Current Electricity Notes for JEE by PW

Current Electricity connects the microscopic movement of charge carriers with the behaviour of electrical circuits. Understanding drift velocity, resistance, resistivity, circuit laws, and measuring instruments is essential for solving Class 12 Physics numericals and competitive examination questions. You must also know when to apply individual formulas, how to track potential differences, and how to simplify complex circuits.

A systematic revision of the chapter helps connect these concepts with practical circuit problems. Physics Wallah supports JEE aspirants through dedicated courses for Class 11, Class 12, and droppers, along with live and recorded classes, handwritten notes, Daily Practice Problems (DPPs), mock tests, previous years’ questions (PYQs), and regular assessments.

Electric Potential, Current, and Charge Flow

Electric potential is measured relative to a reference point. A battery maintains a potential difference between its terminals and establishes an electric field in a connected circuit, driving charge carriers through the circuit.

Electric current is the rate of flow of electric charge.

Instantaneous current: I = dq/dt

Average current: I_avg = ΔQ/Δt

SI unit: 1 A = 1 C/s

Current is a scalar quantity. Conventional current follows the direction in which positive charges would move. In metallic conductors, electrons move opposite to conventional current.

For a time-dependent current such as I(t) = 20 + 3t amperes, the charge transferred from t = 0 to t = 20 s is:

Q = ∫₀²⁰ I(t) dt

Q = ∫₀²⁰ (20 + 3t) dt

Q = [20t + (3t²/2)]₀²⁰

Q = 400 + 600 = 1000 C

Memory tip: In metallic conductors, electron motion is opposite to conventional current.

Microscopic Motion and Drift Velocity

Electrons in a conductor move randomly even when no electric field is applied. Their average velocity is zero because their random motion has no preferred direction.

When an electric field is established, electrons acquire a small average velocity called drift velocity. Electron drift is opposite to the electric field, whereas conventional current is in the direction of the field.

The average time between successive collisions is called relaxation time, represented by τ.

The drift velocity of electrons is: v⃗_d = −eτE⃗/m

Here, e is the magnitude of the electronic charge, τ is relaxation time, E⃗ is the electric field, and m is the electron mass.

The magnitude of drift velocity is: v_d = eEτ/m

The relation between current and drift velocity is: I = neAv_d

where:

• n = number density of free electrons

• e = magnitude of electronic charge

• A = cross-sectional area

• v_d = magnitude of drift velocity

The drift velocity is generally small compared with the random speeds of electrons.

Ohm’s Law, Resistance, and Resistivity

For an ohmic conductor maintained under constant physical conditions, the potential difference across it is directly proportional to the current flowing through it.

Ohm’s law:

V = IR

If voltage is plotted against current, the slope of the straight-line graph gives resistance.

Ohm’s law does not apply universally to every material or operating condition.

Resistance of a uniform conductor: R = ρL/A

where ρ is resistivity, L is conductor length, and A is cross-sectional area.

Resistivity depends on the material and its physical conditions, whereas resistance also depends on the conductor’s dimensions.

The microscopic expression for resistivity in the classical Drude model is: ρ = m/(ne²τ)

Conductivity is the reciprocal of resistivity: σ = 1/ρ

Temperature Dependence of Resistivity

For many metallic conductors over a limited temperature range, resistivity increases with temperature because increased lattice vibrations reduce relaxation time.

For many semiconductors, resistivity decreases as temperature rises because the number of charge carriers increases significantly.

The approximate linear temperature relation for resistivity is:

ρ = ρ₀[1 + α(T − T₀)]

Similarly, for resistance:

R = R₀[1 + α(T − T₀)]

Here, ρ₀ and R₀ are the resistivity and resistance at reference temperature T₀, and α is the temperature coefficient of resistivity or resistance at that reference temperature.

When the reference temperature is 0°C, the expressions are often written as:

ρₜ = ρ₀(1 + αt)

Rₜ = R₀(1 + αt)

These linear relations are approximations and should be used over the temperature range for which they are valid.

Circuit Laws and Resistor Combinations

Circuit analysis requires tracking current directions and potential changes consistently. A potential rise occurs when moving from lower to higher potential, while a potential drop occurs when moving from higher to lower potential.

Kirchhoff’s Current Law (KCL)

Kirchhoff’s current law follows from conservation of charge. The total current entering a junction equals the total current leaving it.

ΣI = 0

For example, if currents I₁ and I₂ enter a junction and I₃ and I₄ leave it: I₁ + I₂ = I₃ + I₄

An assumed current direction is acceptable. A negative calculated value indicates that the actual current flows opposite to the assumed direction.

Kirchhoff’s Voltage Law (KVL)

Kirchhoff’s voltage law follows from conservation of energy. The algebraic sum of potential changes around a closed loop is zero.

ΣΔV = 0

When writing a loop equation, account for the sign of each cell’s emf and the potential change across each resistor according to the direction of traversal.

Resistors in Series

For resistors connected in series:

• The same current flows through every resistor.

• The supply voltage is divided across the resistors.

• The equivalent resistance is the sum of individual resistances.

R_eq = R₁ + R₂ + R₃ + ...

For two resistors in series: V₁/V₂ = R₁/R₂

The equivalent resistance is greater than any individual resistance when all resistances are positive.

Resistors in Parallel

For resistors connected in parallel:

• The potential difference across each branch is the same.

• The total current is the sum of the branch currents.

• Current divides inversely with resistance.

The equivalent resistance is: 1/R_eq = 1/R₁ + 1/R₂ + 1/R₃ + ...

For two resistors in parallel: R_eq = R₁R₂/(R₁ + R₂)

The equivalent resistance is less than the smallest branch resistance when all branch resistances are positive.

Complex Circuits, Bridges, and Symmetry

Points joined by an ideal wire without any intervening circuit element have the same potential. Identifying these nodes helps determine whether resistors are actually in series or parallel.

A resistor connected between two equal-potential points has zero potential difference across it. Therefore, no current flows through that resistor, and it can be ignored when calculating the remaining circuit currents.

Wheatstone Bridge

A Wheatstone bridge is balanced when the ratio of the resistances in one pair of adjacent arms equals the ratio in the other pair.

For the standard labelling of a bridge with arms R₁, R₂, R₃, and R₄: R₁/R₂ = R₃/R₄

At balance, the potential difference between the galvanometer terminals is zero, so no current flows through the galvanometer.

For an unbalanced bridge, use Kirchhoff’s laws, nodal analysis, or an appropriate circuit transformation.

Nodal Analysis

Nodal analysis determines unknown node potentials by applying Kirchhoff’s current law.

For a resistor R connecting nodes at potentials V₁ and V₂, the current from node 1 to node 2 is:

I₁₂ = (V₁ − V₂)/R

Choose a reference node with potential zero, express the other node potentials as variables, and apply KCL at each unknown node.

Symmetry in Circuits

Symmetry can simplify a circuit before applying Kirchhoff’s laws.

• Equal-potential points: No current flows through a resistor connected between them.

• Symmetric branches: Equivalent branches may carry equal currents when their components and connections are identical.

• Infinite repeating circuits: The equivalent resistance can be found by assigning the repeating network an unknown resistance and applying its self-similar structure.

For every symmetry argument, verify that the circuit connections and component values actually preserve the required symmetry.

Cells, Internal Resistance, Heating, and Power

A practical cell has an electromotive force (emf) and internal resistance. The terminal voltage depends on whether the cell is discharging or being charged.

For a cell with emf E and internal resistance r:

During discharge: V = E − Ir

During charging: V = E + Ir

When no current flows: V = E

For a cell connected to an external resistance R during discharge: I = E/(R + r)

Maximum Power Transfer

The power delivered to an external resistance is maximum when the external resistance equals the internal resistance of the source.

R = r

The maximum power delivered to the external resistance is: P_max = E²/(4r)

This result assumes a source with constant emf E and internal resistance r.

Combination of Identical Cells

For n identical cells, each with emf E and internal resistance r:

Cells connected in series:

E_eq = nE

r_eq = nr

Cells connected in parallel:

E_eq = E

r_eq = r/n

If there are m parallel rows, each containing n identical cells in series:

E_eq = nE

r_eq = nr/m

These relations assume identical cells connected with the same polarity.

Heating Effect of Electric Current

Joule’s law of heating gives the heat produced in a resistor over time t.

H = I²Rt

For a resistor with constant resistance, equivalent expressions are:

H = VIt

H = V²t/R

Electrical Power

Electrical power is the rate at which electrical energy is transferred.

P = VI

Using Ohm’s law:

P = I²R

P = V²/R

Power and Resistance of Bulbs

For a bulb operating at its rated voltage and rated power:

R = V_rated²/P_rated

In series, bulbs carry the same current. For bulbs with the same rated voltage operating below their ratings, the bulb with the higher resistance dissipates more power at the same current.

In parallel, each bulb receives the same supply voltage. The bulb’s actual power depends on its operating voltage and resistance; rated-power comparisons assume suitable operating conditions.

Electrical Measuring Instruments

Ammeter

An ammeter measures current and is connected in series with the circuit branch. An ideal ammeter has zero resistance.

A galvanometer of resistance G and full-scale current I_g can be converted into an ammeter of range I by connecting a shunt resistance S in parallel.

S = I_gG/(I − I_g)

Here, I must be greater than I_g.

Voltmeter

A voltmeter measures the potential difference between two points and is connected in parallel with the component. An ideal voltmeter has infinite resistance.

A galvanometer of resistance G and full-scale current I_g can be converted into a voltmeter of range V by connecting a resistance R_s in series.

R_s = V/I_g − G

The total resistance of the converted voltmeter is: R_total = V/I_g

Potentiometer

A potentiometer compares potential differences using a uniform wire carrying a steady current. At the balance point, the galvanometer shows zero deflection, meaning no current flows through the galvanometer branch.

If k is the potential gradient and l is the balancing length: E = kl

For two cells balanced at lengths l₁ and l₂: E₁/E₂ = l₁/l₂

To determine the internal resistance r of a cell, let l₁ be the balancing length for its emf when the external circuit is open, and l₂ be the balancing length for its terminal voltage when it supplies current through an external resistance R.

Then: E/V = l₁/l₂

The internal resistance is: r = R(E/V − 1)

Substituting the balancing-length ratio: r = R(l₁ − l₂)/l₂

This method assumes the potentiometer current and potential gradient remain constant during the measurements.

Physics Wallah supports JEE preparation through dedicated courses for Class 11, Class 12, and droppers, along with live and recorded classes, handwritten notes, DPPs, mock tests, PYQs, and regular assessments. Revise the formulas topic by topic, practise circuit-based numericals, and use assessments to identify concepts that need more attention.

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