Unit 10: Electrostatics and Current Electricity

This is the highest-weight unit in NEET Physics — expect 4–5 MCQs combined. Electrostatics yields questions on point-charge force, dipoles, Gauss's law applications, and capacitor networks. Current Electricity tests Ohm's law, resistor networks, Kirchhoff's laws, and Wheatstone/potentiometer.

Concept Map

Electrostatics

• Coulomb's law, principle of superposition
• Electric field of a point charge, dipole, ring, infinite line, plane
• Gauss's law and applications
• Electric potential and equipotential surfaces
• Capacitors: parallel-plate, dielectrics, energy, combinations

Current Electricity

• Drift velocity and Ohm's law
• Resistivity, temperature dependence
• Resistor combinations
• EMF, internal resistance
• Kirchhoff's voltage/current laws
• Wheatstone bridge, meter bridge, potentiometer
• Heating effect (Joule's law)

Topic 1: Electric Charge and Coulomb's Law

Sub-topic A: Properties of Charge

• Charge is quantised: $q = n e$ with $e = 1.6 \times 10^{-19}\ \text{C}$.
• Charge is conserved: net charge of isolated system is constant.
• Charge is invariant: same in all inertial frames.
• Like charges repel, unlike attract.

Sub-topic B: Coulomb's Law

Force between two point charges $q_1$ and $q_2$ separated by $r$:

In a medium of dielectric constant $K$ (relative permittivity), force is reduced by factor $K$.

Topic 2: Electric Field

Sub-topic A: Definition

with $q_0$ a small positive test charge. Unit: N/C or V/m.

Sub-topic B: Field of a Point Charge

Sub-topic C: Field of a Dipole

Two charges $\pm q$ separated by small distance $2a$. Dipole moment $\vec p = q \cdot 2a$ from $-q$ to $+q$.

• Axial point (along dipole, distance $r$ from centre):

• Equatorial point (perpendicular to dipole):

Note: $E_\text{axial} = 2 E_\text{eq}$.

Sub-topic D: Field of Continuous Distributions

For an infinite line of linear charge density $\lambda$:

For an infinite sheet of surface charge density $\sigma$:

For a conductor surface with charge density $\sigma$:

For a uniformly charged ring of radius $R$ on axis at distance $x$:

Maximum at $x = R/\sqrt{2}$.

Topic 3: Gauss's Law

Sub-topic A: Applications

• Spherical shell of charge $Q$:

  • Outside ($r > R$): $E = kQ/r^2$.
  • Inside ($r < R$): $E = 0$ (cavity).

• Solid uniformly charged sphere of radius $R$ and total charge $Q$:

  • Outside: $E = kQ/r^2$.
  • Inside: $E = kQr/R^3$ (linear in $r$).

• Infinite cylinder: similar logic, gives $E = \lambda/(2\pi\varepsilon_0 r)$ for $r > R$.

Sub-topic B: Conductors in Electrostatics

• Inside a conductor: $E = 0$ in equilibrium.
• Charge resides on the surface.
• Field just outside conductor: $E = \sigma/\varepsilon_0$ (perpendicular).
• Cavity inside conductor: $E = 0$ inside cavity (Faraday cage).

Topic 4: Electric Potential

Sub-topic A: Definition

For a point charge: $V(r) = kq/r$.

Unit: volt (V) $=$ J/C.

Sub-topic B: Work and Energy

Work done by an external agent to bring charge $q$ from $\infty$ to $\vec r$: $W = qV(\vec r)$.

PE of two charges: $U = k q_1 q_2/r$.

Sub-topic C: Dipole in External Field

Torque: $\vec\tau = \vec p \times \vec E$, $\vert \tau\vert = pE\sin\theta$.

Potential energy: $U = -\vec p \cdot \vec E = -pE\cos\theta$.

Stable equilibrium at $\theta = 0$; unstable at $\theta = \pi$.

Sub-topic D: Equipotential Surfaces

• Surfaces of constant $V$.
• $\vec E \perp$ equipotential surface.
• No work done in moving charge along equipotential.
• For point charge: concentric spheres.
• For uniform field: planes perpendicular to field.

Topic 5: Capacitance

Sub-topic A: Definition

Unit: farad (F).

Sub-topic B: Parallel-Plate Capacitor

For plates of area $A$ separated by $d$ in vacuum:

With dielectric of constant $K$ filling the gap:

Partial filling (thickness $t$ of dielectric):

Sub-topic C: Energy Stored

Energy per unit volume in field: $u = \tfrac{1}{2}\varepsilon_0 E^2$.

Sub-topic D: Combinations

In series, charges are equal across each capacitor. In parallel, voltages are equal.

Sub-topic E: Connection / Sharing of Charge

Two capacitors with $Q_1$ and $Q_2$ on $C_1$ and $C_2$ connected in parallel:

Heat dissipated:

Topic 6: Current Electricity

Sub-topic A: Current and Drift Velocity

with $n$ free electron density, $A$ cross-section, $v_d$ drift speed (~ $10^{-4}\ \text{m/s}$).

Drift velocity:

Sub-topic B: Ohm's Law

Resistivity $\rho$ is material property; $R$ is geometry-dependent.

Sub-topic C: Temperature Dependence

with $\alpha > 0$ for metals (resistance increases with T). For semiconductors, $\alpha < 0$ (resistance falls with T).

Sub-topic D: Resistor Combinations

Topic 7: EMF and Internal Resistance

Sub-topic A: Cell Model

A real cell has emf $\varepsilon$ and internal resistance $r$. Terminal voltage:

When charging: $V = \varepsilon + Ir$.

Power delivered to load $R$:

Maximum at $R = r$ (impedance matching): $P_\text{max} = \varepsilon^2/(4r)$, efficiency 50%.

Sub-topic B: Cells in Series and Parallel

• Series (same direction): $\varepsilon_\text{net} = \varepsilon_1 + \varepsilon_2$, $r_\text{net} = r_1 + r_2$.
• Parallel (identical cells, $n$ of them): $\varepsilon_\text{net} = \varepsilon$, $r_\text{net} = r/n$.

Topic 8: Kirchhoff's Laws

Sub-topic A: Junction Rule (KCL)

At any junction, $\sum I_\text{in} = \sum I_\text{out}$ (conservation of charge).

Sub-topic B: Loop Rule (KVL)

Around any closed loop, $\sum \Delta V = 0$ (conservation of energy).

Sign convention: traverse loop and add $+\varepsilon$ if from $-$ to $+$ of cell, subtract IR drops if traversing in current direction.

Topic 9: Wheatstone Bridge and Meter Bridge

Sub-topic A: Wheatstone Bridge

Four resistors $P, Q, R, S$ in a bridge. Balance condition:

When balanced, no current through galvanometer; bridge insensitive to galvanometer resistance and cell EMF.

Sub-topic B: Meter Bridge

A 1 m wire of uniform resistance. Bridge balance at length $\ell$ from one end gives

Sub-topic C: Potentiometer

A long uniform wire with constant current. Potential gradient $k = V/L$. To compare two emfs $\varepsilon_1$ and $\varepsilon_2$ with balance lengths $\ell_1$ and $\ell_2$:

Advantage of potentiometer over voltmeter: at balance no current is drawn from the cell — so it measures true emf, not terminal voltage.

To find internal resistance: balance with cell alone (length $\ell_1$), then with cell in parallel with external $R$ (length $\ell_2$):

Topic 10: Heating Effect of Current (Joule's Law)

Power dissipated in resistor:

Energy: $E = Pt$ (joules).

In household: 1 unit = 1 kWh = $3.6 \times 10^6$ J.

NEET Pattern MCQ Tips

• Coulomb's law / 3-charge: forces along triangle.
• Dipole field: axial vs equatorial (factor 2).
• Gauss's law: choose Gaussian surface aligning with symmetry.
• Capacitor energy: before vs after dielectric insertion.
• Capacitor sharing: heat lost = $C_1 C_2 (V_1 - V_2)^2/[2(C_1 + C_2)]$.
• Resistor network: identify series/parallel.
• Kirchhoff numerical: 2-loop circuit.
• Wheatstone / Meter bridge: balance condition.
• Potentiometer: cell emf comparison.
• Drift velocity: $I = nAev_d$.
• Assertion-Reason: equipotential surfaces, current direction.

Common Confusions and Traps

• Like charges of $+q$ have $U = +kq^2/r > 0$; unlike have $U < 0$.
• Field inside a charged spherical shell is zero but potential is non-zero (constant equal to surface value).
• Dielectric constant $K$ increases capacitance ($C = KC_0$).
• When a battery is disconnected and dielectric inserted: $Q$ stays same, $V$ decreases, $E$ decreases, energy decreases.
• When battery remains connected and dielectric inserted: $V$ stays same, $Q$ increases, $E$ stays same, energy increases.
• The drift velocity is very small (~ $10^{-4}\ \text{m/s}$) yet current travels nearly at $c$ because the electric field establishes itself quickly.
• Resistivity is a property of the material; resistance is a property of the piece.
• A galvanometer does not read EMF directly; one needs a potentiometer.
• Internal resistance lowers the terminal voltage below EMF when current flows.

Quick Revision Card

• $F = kq_1q_2/r^2$; $k = 9 \times 10^9$ Nm²/C².
• $E_\text{point} = kq/r^2$; $V_\text{point} = kq/r$.
• Dipole axial: $2kp/r^3$; equatorial: $kp/r^3$.
• Infinite line: $E = \lambda/(2\pi\varepsilon_0 r)$.
• Infinite sheet: $E = \sigma/(2\varepsilon_0)$.
• Charged shell: $E_\text{in} = 0$, $E_\text{out} = kQ/r^2$.
• $C_\text{parallel-plate} = \varepsilon_0 A/d$; with dielectric $\times K$.
• Series caps: $1/C_\text{eq}$; parallel: add.
• Energy: $U = Q^2/(2C) = CV^2/2$.
• $I = nAev_d$; $\rho = m/(ne^2\tau)$.
• Series R: add; parallel: $1/R_\text{eq}$.
• Wheatstone balance: $P/Q = R/S$.
• Potentiometer: $\varepsilon_1/\varepsilon_2 = \ell_1/\ell_2$.
• $P = I^2 R = V^2/R$.

Worked NEET Examples

Example 1: Force Between Charges

Two charges $+10\ \mu$C and $+20\ \mu$C separated by 10 cm in vacuum. Force:

Example 2: Field at Centre of Square

Four equal charges +q at corners of a square of side $a$. Net field at centre = 0 (by symmetry).

If the four charges are +q, +q, +q, -q, then by symmetry only the diagonals matter. Net field has component from the imbalance, pointing toward -q.

Example 3: Capacitor with Dielectric

A parallel plate capacitor (1 μF in vacuum) is connected to a 100 V battery. Dielectric of $K = 4$ is inserted while battery is connected. New charge:

Before: $Q_0 = CV = 10^{-6} \times 100 = 10^{-4}$ C. After: $C' = 4 \mu$F. $Q' = C'V = 4 \times 10^{-4}$ C. So Q quadruples; energy quadruples; field unchanged.

If battery is disconnected first, then dielectric inserted: $Q$ unchanged, $C$ → 4C, $V$ → V/4, $E$ → E/4, energy → energy/4.

Example 4: Drift Velocity

Copper wire 1 mm² cross-section carrying 1 A. Free electron density $n = 8.5 \times 10^{28}/$m³.

Very small drift velocity — yet current flows nearly instantly because the electric field is established at the speed of light.

Example 5: Wheatstone Bridge

In a Wheatstone bridge: $P = 4\Omega, Q = 8\Omega, R = 6\Omega$. For balance, $P/Q = R/S$, so $S = R Q/P = 6 \times 8/4 = 12\ \Omega$.

Derivations

Field of an Infinite Line of Charge (Gauss)

Cylinder of radius $r$, length $L$ coaxial with wire. By symmetry, E is radial.

So $E = \lambda/(2\pi\varepsilon_0 r)$.

Field Near Infinite Sheet (Gauss)

Pillbox crossing the sheet, area $A$ each face. Flux: $2 E A = \sigma A/\varepsilon_0$, so $E = \sigma/(2\varepsilon_0)$ — independent of distance.

Field Inside a Solid Charged Sphere

For uniform volume density $\rho$ in a sphere of radius $R$. At radius $r < R$, enclosed charge: $q_\text{en} = \rho \cdot (4/3)\pi r^3$.

$E = \rho r/(3\varepsilon_0)$. Equivalently $E = kQ r/R^3$.

Capacitance of Parallel Plates

For two oppositely-charged plates with $\sigma = Q/A$:

$E$ between plates = $\sigma/\varepsilon_0 = Q/(\varepsilon_0 A)$.

Voltage: $V = E d = Qd/(\varepsilon_0 A)$.

Capacitance: $C = Q/V = \varepsilon_0 A/d$.

Energy in a Capacitor

Work done to charge from 0 to $Q$: $W = \int_0^Q V\,dq = \int_0^Q (q/C)\,dq = Q^2/(2C)$.

Equivalently: $U = (1/2)CV^2 = (1/2)QV$.

Drift Velocity from Ohm's Law

In a wire with electric field $E$, electrons accelerate, then collide. Between collisions, gained velocity $\Delta v = eE\tau/m$ (where $\tau$ is mean free time). On average, $v_d = eE\tau/m$.

Current density: $j = ne v_d = ne^2\tau E/m$. So conductivity $\sigma = ne^2\tau/m$ and resistivity $\rho = m/(ne^2\tau)$.

Special Topics

Field of a Uniformly Charged Ring

On the axis at distance $x$: $E = kQx/(R^2 + x^2)^{3/2}$. Maximum at $x = R/\sqrt 2$, value: $kQ/(3\sqrt 3 R^2/2)$.

Field of a Uniformly Charged Disc

On the axis at distance $x$ (disc radius $R$):

Limits: $x \to 0$ → $E = \sigma/(2\varepsilon_0)$ (infinite sheet result). $x \gg R$ → $E \to kQ/x^2$ (point charge).

Energy Density of Electric Field

Energy stored per unit volume in an electric field:

For a charged capacitor, integrating $u$ over the volume between plates recovers $(1/2)CV^2$.

Kirchhoff Solving Strategy

Label currents in each branch. Apply junction rule (KCL) at each node. Apply loop rule (KVL) for independent loops. Solve the linear system.

For a 2-loop circuit with two unknowns, you get two equations.

Maximum Power Transfer in Battery

Power delivered to external resistance $R$ by cell with EMF $\varepsilon$, internal $r$:

$dP/dR = 0$ gives $R = r$, $P_\text{max} = \varepsilon^2/(4r)$, efficiency 50%.

Cells in Combinations

N Identical Cells in Series

$\varepsilon_\text{net} = N\varepsilon$, $r_\text{net} = Nr$. Current through external $R$: $I = N\varepsilon/(R + Nr)$.

For $R \gg r$: $I \approx N\varepsilon/R$ (use series).

N Identical Cells in Parallel

$\varepsilon_\text{net} = \varepsilon$, $r_\text{net} = r/N$. Current: $I = \varepsilon/(R + r/N) = N\varepsilon/(NR + r)$.

For $R \ll r$: $I \approx N\varepsilon/r$ (use parallel).

N Cells in M Rows

A 2D array of $N$ cells (each in series within a row, $M$ rows in parallel): condition for max current is $NR_\text{external} = Mr$.

Heating and Power Calculations

A common NEET question: a 100 W, 220 V bulb run on 110 V dissipates $100 (110/220)^2 = 25$ W.

Formula Sheet