Current Electricity: NEET notes

1. Electric current

NCERT § "Electric Current"

• Electric current through a cross-section is the net charge crossing it per unit time: I = q/t for a steady current, or I = dq/dt in general.
• The SI unit is the ampere (A), a base unit; 1 A corresponds to 1 C of charge passing per second.
• By convention the direction of current is the direction in which positive charge would flow, opposite to the motion of electrons in a metal.
• Current has a direction along the wire but is a scalar, because currents at a junction add algebraically, not by vector addition.
• Examples span a huge range: lightning carries very large currents for a short time, while currents in electronic devices can be of the order of microamperes or less.

2. Electric currents in conductors and Ohm's law

NCERT § "Ohm's Law"

• In metals the charge carriers are free electrons; in electrolytes both positive and negative ions carry current.
• Without an applied field, free electrons move randomly in all directions and there is no net flow; a steady current needs a steady electric field maintained by a source such as a cell.
• Ohm's law: the current through a conductor is proportional to the potential difference across it, V = IR, where R is its resistance (unit ohm, Ω).
• Resistance depends on the dimensions of the conductor: R = ρl/A, directly proportional to length and inversely to area of cross-section.
• Resistivity ρ depends on the material and the temperature, not on the size or shape; its SI unit is Ω m.
• Current density j is current per unit area normal to the flow (unit A m⁻²); Ohm's law in vector form is j = σE, where σ = 1/ρ is the conductivity.
• When a wire of fixed volume is stretched to n times its length, its area becomes 1/n times, so its resistance becomes n² times.

3. Electron drift and where resistivity comes from

NCERT § "Drift of Electrons and the Origin of Resistivity"

• Free electrons collide frequently with the ions of the lattice. The average time between successive collisions is the relaxation time τ.
• An applied field E gives each electron an acceleration −eE/m between collisions, producing a small average velocity opposite to E, the drift velocity v_d = −eEτ/m.
• The current through a conductor of area A with n free electrons per unit volume is I = neAv_d.
• Drift speeds are very small, typically a fraction of a millimetre to a few millimetres per second, while the random thermal speeds of electrons are very large.
• A lamp lights almost at once when switched on because the electric field is set up along the whole wire nearly instantly, starting all the electrons drifting together; it is not because electrons rush from the switch.
• Combining the two results gives the resistivity in terms of microscopic quantities: ρ = m/(ne²τ).
• For a wire of non-uniform cross-section carrying a steady current, the current is the same at every section, so the drift speed is larger where the wire is thinner.

4. Mobility

NCERT § "Mobility"

• Mobility μ is the magnitude of drift velocity per unit electric field: μ = |v_d|/E.
• Its SI unit is m² V⁻¹ s⁻¹.
• For electrons, μ = eτ/m, so a longer relaxation time gives a higher mobility.
• Conductivity can be written in terms of mobility: σ = neμ for one type of carrier.
• Mobility is defined as a positive quantity for every carrier; the direction of drift is decided separately by the sign of the charge.

5. Limitations of Ohm's law

NCERT § "Limitations of Ohm's Law"

• Ohm's law is an empirical rule that many metals obey closely, not a fundamental law of nature.
• Deviation 1: the V-I relation may stop being linear, for example as a conductor heats up at larger currents.
• Deviation 2: the relation may depend on the sign of V, so reversing the voltage does not simply reverse the current with the same size (as in a diode).
• Deviation 3: the relation may not be single-valued, with more than one current for the same voltage (as in gallium arsenide).
• Materials and devices that do not follow V = IR are called non-ohmic.

6. Resistivity of materials and its temperature dependence

NCERT § "Resistivity of Various Materials" and § "Temperature Dependence of Resistivity"

• By resistivity, materials fall into conductors (metals, low ρ), semiconductors (intermediate ρ) and insulators (very high ρ).
• For metals over a limited range, ρ_T = ρ₀[1 + α(T − T₀)], where α is the temperature coefficient of resistivity; α is positive for metals.
• Metals become more resistive when heated mainly because the relaxation time τ falls (more frequent collisions); n hardly changes.
• Alloys such as nichrome, manganin and constantan have high resistivity and very small temperature coefficient, so they are used in heating elements and standard resistors.
• For semiconductors resistivity decreases with rising temperature (negative α), because the number density of charge carriers n increases with temperature.
• For insulators and semiconductors, the rise in n with temperature outweighs the fall in τ, which is why their behaviour is opposite to metals.
• In problems, ignore the change in the wire's dimensions on heating unless told otherwise; then R changes in the same ratio as ρ.

7. Electrical energy and power

NCERT § "Electrical Energy, Power"

• When charge moves through a potential difference V, the energy transferred per unit time is P = VI.
• For a resistor, the power dissipated as heat is P = I²R = V²/R (Joule heating).
• Two appliances rated for the same voltage: the one with the higher power rating has the lower resistance, since R = V²/P.
• In series, the same current flows, so the higher-resistance device dissipates more power; a lower-rated bulb glows brighter when connected in series with a higher-rated one.
• In parallel, the same voltage is applied, so the lower-resistance device dissipates more power.
• Power is sent over long distances at high voltage: for a fixed power P delivered through cables of resistance R_c, the loss P²R_c/V² falls sharply as V is raised.
• The kilowatt hour is the commercial unit of electrical energy, 1 kWh = 3.6 × 10⁶ J.

8. Cells, emf and internal resistance

NCERT § "Cells, emf, Internal Resistance"

• The emf ε of a cell is the potential difference between its terminals when no current is drawn; despite the name, it is a potential difference, not a force.
• Charge moving through the electrolyte meets an internal resistance r.
• When the cell drives a current I through an external resistor R, I = ε/(R + r) and the terminal voltage is V = ε − Ir, which is less than ε.
• The terminal voltage equals ε only on open circuit (I = 0), and falls to zero when the terminals are shorted, where the maximum current ε/r flows.
• Two readings with different external resistors give two equations that can be solved for both ε and r.
• The internal resistance of a car battery is very small, so it can supply the large currents a starter motor needs; dry cells have larger internal resistance.

9. Cells in series and in parallel

NCERT § "Cells in Series and in Parallel"

• Two cells in series (positive of one to negative of the next) behave like one cell of emf ε_eq = ε₁ + ε₂ and internal resistance r_eq = r₁ + r₂.
• If one cell in a series pair is reversed, its emf is subtracted: ε_eq = ε₁ − ε₂, while the internal resistances still add.
• For n identical cells in series: emf nε and internal resistance nr.
• Two cells in parallel (like terminals together) behave like one cell with ε_eq = (ε₁r₂ + ε₂r₁)/(r₁ + r₂) and r_eq = r₁r₂/(r₁ + r₂).
• The parallel formula can be written 1/r_eq = 1/r₁ + 1/r₂ and ε_eq/r_eq = ε₁/r₁ + ε₂/r₂; if one cell is reversed, its ε changes sign.
• For n identical cells in parallel: emf ε and internal resistance r/n.
• When two cells of unequal emf are joined in parallel with no external load, a current circulates between them, driven through the lower-emf cell by the higher-emf one.

10. Kirchhoff's rules

NCERT § "Kirchhoff's Rules"

• Junction rule: the total current flowing into any junction equals the total current flowing out of it. It follows from conservation of charge in steady state.
• Loop rule: going around any closed loop, the changes in potential add up algebraically to zero. It follows from the electrostatic force being conservative (energy conservation).
• Assign a current with an assumed direction to each branch; if the solution gives a negative value, the actual current flows the other way, and the magnitude is still correct.
• Going through a resistor in the direction of the assumed current, the potential falls by IR; going against it, it rises by IR.
• Going through a cell from its negative to positive terminal, the potential rises by ε; from positive to negative, it falls by ε.
• Use the junction rule first to reduce the number of unknown currents, then write as many independent loop equations as are needed.
• Symmetry in a network (equal potentials at symmetric points) can remove resistors that carry no current and simplify the work.

11. Wheatstone bridge

NCERT § "Wheatstone Bridge"

• Four resistors form a closed loop; a battery is connected across one pair of opposite corners and a galvanometer across the other pair.
• The bridge is balanced when no current flows through the galvanometer; the two ends of the galvanometer are then at the same potential.
• Balance condition, using our own labels: the battery is across corners A and C; path A→B→C has P (from A to B) then Q (B to C), and path A→D→C has R (A to D) then S (D to C); the galvanometer joins B and D. The bridge is balanced when P/Q = R/S, which is the same as PS = QR (products of opposite arms equal).
• Whatever letters a figure uses, check the condition this way: take the two resistors meeting at one battery corner; each one's ratio to the resistor that follows it on its own path must be the same for both paths.
• At balance, the galvanometer branch (or any resistor put in its place) can be removed, and the network reduces to two parallel series-pairs.
• The balance condition does not depend on the emf of the battery or the resistance of the galvanometer.
• If the ratios are not equal, the bridge is unbalanced and the middle resistor carries current; then Kirchhoff's rules must be used.
• The bridge lets an unknown resistance be found from three known ones by adjusting one of them until the galvanometer shows no deflection.

Must-know facts

1. I = dq/dt; ampere is an SI base unit; current is a scalar.
2. Conventional current flows opposite to electron motion.
3. V = IR; R = ρl/A; ρ in Ω m depends on material and temperature only.
4. j = σE with σ = 1/ρ; unit of j is A m⁻².
5. Drift velocity v_d = −eEτ/m; I = neAv_d.
6. ρ = m/(ne²τ); mobility μ = |v_d|/E = eτ/m, unit m² V⁻¹ s⁻¹.
7. Drift speed is tiny (about mm/s or less); signals travel fast because the field is set up almost instantly.
8. In a tapering wire with steady current, current is the same everywhere; drift speed and current density are larger at the narrow end.
9. Stretching a wire to n times its length at constant volume makes R n² times.
10. Metals: α > 0 because τ decreases on heating; semiconductors: α < 0 because n increases on heating.
11. Nichrome, manganin and constantan: high resistivity, low temperature coefficient.
12. P = VI = I²R = V²/R; rated resistance R = V²/P.
13. In series the lower-wattage bulb glows brighter; in parallel the higher-wattage bulb glows brighter.
14. Terminal voltage V = ε − Ir while discharging; V = ε on open circuit.
15. Series cells: emfs and internal resistances add (reversed cell's emf subtracts).
16. n identical cells in parallel: emf ε, internal resistance r/n.
17. Junction rule = charge conservation; loop rule = energy conservation.
18. Balanced Wheatstone bridge: P/Q = R/S (P, Q along one path; R, S along the other, each listed from the same battery corner), i.e. products of opposite arms are equal; the galvanometer arm carries no current.

Common traps

Formulas

Key terms

Electric current

The net rate of flow of charge through a cross-section.

Current density

Current per unit area, with the area taken normal to the direction of flow.

Resistivity

A material's inherent opposition to current, independent of the sample's dimensions.

Conductivity

The reciprocal of resistivity.

Relaxation time

The average time between successive collisions of a free electron.

Drift velocity

The small average velocity of free electrons along a conductor due to an applied field.

Mobility

Drift speed of a charge carrier per unit electric field.

Non-ohmic device

A material or device whose current does not vary in direct proportion to voltage.

Temperature coefficient of resistivity

The fractional change in resistivity per unit change in temperature.

Joule heating

Conversion of electrical energy into heat in a resistor, at a rate I²R.

emf

The potential difference between the terminals of a cell when it supplies no current.

Internal resistance

The resistance to current inside a cell, mainly from its electrolyte.

Terminal voltage

The potential difference across a cell's terminals when it is in a circuit.

Junction

A point in a circuit where three or more conductors meet.

Wheatstone bridge

A four-resistor arrangement used to compare resistances by finding a null in a galvanometer.