$ A crossing the bridge from D to B — a direction that emerges from the solution rather than being assumed at the start.

Summary

• $I=Q/t$; conventional current runs opposite to the electron drift.
• $I=nAe{v}_d$, with drift speeds around $10^{-4}$ m/s — an electron takes hours to cross a wire.
• A lamp lights instantly because the field propagates at nearly light speed, not the electrons.
• $V=IR$ holds for ohmic conductors at constant temperature; $R=\rho L/A$ combines material and geometry.
• $\vec{j}=\sigma \vec{E}$ is the microscopic form of Ohm's law.
• $R_2=R_1[1+\alpha (T_2-T_1)]$, with $R_1$ taken at the reference temperature; convert currents to resistances first when needed.
• $P=VI=I^{2}R=V^2/R$.
• Discharging: $V=\epsilon -Ir$. Charging: $V=\epsilon +Ir$ — the internal drop changes sign.
• $I_{\max }=\epsilon /r$ when the terminals are shorted.
• Series resistors add with common current; parallel resistors add reciprocally with common voltage — the reverse of capacitors.
• The junction rule expresses conservation of charge; the loop rule expresses conservation of energy.
• Assumed current directions may be chosen freely; a negative answer just reverses the direction.
• Wheatstone balance is $P/Q=R/S$ with zero galvanometer current — test the ratio before assuming it.
• An unbalanced bridge carries current in its middle arm and requires Kirchhoff's rules throughout.
• The Additional Exercises block has been removed, leaving Exercises 3.1 to 3.11.