Current Electricity is one of the most reliably scoring chapters in NEET Physics — almost every year brings questions on resistivity, combinations of cells, Kirchhoff's rules, Wheatstone bridge and the potentiometer. The concepts are few, but they must be applied fast and without algebraic slips.
Electric Current, Drift Velocity and Mobility
An electric current exists whenever charge crosses a surface in a net, directed way. If a charge $\Delta q$ passes through a cross-section in time $\Delta t$, the average current is $I = \Delta q/\Delta t$, and the instantaneous current is $I = dq/dt$. Current is a scalar with the SI unit ampere (1 A = 1 C s⁻¹); the conventional direction is that of positive charge flow, i.e. opposite to electron motion in a metal.
In an isolated metal, the free electrons (roughly one per atom, giving number densities of order $10^{29}$ m⁻³) move randomly at high thermal speeds — about $10^5$ m s⁻¹ — with no preferred direction, so there is no current. Switch on a field $E$ inside the wire and each electron picks up an acceleration $a = eE/m$ between collisions with the vibrating lattice. Collisions randomise velocity, so the electron never builds up speed indefinitely; averaged over the whole population, a small extra velocity opposite to $E$ appears:
$$v_d = \frac{eE\tau}{m}$$
where $\tau$ is the relaxation time (mean time between collisions, of order $10^{-14}$ s). This drift velocity is astonishingly small — typically a fraction of a millimetre per second. The reason a bulb lights instantly is that the electric field (and hence the drift) is set up throughout the circuit at nearly the speed of light; the electrons themselves crawl.
Relating current to drift: in time $\Delta t$ the electrons within a length $v_d\Delta t$ of the wire cross a given section, so
$$I = neAv_d, \qquad J = \frac{I}{A} = nev_d$$
where $n$ is free-electron density, $A$ the area of cross-section and $J$ the current density (a vector, along $E$, unit A m⁻²). Mobility is drift speed per unit field, $\mu = v_d/E = e\tau/m$, measured in m² V⁻¹ s⁻¹. For a solution or plasma where both signs of carrier move, currents add: $I = (n_+e_+\mu_+ + n_-e_-\mu_-)EA$.
Two useful consequences: (i) in a wire of non-uniform cross-section carrying steady current, $I$ is the same everywhere, so $v_d$ and $J$ are larger where the wire is thinner; (ii) $v_d$ does not depend on the wire's length except through the field $E = V/L$.
Ohm's Law, Resistivity and Its Temperature Dependence
Substituting $v_d = eE\tau/m$ into $J = nev_d$ gives the microscopic form of Ohm's law:
$$J = \frac{ne^2\tau}{m}E = \sigma E, \qquad \sigma = \frac{ne^2\tau}{m}$$
Here $\sigma$ is conductivity and $\rho = 1/\sigma = m/(ne^2\tau)$ is resistivity. Putting $E = V/L$ and $J = I/A$ for a uniform conductor gives the familiar macroscopic statements
$$V = IR, \qquad R = \rho\frac{L}{A}$$
Resistance is measured in ohms, resistivity in ohm-metre. Note the essential distinction: $\rho$ is a property of the material (depends on $n$, $\tau$, temperature) while $R$ also depends on geometry. If a wire is stretched or drawn so that volume stays constant, $R \propto L^2 \propto 1/A^2$ — a much-loved exam result.
Ohm's law is not a universal law of nature; it is obeyed only by "ohmic" conductors over limited conditions. Deviations include:
- $V$–$I$ graphs that are non-linear (e.g. a semiconductor diode, a filament lamp at high current).
- Materials where $V$ is not a single-valued function of $I$ (e.g. GaAs shows a negative-resistance region).
- Devices where the response depends on the sign of $V$ (a diode conducts one way).
Temperature effects follow from $\rho = m/(ne^2\tau)$:
- Metals: $n$ is essentially fixed, but heating increases lattice vibrations, reducing $\tau$; hence $\rho$ rises. Empirically $\rho_T = \rho_0[1 + \alpha(T - T_0)]$ with $\alpha$ positive and small ($\sim 10^{-3}$ K⁻¹ for copper). $\alpha$ is the temperature coefficient of resistivity.
- Semiconductors and insulators: heating liberates far more carriers, so $n$ grows exponentially and $\rho$ falls sharply — $\alpha$ is negative and large in magnitude.
- Alloys like nichrome, manganin and constantan have high $\rho$ and very small $\alpha$, which is why they are used for heating elements and standard resistors.
Cells, EMF, Internal Resistance and Combinations
A cell converts chemical energy into electrical energy. Its emf $\varepsilon$ is the work done per unit charge by the cell in driving charge around the whole circuit — numerically the terminal potential difference when no current flows (open circuit). Because the electrolyte itself resists charge flow, the cell has an internal resistance $r$. For a cell discharging through external resistance $R$:
$$I = \frac{\varepsilon}{R + r}, \qquad V = \varepsilon - Ir$$
So the terminal voltage is always less than the emf while discharging, equals $\varepsilon$ at $I = 0$, and exceeds $\varepsilon$ when the cell is being charged ($V = \varepsilon + Ir$). Internal resistance can be found from $r = R\left(\dfrac{\varepsilon}{V} - 1\right)$.
Resistors in series: same current, voltages add, $R_s = R_1 + R_2 + \dots$ The equivalent is larger than the largest member. Resistors in parallel: same voltage, currents add, $1/R_p = 1/R_1 + 1/R_2 + \dots$ The equivalent is smaller than the smallest member. For two resistors, $R_p = R_1R_2/(R_1+R_2)$, and current divides as $I_1 = I,R_2/(R_1+R_2)$.
Cells in series (n identical cells, emf $\varepsilon$, internal resistance $r$ each, all like poles connected head-to-tail): $$\varepsilon_{eq} = n\varepsilon, \quad r_{eq} = nr, \quad I = \frac{n\varepsilon}{R + nr}$$ This is advantageous when $R \gg nr$. If one cell is reversed, subtract $2\varepsilon$ from the total emf but keep adding its $r$.
Cells in parallel (m identical cells): $$\varepsilon_{eq} = \varepsilon, \quad r_{eq} = r/m, \quad I = \frac{m\varepsilon}{mR + r}$$ Useful when $R \ll r$. For two different cells in parallel, the general results are $$\varepsilon_{eq} = \frac{\varepsilon_1 r_2 + \varepsilon_2 r_1}{r_1 + r_2}, \qquad r_{eq} = \frac{r_1 r_2}{r_1 + r_2}$$
For a mixed grouping of $N = mn$ cells ($n$ in series per row, $m$ rows in parallel), the current is maximum when the total internal resistance equals the external resistance, $nr/m = R$.
Kirchhoff's Rules, Wheatstone Bridge and Potentiometer
Networks that cannot be reduced by series–parallel logic need Kirchhoff's two rules.
- Junction rule (conservation of charge): the algebraic sum of currents at any junction is zero; total current entering = total leaving. In steady state charge cannot pile up at a point.
- Loop rule (conservation of energy): around any closed loop, the algebraic sum of potential changes is zero, $\sum \Delta V = 0$.
Sign conventions that prevent most errors: traversing a resistor along the assumed current direction gives a drop $-IR$; against it, $+IR$. Crossing a cell from $-$ to $+$ terminal gives $+\varepsilon$, from $+$ to $-$ gives $-\varepsilon$, regardless of current direction. Assume current directions arbitrarily; a negative answer simply means the real direction is opposite.
Wheatstone bridge: four resistors $P, Q, R, S$ form a quadrilateral, with a cell across one diagonal and a galvanometer across the other. The bridge is balanced when no current flows through the galvanometer, and applying the loop rule to the two halves gives
$$\frac{P}{Q} = \frac{R}{S}$$
At balance, the galvanometer branch may be removed or replaced by anything without changing the currents — the key trick for simplifying symmetric networks. The metre bridge is a practical Wheatstone bridge: a 1 m uniform wire replaces two arms, so if the null point is at length $\ell$ from one end, $R/S = \ell/(100-\ell)$, i.e. an unknown $S = R(100-\ell)/\ell$. Balancing near the middle of the wire minimises fractional error.
Potentiometer: a long uniform wire carrying a steady current from a driver cell develops a uniform potential gradient $k = V/L$ (volt per metre). A cell whose emf is to be measured is connected so that its emf opposes the tapped potential drop; at the null point (galvanometer shows zero) no current is drawn from the test cell, so the balancing length measures the emf, not the terminal voltage. This is the potentiometer's great advantage over a voltmeter.
- Comparing two emfs: $\varepsilon_1/\varepsilon_2 = \ell_1/\ell_2$.
- Internal resistance: balance the cell on open circuit ($\ell_1$), then with a shunt resistance $R$ across it ($\ell_2$); then $r = R(\ell_1 - \ell_2)/\ell_2$.
- Sensitivity improves with a longer wire or smaller potential gradient. The driver cell's emf must exceed the emf being measured, else no null point exists.
Heating Effect, Power and Energy
Work done by the source in moving charge $q$ through potential difference $V$ is $qV$; dividing by time, the electrical power delivered is $P = VI$. For an ohmic resistor this energy appears entirely as heat (Joule heating):
$$P = VI = I^2R = \frac{V^2}{R}$$
and the heat produced in time $t$ is $H = I^2Rt$ joules — Joule's law. The commercial unit of energy is the kilowatt-hour: 1 kWh = $3.6\times10^6$ J.
Which formula to use depends on what is constant:
- Resistors in series carry the same current, so use $P = I^2R$: the largest resistance dissipates the most power.
- Resistors in parallel share the same voltage, so use $P = V^2/R$: the smallest resistance dissipates the most power — the inverse of the series rule, and a frequent source of error when a problem switches between the two combinations mid-way. A useful check in any network: the total power delivered by the source must always equal the sum of the powers dissipated in every resistor, which is a quick sanity test in bridge or mixed circuits.
Two everyday devices lean on these ideas. A fuse wire is deliberately thin, so its high resistance and low melting point let $I^2R$ heating melt it and break the circuit before excess current damages anything downstream; its rated current scales roughly as $I \propto r^{3/2}$ with the wire's radius. A filament bulb or heater is designed so that its working resistance, at operating temperature, gives the rated power at the rated voltage — which is why a bulb's resistance measured cold with a multimeter reads noticeably lower than its resistance when lit.
Common Mistakes and Exam Traps
Applying $V = IR$ across a real cell without its internal resistance. Between a cell's terminals, $V = \varepsilon - Ir$ while it discharges and $V = \varepsilon + Ir$ while it is being charged; plugging the emf straight into $V = IR$ for the external resistor, ignoring $r$, is the single most common slip in cell-circuit problems.
Swapping the series and parallel power rules. Remembering only 'larger resistance means more heat' without first checking whether the resistors share current (series, so use $I^2R$) or voltage (parallel, so use $V^2/R$) leads to the opposite conclusion about as often as the right one.
Losing track of sign convention while traversing a loop in Kirchhoff's-law problems. A resistor contributes $-IR$ when traversed along the assumed current and $+IR$ against it; a cell contributes $+\varepsilon$ going from $-$ to $+$ and $-\varepsilon$ the other way. Inconsistent sign-tracking across a multi-loop circuit, not a conceptual misunderstanding, is usually why marks are lost here.
Reading a potentiometer's null point as if it were a voltmeter reading. At balance the test cell draws zero current, so the balancing length measures its emf, not its terminal voltage — treating the two as equivalent (as they would be for an ideal voltmeter only) is a classic conceptual trap, especially in questions comparing the two instruments directly.