Current Electricity: common doubts, answered
The questions students ask most often about Current Electricity, each with a short answer. For the full chapter, read the Current Electricity notes.
About the chapter
What unit conversion mistake should I avoid in resistance and drift speed problems?
Square the conversion factor for areas: 1 mm² is 10⁻⁶ m², not 10⁻³ m², and 1 cm² is 10⁻⁴ m². Area contains two lengths, so the millimetre-to-metre factor applies twice. Getting this wrong changes a resistance from R = ρl/A or a drift speed from I = neAv_d by a factor of a thousand, which is one of the commonest slips in numerical questions on this chapter.
Electric current
Read this section in the notes →Why is electric current a scalar even though it has a direction?
Because currents do not add by the vector rule. At a junction the currents simply add as numbers, whatever the angles between the wires, which is exactly what Kirchhoff's junction rule uses. The arrow on a current only shows the sense of flow along the conductor. Current density j = I/A, on the other hand, is a vector that points along the flow of positive charge at each point.
Electric currents in conductors and Ohm's law
Read this section in the notes →What is the difference between resistance and resistivity?
Resistance belongs to a particular piece of material, while resistivity belongs to the material itself. In R = ρl/A, the resistance depends on the length and cross-section of the wire, but ρ depends only on the substance and its temperature. A long thin copper wire has more resistance than a short thick one, yet both have the same resistivity. R is in ohm, ρ in ohm metre.
Why does stretching a wire to n times its length increase its resistance n² times?
Because stretching keeps the volume fixed, so the area falls by the same factor n that the length rises. In R = ρl/A both changes push R upward, giving n² times the original. A wire drawn to twice its length has four times the resistance, and one stretched by 25% has 1.5625 times. The resistivity does not change, since the material is the same; only the shape has.
Electron drift and where resistivity comes from
Read this section in the notes →What is the difference between drift velocity and the random motion of electrons?
Random motion is fast but points every which way, averaging to zero, while drift velocity is the small steady average velocity the field adds. Between collisions the field accelerates each electron slightly opposite to E, and each collision scrambles its direction again. Averaged over all electrons this gives v_d = −eEτ/m, typically of the order of a millimetre per second or less, and only this drift carries current.
If drift velocity is so small, why does a bulb light up as soon as the switch is pressed?
Because the electric field, not the electrons themselves, spreads through the circuit almost instantly. The wires are already full of free electrons. When the circuit is closed, the field is set up along the whole conductor very quickly, and electrons everywhere, including those in the filament, start drifting at almost the same moment. No electron has to travel from the switch to the bulb.
How does drift speed change where a wire becomes thinner?
Drift speed is larger in the thinner part, while the current is the same everywhere. In a steady current charge cannot pile up anywhere, so the same I crosses every cross-section. Since I = neAv_d and n is fixed for the material, a smaller area A must mean a larger v_d. The current density j = I/A is also larger where the wire narrows.
Mobility
Read this section in the notes →What is mobility of charge carriers, and what is its SI unit?
Mobility is the drift speed a carrier gains per unit electric field: μ = |v_d|/E = eτ/m. It measures how freely carriers move through a material; a longer relaxation time or a smaller mass gives a higher mobility. Its SI unit is m² V⁻¹ s⁻¹. In semiconductors electrons and holes have different mobilities, so the two kinds of carrier do not contribute equally to the current.
Limitations of Ohm's law
Read this section in the notes →When does Ohm's law fail?
Ohm's law fails whenever V is not simply proportional to I. That happens when the V-I graph is not a straight line, when reversing the voltage changes the size of the current and not just its direction, as in a diode, or when one voltage can give more than one current, as in GaAs. Ohm's law describes many materials well, especially metals, but it is not a basic law of nature.
Resistivity of materials and its temperature dependence
Read this section in the notes →Why does the resistance of a metal increase with temperature?
Because electrons collide more often when the metal is hotter, so the relaxation time τ falls. The number of free electrons per unit volume in a metal hardly changes with temperature. The ions vibrate more strongly and scatter electrons more frequently, and since ρ = m/(ne²τ), a shorter τ means a larger resistivity. Over a limited range this gives ρ_T = ρ₀[1 + α(T − T₀)] with α positive.
Why does the resistivity of a semiconductor decrease when it is heated?
Because heating frees many more charge carriers, and this outweighs the extra collisions. In ρ = m/(ne²τ), the relaxation time does fall with temperature, but the number density n of carriers rises sharply as more electrons gain enough energy to conduct. The net result is a falling resistivity, so a semiconductor has a negative temperature coefficient α, unlike a metal.
Electrical energy and power
Read this section in the notes →Which glows brighter in series, a 100 W bulb or a 25 W bulb?
The 25 W bulb glows brighter in series. Both ratings are for the same rated voltage, and R = V²/P, so the 25 W bulb has four times the resistance of the 100 W one. In series the same current flows through both, and the power I²R is larger in the larger resistance. Connected in parallel across the rated voltage, the 100 W bulb is the brighter one.
Why is electric power transmitted over long distances at very high voltage?
To reduce the heat wasted in the transmission cables. For a fixed power P sent at voltage V, the current is P/V, so the loss in cables of resistance R_c is P_c = P²R_c/V². Raising the voltage ten times cuts the loss a hundred times. Transformers step the voltage up at the power station and back down near homes, where high voltage would be dangerous.
Cells, emf and internal resistance
Read this section in the notes →What is the difference between the emf of a cell and its terminal voltage?
Emf is the voltage across the cell when no current is drawn, while terminal voltage is the voltage across it while it supplies current. A real cell has internal resistance r, so when it drives a current I the terminal voltage is V = ε − Ir, less than ε. The two are equal only on open circuit. With the terminals shorted, the current reaches its largest value, ε/r.
Can the terminal voltage of a cell ever be greater than its emf?
Yes, while the cell is being charged. If an outside source drives current into the cell against its emf, the voltage drop across the internal resistance adds to the emf instead of subtracting from it, giving V = ε + Ir. During normal use, when the cell itself drives the current, V = ε − Ir and the terminal voltage is less than the emf.
Cells in series and in parallel
Read this section in the notes →How do I find the equivalent emf and internal resistance of two cells in parallel?
With like terminals joined, use ε_eq = (ε₁r₂ + ε₂r₁)/(r₁ + r₂) and r_eq = r₁r₂/(r₁ + r₂). The internal resistances combine like parallel resistors, and the equivalent emf is a weighted average of the two. If one cell is connected the other way round, change the sign of its emf. In series it is simpler: emfs add with their signs and internal resistances just add.
Kirchhoff's rules
Read this section in the notes →Which conservation laws are behind Kirchhoff's two rules?
The junction rule comes from conservation of charge, and the loop rule from conservation of energy. In a steady state charge cannot collect at a junction, so the total current flowing in equals the total flowing out. A charge carried once round a closed loop returns to the potential it started at, so all the rises and drops in potential along the loop must add up to zero.
What sign should I use for a cell and a resistor in Kirchhoff's loop rule?
Crossing a cell from its negative to its positive terminal counts as +ε, and from positive to negative as −ε, whatever the current is doing. Crossing a resistor in the direction of the assumed current counts as −IR, and against it as +IR. Pick one direction to travel round the loop, apply these rules the same way throughout, and set the total to zero.
What happens if I assume the wrong direction for a current in a Kirchhoff problem?
Nothing goes wrong; that current simply comes out negative. A negative answer means the current actually flows opposite to the arrow you drew, with the magnitude you found. So you can choose current directions freely at the start. Just keep the same choice in every equation, and reverse the arrow only when you state the final answer.
Wheatstone bridge
Read this section in the notes →Why does no current flow through the galvanometer in a balanced Wheatstone bridge?
At balance the two ends of the galvanometer are at the same potential, so nothing drives charge through it. This happens when P/Q = R/S: each path divides the battery's voltage in the same ratio, so the two midpoints match. The condition does not involve the battery's emf or the galvanometer's resistance, which is why a balanced bridge can measure an unknown resistance accurately.
How can I tell whether a bridge-shaped network of five resistors is balanced?
Compare the ratio of the two resistors on one path with the ratio of the matching two on the other path. If the ratios are equal, the middle resistor carries no current and can be removed, leaving simple series and parallel groups. If they differ, the middle arm does carry current, and you have to solve the network with Kirchhoff's rules instead.
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