Physics — NEET G12 Benchmarking Test-01

1 · Electrostatic Potential

P1 · every year

The definition that everything hangs on

Potential at a point = work done by an external agent to bring a unit positive charge from infinity to that point, slowly (no kinetic energy gained).

V = W_ext / q₀ unit: volt = joule/coulomb = J C⁻¹ W_ext = q(V_B − V_A) W_field = −W_ext = q(V_A − V_B) Potential is a scalar — add with signs, never with vectors.

Standard results — memorise the whole column

Charge configuration	Potential V	Field E	Watch out
Point charge q at distance r	kq/r	kq/r²	V carries the sign of q; E is magnitude
Group of charges	Σ kq_i/r_i (scalar sum)	vector sum	V can be 0 where E ≠ 0, and vice-versa
Dipole, general point (r ≫ a)	kp cosθ / r²	(kp/r³)√(1+3cos²θ)	θ measured from +q side of dipole axis
Dipole — axial (θ = 0)	kp/r²	2kp/r³	maximum V
Dipole — equatorial (θ = 90°)	0	kp/r³	V = 0 but E ≠ 0 — classic trap
Charged spherical shell (R), outside	kq/r	kq/r²	behaves like point charge at centre
Shell, on surface	kq/R	kq/R²	V continuous, E jumps
Shell, inside	kq/R (constant)	0	V ≠ 0 inside even though E = 0
Solid non-conducting sphere, inside	kq(3R²−r²)/2R³	kqr/R³	V_centre = 1.5 × V_surface
Infinite line charge λ	−2kλ ln r + C	2kλ/r	no zero at infinity — only ΔV meaningful

k = 1/4πε₀ = 9 × 10⁹ N m² C⁻² · ε₀ = 8.854 × 10⁻¹² C² N⁻¹ m⁻²

Trap — V and E are independent Four cases NEET rotates between:

E = 0, V ≠ 0 → inside a charged shell / conductor
E ≠ 0, V = 0 → equatorial point of a dipole; midpoint between +q and −q
E = 0, V = 0 → far away (infinity)
E ≠ 0, V ≠ 0 → ordinary point near a single charge

Dipole: potential vanishes on the equatorial plane, the field does not.

2 · Equipotential Surfaces & E = −dV/dr

Five properties — one is asked almost every year

Work done in moving a charge on an equipotential surface = zero (any path).
Field is always perpendicular to the equipotential surface.
Field points from high V to low V (direction of steepest fall).
Two equipotential surfaces can never intersect (a point would need two potentials).
Surfaces are closer together where E is stronger; equally spaced & parallel in a uniform field.

E = −dV/dr → E_x = −∂V/∂x, E_y = −∂V/∂y, E_z = −∂V/∂z dV = −E·dr → V_B − V_A = −∫_A^B E·dr Uniform field: V = −Ed (d measured along E) · 1 V m⁻¹ = 1 N C⁻¹

Field lines (red) always cut equipotentials (green) at 90°.

3 · Electrostatic Potential Energy

Two charges: U = k q₁q₂ / r₁₂ (with signs — attraction gives U < 0) Three charges: U = k[ q₁q₂/r₁₂ + q₂q₃/r₂₃ + q₁q₃/r₁₃ ] — one term per pair, ⁿC₂ pairs Charge in external field: U = qV · system + field: U = kq₁q₂/r + q₁V(r₁) + q₂V(r₂)

Dipole in a uniform external field — the whole set

τ = p × E → τ = pE sin θ (max at θ = 90°, zero at 0° and 180°) U(θ) = −p·E = −pE cos θ W (rotating θ₁ → θ₂) = pE(cos θ₁ − cos θ₂) Net force on a dipole in a uniform field = 0; it only rotates. In a non-uniform field it also translates.

θ	0°	90°	180°
U	−pE (minimum)	0	+pE (maximum)
τ	0	pE (maximum)	0
Equilibrium	stable	—	unstable

Memory hook W = pE(cos θ₁ − cos θ₂) — the initial angle comes first. Aamirah, if you write (cos θ₂ − cos θ₁) you get the sign backwards and pick the negative option. Initial minus final.

4 · Conductors, Dielectrics & Polarisation

Conductor in an electrostatic field

E_inside = 0 (free electrons rearrange until they cancel the applied field)
Entire conductor is one equipotential volume; surface is an equipotential
All excess charge resides on the outer surface
Just outside: E = σ/ε₀, normal to the surface
σ is largest where curvature is largest (sharp points) → corona discharge, lightning rods
Electrostatic shielding: field inside a cavity (no charge in it) is zero, whatever happens outside

Dielectrics

Non-polar (H₂, O₂, CO₂, CH₄, benzene): zero dipole moment until a field is applied
Polar (H₂O, HCl, NH₃): permanent dipole moment, randomly oriented until a field aligns them
Polarisation P = χ_e ε₀ E; K = 1 + χ_e (K ≥ 1 always)
Net field inside: E = E₀/K — reduced, never reversed
Bound surface charge σ_b = P; E_induced = E₀(1 − 1/K)
Conductor = dielectric with K → ∞

Polarisation weakens the internal field, so the same charge sits at a lower voltage → C rises K-fold.

5 · Capacitance

P1 · numerical-heavy

C = Q/V · unit farad (F) = C V⁻¹ · 1 μF = 10⁻⁶ F, 1 pF = 10⁻¹² F C depends only on geometry and the medium — not on Q or V.

Capacitor	Capacitance	Key point
Isolated sphere, radius R	C = 4πε₀R = R/k	Earth: C ≈ 711 μF
Parallel plate, vacuum	C₀ = ε₀A/d	E = σ/ε₀ between plates, V = Ed
Fully filled with dielectric K	C = Kε₀A/d = KC₀	rises K-fold
Dielectric slab thickness t < d	C = ε₀A / (d − t + t/K)	position of slab is irrelevant
Conducting slab thickness t	C = ε₀A / (d − t)	put K → ∞ in the row above
Two dielectrics stacked (⊥ to plates)	series: d₁/K₁ + d₂/K₂ in denominator	same charge through both
Two dielectrics side-by-side (∥)	C = ε₀(K₁A₁ + K₂A₂)/d	same voltage across both
Spherical capacitor (a inside, b outside)	C = 4πε₀ ab/(b − a)	—

Trap — "position of the slab doesn't matter" For a slab of thickness t inside a plate separation d, sliding it up or down changes nothing. Only t and K matter. NEET loves offering "depends on position" as a distractor.

6 · Series and Parallel Combinations

Quantity	Series	Parallel
Same for all	Charge Q	Voltage V
Equivalent C	1/C = 1/C₁ + 1/C₂ + …	C = C₁ + C₂ + …
Two capacitors	C = C₁C₂/(C₁+C₂)	C = C₁ + C₂
C_eq compared to members	smaller than the smallest	larger than the largest
Voltage division	V₁ = V·C₂/(C₁+C₂) — inverse to C	equal
Charge division	equal	Q₁ = Q·C₁/(C₁+C₂) — direct to C
n identical capacitors C	C/n	nC

Memory hook — capacitors are the mirror of resistors Resistors add in series; capacitors add in parallel. Whenever you feel unsure, ask "which quantity is common?" — series shares charge, parallel shares voltage. Everything else follows.

Redistribution when two charged capacitors are joined

Common potential V = (C₁V₁ + C₂V₂)/(C₁ + C₂) = Q_total/C_total Heat lost ΔU = C₁C₂(V₁ − V₂)² / 2(C₁ + C₂) Energy is always lost (as heat/radiation) unless V₁ = V₂. Charge is conserved; energy is not.

7 · Energy Stored & Energy Density

U = ½QV = ½CV² = Q²/2C Energy density u = ½ε₀E² (with dielectric: u = ½Kε₀E² = ½ε₀E₀²/K) Force between plates F = Q²/2ε₀A = ½QE (always attractive) Charging a capacitor with a battery: battery supplies QV, capacitor stores ½QV, the other ½QV is lost as heat in the wires — efficiency is exactly 50%.

8 · Capacitor Lab — the question you keep losing

Fix this tonight

The single most common capacitor mistake is answering an "isolated capacitor" question with "battery-connected" logic. Slide the parameters and toggle the switch to watch which quantity stays pinned.

Parallel-plate capacitor — what stays constant?

Base state: A = 100 cm², d = 2.0 mm, air, connected to a 12 V battery.

Plate separation d 2.0 mm

Dielectric constant K 1.0

Plate area A 100 cm²

—

E field

—

Energy U

—

Action	Battery CONNECTED (V constant)	Battery REMOVED (Q constant)
Insert dielectric K	C ↑K · Q ↑K · V — · E — · U ↑K	C ↑K · Q — · V ↓K · E ↓K · U ↓K
Increase separation d	C ↓ · Q ↓ · V — · E ↓ · U ↓	C ↓ · Q — · V ↑ · E unchanged · U ↑
Increase area A	C ↑ · Q ↑ · V — · E — · U ↑	C ↑ · Q — · V ↓ · E ↓ · U ↓

Trap — the one that catches everyone With the battery removed, pulling the plates apart leaves E unchanged (E = σ/ε₀ = Q/ε₀A, and Q and A are both fixed) while V rises and U rises. The extra energy is the work you did against the attraction between the plates. If the battery is still connected, E = V/d falls instead.

9 · Current Electricity — Drift Velocity

P1 · in portion

The physical picture

Free electrons in a metal move at ~10⁵–10⁶ m s⁻¹ randomly (thermal speed), colliding with lattice ions every relaxation time τ ≈ 10⁻¹⁴ s. With no field the average velocity is zero. Switch on a field and a small steady drift of ~10⁻⁴ m s⁻¹ superposes on the random motion — that drift is the current.

v_d = −(eE/m)τ → |v_d| = eEτ/m = eVτ/(mL) I = neAv_d · J = I/A = ne v_d = σE (J is a vector, I is a scalar) Mobility μ = |v_d|/E = eτ/m · units m² V⁻¹ s⁻¹ · σ = neμ ρ = m/(ne²τ) · σ = ne²τ/m · R = ρL/A

Trap — the four dependences of v_d

v_d ∝ I and v_d ∝ E — directly
v_d ∝ 1/A — thinner wire, faster drift (same current)
v_d ∝ 1/n — more free electrons, slower drift
v_d is independent of the length if E is fixed; but for a fixed V, v_d ∝ 1/L
Series wires of different area carry the same I, so v_d differs; the thin part has larger v_d, E and J

Memory hook — "the bulb lights instantly but electrons crawl" The electric field travels at nearly the speed of light (~3 × 10⁸ m s⁻¹) and sets every electron in the circuit moving at once. The electrons themselves would take hours to travel the wire. Signal speed ≠ drift speed. Asked as an assertion–reason question repeatedly.

Two electrons, two random paths, the same tiny average drift.

Number-sense worth carrying into the hall

Copper: n ≈ 8.5 × 10²⁸ m⁻³ free electrons (one per atom)
For I = 1 A in a 1 mm² copper wire: v_d ≈ 0.07 mm s⁻¹ — of order 10⁻⁴ m s⁻¹
Thermal speed ≈ 10⁵ m s⁻¹ → drift is ~10⁹ times smaller
τ ≈ 10⁻¹⁴ s, mean free path ≈ few nm
1 A = 1 C s⁻¹ = 6.25 × 10¹⁸ electrons per second

10 · Ohm's Law — and where it breaks

V = IR · R = ρL/A · microscopic form: J = σE Ohm's law is not a universal law — it's an empirical property of some materials at constant temperature.

V–I characteristic — ohmic vs non-ohmic

Tap a conductor. Ohmic devices give a straight line through the origin; anything else is non-ohmic.

Limitations of Ohm's law — four standard graphs

Case	Behaviour	Example
V not ∝ I	curve, not a straight line	diode, bulb filament (R rises with heat)
V–I relation depends on sign of V	asymmetric — conducts one way only	p-n junction diode
More than one I for the same V	S-shaped / negative-resistance region	GaAs, thyristor
Non-linear + direction-dependent	—	electrolytes, vacuum tubes

Resistivity — what it does and doesn't depend on

	Resistance R	Resistivity ρ
Depends on length & area?	Yes (R = ρL/A)	No — material property
Depends on temperature?	Yes	Yes
SI unit	ohm (Ω)	Ω m
Formula	R = ρL/A	ρ = m/(ne²τ)

ρ_T = ρ₀[1 + α(T − T₀)] · α = temperature coefficient of resistivity (K⁻¹)

Material	α	On heating	Reason
Metals (Cu, Ag, Al)	positive, ~10⁻³ K⁻¹	ρ increases	τ falls (more lattice vibration); n almost constant
Semiconductors (Si, Ge, C)	negative	ρ decreases	n rises exponentially — beats the fall in τ
Insulators	negative, large	ρ decreases	same reason as semiconductors
Alloys — nichrome, manganin, constantan	nearly zero	almost unchanged	used for standard resistors & heating elements

Trap — stretching a wire Volume stays constant. Stretch to n times the length → A becomes A/n → R becomes n²R. Halve the length → R becomes R/4. ρ never changes.

Only this much of Current Electricity is in the test Drift velocity, the I = neAv_d relation and Ohm's law. Kirchhoff's laws, Wheatstone bridge, metre bridge, potentiometer, cells in series/parallel and internal resistance are not in this paper — don't burn tonight's hours on them.

60-Second Recall — read this in the car

Point chargeV = kq/r E = kq/r²

Dipole equatorialV = 0, E = kp/r³

Inside a shellE = 0, V = kq/R ≠ 0

Field from potentialE = −dV/dr

Dipole energy & torqueU = −pE cosθ, τ = pE sinθ

Rotation workW = pE(cosθ₁ − cosθ₂)

Just outside a conductorE = σ/ε₀

DielectricE = E₀/K, K = 1 + χₑ

Parallel plateC = Kε₀A/d

Slab thickness tC = ε₀A/(d − t + t/K)

Series sharesQ · parallel shares V

EnergyU = ½CV² = Q²/2C, u = ½ε₀E²

Common potentialV = (C₁V₁+C₂V₂)/(C₁+C₂)

Battery ONV fixed · battery OFF: Q fixed

CurrentI = neAv_d, J = σE

Driftv_d = eEτ/m ≈ 10⁻⁴ m/s

Mobilityμ = eτ/m, σ = neμ

Resistivityρ = m/ne²τ, R = ρL/A

Stretch n×R → n²R

Zero-α alloysnichrome, manganin, constantan