Physics — Complete Formula Sheet

$F$: Electrostatic force (N) · $k$: Coulomb's constant = 9×10⁹ N·m²/C² · ${\epsilon }_0$: Permittivity of free space = 8.85×10⁻¹² C²/(N·m²) · $q_1,q_2$: Point charges (C) · $r$: Distance between charges (m)

Force between two point charges. Attractive if charges opposite, repulsive if same.

Force per unit positive test charge. Vector — direction is radially away from +Q.

Work done per unit positive charge to bring it from infinity to point r. Scalar quantity.

Field points from high to low potential. Negative gradient of V.

$p$: Dipole moment (C·m) · $2a$: Distance between the two charges (m)

Magnitude = q × 2a (charge × separation). Direction: from −q to +q.

Direction: same as dipole moment p.

Direction: antiparallel to p. Note: axial field = 2× equatorial field.

Total electric flux through closed surface = enclosed charge / ε₀.

Uses Gauss's law with cylindrical surface. λ = linear charge density (C/m).

Independent of distance from sheet. σ = surface charge density (C/m²).

Charge stored per unit potential difference. Unit: farad (F).

$A$: Plate area (m²) · $d$: Separation between plates (m)

Larger area or smaller separation ⇒ higher capacitance.

Dielectric constant K > 1 increases capacitance K-fold.

Three equivalent forms — pick whichever avoids the unknown quantity.

Same charge on each, voltages add. Equivalent capacitance is SMALLER than smallest.

Same voltage across each, charges add. Equivalent is LARGER than largest.

$I$: Current (A = C/s) · $q$: Charge crossing a section (C) · $t$: Time (s)

Rate of flow of charge. Conventional current is opposite to electron flow.

$n$: Free-electron density (m⁻³) · $e$: Electron charge = 1.6×10⁻¹⁹ C · $A$: Cross-sectional area (m²) · $v_d$: Drift velocity (m/s)

Macroscopic current from microscopic drift of free electrons under an applied field.

Drift velocity is proportional to applied electric field. \tau is the relaxation time.

Current density J is proportional to field E. Conductivity \sigma depends only on the material.

Voltage across a conductor is proportional to current, provided temperature is constant.

$\rho$: Resistivity (\Omega\cdot m) · $L$: Length (m) · $A$: Cross-section area (m²)

Longer conductor ⇒ more resistance. Thicker conductor ⇒ less resistance.

For metals \alpha > 0 (resistivity rises with T). For semiconductors \alpha < 0.

Conservation of charge at any junction — no charge accumulates.

Conservation of energy around a closed loop. Choose direction, be consistent with signs.

$\epsilon$: EMF of cell (V) · $r$: Internal resistance (\Omega) · $V$: Terminal voltage (V)

Terminal voltage V is less than EMF \varepsilon when current is drawn from the cell.

Balance ⇒ no current through galvanometer ⇒ B and D at same potential.

Series: EMFs add, internal resistances add. Parallel: use the reciprocal-like formula for identical cells (r_eq = r/n).

Three equivalent forms — pick the one avoiding the unknown quantity.

$q$: Charge (C) · $\vec{v}$: Velocity of charge (m/s) · $\vec{B}$: Magnetic field (T)

Total electromagnetic force on a charge. Magnetic part is perpendicular to v — does no work.

Field due to a small current element. Analogue of Coulomb's law for magnetism.

Field lines are concentric circles around the wire; right-hand thumb along I.

For N turns, multiply by N: B = \mu_0 NI/(2R). Direction perpendicular to plane of loop.

At the centre (x = 0) reduces to \mu_0 I/(2R). Falls as 1/x³ far away.

Line integral of B around a closed path = \mu_0 × enclosed current. Only useful with symmetry.

n = turns per unit length (m⁻¹). Field is uniform inside, ≈ zero outside (ideal case).

N = total number of turns. Field exists only within the toroidal core.

Maximum when L is perpendicular to B (\theta = 90°). Zero when parallel.

Attractive if currents parallel, repulsive if antiparallel. Defines the ampere.

Magnetic dipole moment m = NIA. Torque is zero when m is aligned with B.

Independent of speed and radius (in non-relativistic limit) — this is why cyclotrons work.

Faster or heavier particles orbit in larger circles. Doubling B halves the radius.

$B_{axial}$: Magnetic field on the axis (T) · $m$: Magnetic dipole moment (A m² or J T⁻¹) · $r$: Distance from the centre of the magnet (m) · ${\mu }_0$: Permeability of free space = 4π×10⁻⁷ T m A⁻¹

Valid for a short magnet, i.e. r much larger than the magnet's length. B points along m (from the S-end to the N-end direction). Unit: tesla (T).

${\vec{B}}_{eq}$: Magnetic field on the equatorial line (T) · $\vec{m}$: Magnetic dipole moment (A m²) · $r$: Distance from the centre of the magnet (m)

Magnitude is half the axial field at the same distance. The minus sign means B is antiparallel to m. Valid only for r much larger than the magnet's length.

$\tau$: Torque (N m) · $m$: Magnetic dipole moment (J T⁻¹) · $B$: Uniform magnetic field (T) · $\theta$: Angle between m and B

The torque tries to align m with B. In a uniform field the net force is zero, so the magnet only rotates. Unit: N m.

$U_m$: Magnetic potential energy (J) · $\theta$: Angle between m and B

Zero of energy is taken at θ = 90°. Minimum −mB at θ = 0° (stable), maximum +mB at θ = 180° (unstable). Unit: joule (J).

$W$: Work done (J) · ${\theta }_1,{\theta }_2$: Initial and final angles between m and B

Work done by an external agent turning the dipole from θ₁ to θ₂; equals U(θ₂) − U(θ₁). From 0° to 180°, W = 2mB.

$\vec{B}$: Magnetic field (T) · $d\vec{S}$: Area element of the closed surface (m²)

Net magnetic flux through any closed surface is zero, because isolated magnetic poles (monopoles) do not exist. Magnetic field lines always form closed loops.

$\vec{M}$: Magnetisation (A m⁻¹) · ${\vec{m}}_{net}$: Net magnetic moment of the sample (A m²) · $V$: Volume of the sample (m³)

Net magnetic moment per unit volume of the material. Same unit as H: A m⁻¹.

$H$: Magnetic intensity (A m⁻¹) · $n$: Number of turns per unit length (m⁻¹) · $I$: Current in the winding (A)

H inside a long solenoid depends only on turns per unit length and current, not on the core material. Unit: A m⁻¹.

$\vec{B}$: Magnetic field (T) · $\vec{H}$: Magnetic intensity (A m⁻¹) · $\vec{M}$: Magnetisation (A m⁻¹)

H is set by external (free) currents; M is the material's response. B (in T) is the total field.

$\chi$: Magnetic susceptibility (no unit)

χ is dimensionless. Diamagnetic: χ small and negative (−1 ≤ χ < 0). Paramagnetic: χ small and positive. Ferromagnetic: χ very large and positive.

${\mu }_r$: Relative permeability (no unit) · $\mu$: Permeability of the material (T m A⁻¹)

Diamagnetic: μr slightly less than 1. Paramagnetic: μr slightly more than 1. Ferromagnetic: μr ≫ 1. μr and χ have no unit; μ has the unit of μ₀ (T m A⁻¹).

$C$: Curie constant of the material · $T$: Absolute temperature (K)

Susceptibility of a paramagnet is inversely proportional to absolute temperature. Equivalent form: M = C B₀/T. Holds while the magnetisation is far from saturation. The rationalised NCERT text describes this temperature effect only in words, without the equation.

$\Phi$: Magnetic flux (Wb = T·m²) · $\theta$: Angle between B and area vector

Flux is maximum when B is normal to the surface (\theta = 0), zero when parallel.

$\epsilon$: Induced EMF (V) · $N$: Number of turns · $d\Phi /dt$: Rate of change of flux (Wb/s)

Induced EMF is proportional to the rate of change of flux linkage.

Induced current flows in a direction that opposes the change in flux — direct consequence of energy conservation.

EMF induced in a conductor of length L moving with velocity v perpendicular to B.

$L$: Self-inductance (H = Wb/A) · $dI/dt$: Rate of change of current (A/s)

A changing current in a coil induces an EMF in the same coil (opposing the change).

Depends only on geometry and the medium. Larger N, area, or ferromagnetic core → higher L.

$M$: Mutual inductance (H) · $I_1$: Current in primary coil (A)

EMF in coil 2 caused by changing current in coil 1. M depends on geometry and relative position.

For an inner solenoid completely surrounded by an outer solenoid.

Energy is stored in the magnetic field of the inductor.

Energy per unit volume stored in a magnetic field. Analogue of \tfrac{1}{2}\varepsilon_0 E².

Peak EMF depends on turns, area, field and angular speed.

Add like resistors — direct sum in series, reciprocal sum in parallel.

$v$: Instantaneous voltage (V) · $v_m$: Peak voltage (V) · $\omega$: Angular frequency (rad s⁻¹) · $\nu$: Frequency (Hz)

v_m is the peak value (amplitude). For Indian mains ν = 50 Hz, so ω ≈ 314 rad s⁻¹.

$i_m$: Peak current (A) · $R$: Resistance (Ω)

Current and voltage are in phase. Ohm's law holds for instantaneous, peak and rms values.

$I$: rms current (A) · $V$: rms voltage (V)

The rms value is the steady (DC) value that produces the same average power in a resistor. AC meters read rms values.

The average of sin²ωt over a cycle is ½, while the average of sin ωt is zero. Unit: watt (W).

$X_L$: Inductive reactance (Ω) · $L$: Inductance (henry, H)

In a pure inductor the current lags the voltage by π/2: i = i_m sin(ωt − π/2) with i_m = v_m/X_L. Unit: ohm (Ω).

$X_C$: Capacitive reactance (Ω) · $C$: Capacitance (farad, F)

In a pure capacitor the current leads the voltage by π/2: i = i_m sin(ωt + π/2) with i_m = v_m/X_C. As ν → 0, X_C → ∞, so a capacitor blocks DC.

With a phase difference of π/2, the average of sin ωt cos ωt over a cycle is zero. Energy is stored and returned every half-cycle; such a current is called wattless current.

$Z$: Impedance (Ω)

Z plays the role of resistance for peak or rms values. Because the difference is squared, (X_C − X_L)² = (X_L − X_C)². Unit: ohm (Ω).

$\varphi$: Phase angle between current and voltage

With v = v_m sin ωt, the current is i = i_m sin(ωt + φ). If X_C > X_L, φ > 0 and the current leads (capacitive circuit). If X_L > X_C, φ < 0 and the current lags (inductive circuit).

${\omega }_0$: Resonant angular frequency (rad s⁻¹) · ${\nu }_0$: Resonant frequency (Hz)

At resonance X_L = X_C, so Z = R (its minimum) and the current amplitude v_m/R is maximum. Resonance needs both L and C in the circuit.

$P$: Average power (W) · $\cos \varphi$: Power factor

V and I are rms values. Only the resistance dissipates power on average.

Pure R: cos φ = 1. Pure L or pure C: cos φ = 0 (wattless current). Series LCR at resonance: cos φ = 1, maximum power.

$V_p,V_s$: Primary and secondary voltages (V) · $N_p,N_s$: Number of turns in primary and secondary

Step-up: N_s > N_p; step-down: N_s < N_p. Assumes negligible primary resistance and no flux leakage. A transformer works only with AC.

$I_p,I_s$: Primary and secondary currents (A)

In an ideal (100% efficient) transformer input power equals output power. Stepping voltage up steps current down by the same ratio.

$i_d$: Displacement current (A) · ${\epsilon }_0$: Permittivity of free space = 8.854×10⁻¹² C² N⁻¹ m⁻² · ${\Phi }_E$: Electric flux (N m² C⁻¹)

A current associated with a changing electric flux, e.g. between the plates of a charging capacitor. No charge actually flows across the gap. Unit: ampere (A).

$A$: Plate area (m²) · $C$: Capacitance (F) · $\frac{dV}{dt}$: Rate of change of voltage across the plates (V s⁻¹)

Uses Φ_E = EA, E = V/d and C = ε₀A/d. Inside a charging capacitor i_d equals the conduction current in the connecting wires.

$i_c$: Conduction current (A) · ${\mu }_0$: Permeability of free space = 4π×10⁻⁷ T m A⁻¹

The source of B is the total current: conduction current i_c plus displacement current i_d. A changing electric field produces a magnetic field.

In order: Gauss's law for electricity, Gauss's law for magnetism, Faraday's law and the Ampère–Maxwell law. Together they predict electromagnetic waves.

$c$: Speed of light in vacuum (m s⁻¹)

Maxwell obtained this speed from purely electric and magnetic constants. It matched the measured speed of light, showing that light is an electromagnetic wave.

$v$: Speed in the medium (m s⁻¹) · $\mu$: Permeability of the medium (T m A⁻¹) · $\epsilon$: Permittivity of the medium (C² N⁻¹ m⁻²)

μ and ε are the permeability and permittivity of the medium. With μ = μ_rμ₀ and ε = ε_rε₀, the refractive index is n = c/v = √(μ_rε_r).

$E_0$: Electric field amplitude (V m⁻¹) · $B_0$: Magnetic field amplitude (T) · $k$: Wave number (rad m⁻¹) · $\omega$: Angular frequency (rad s⁻¹)

E along x, B along y, propagation along +z. E and B oscillate in phase and are perpendicular to each other and to the direction of travel, so the wave is transverse.

Also true at every instant: E = cB. B is numerically tiny: E₀ = 3 V m⁻¹ corresponds to B₀ = 10⁻⁸ T.

$\lambda$: Wavelength (m) · $\nu$: Frequency (Hz)

Dividing ω by k gives the wave speed: ω/k = νλ = c.

All EM waves travel at c in vacuum. In a medium ν stays the same while both speed and wavelength drop by the factor n.

$f$: Focal length (m or cm) · $R$: Radius of curvature (m or cm)

Holds for paraxial rays. With the New Cartesian convention f and R are negative for a concave mirror and positive for a convex mirror.

$u$: Object distance from the pole · $v$: Image distance from the pole · $f$: Focal length

All distances are measured from the pole. A real object in front of the mirror always has u < 0. Concave mirror: f < 0; convex mirror: f > 0.

$h^{\prime }$: Height of the image · $h$: Height of the object

m negative: real, inverted image. m positive: virtual, erect image. |m| > 1: enlarged.

$i$: Angle of incidence · $r$: Angle of refraction · $n_{21}$: Refractive index of medium 2 relative to medium 1

n₂₁ is the refractive index of medium 2 with respect to medium 1. Entering a denser medium (n₂₁ > 1) the ray bends towards the normal.

$h$: Real depth · $h_1$: Apparent depth · $n$: Refractive index of the medium relative to air

Seen from air, close to the normal, an object at real depth h in a medium of refractive index n appears at depth h₁. The apparent rise is h(1 − 1/n).

$i_c$: Critical angle · $n$: Refractive index of the denser medium relative to the rarer medium

Total internal reflection needs light going from the denser to the rarer medium AND an angle of incidence greater than i_c. For glass (n = 1.5) to air, i_c ≈ 42°. Optical fibres work by repeated TIR.

$n_1$: Refractive index of the medium containing the object · $n_2$: Refractive index of the medium into which light refracts · $R$: Radius of curvature of the surface

Light goes from medium n₁ into medium n₂. R is positive when the centre of curvature is on the side where the light emerges. This is the building block of the lens maker's formula.

$R_1,R_2$: Radii of curvature of the first and second surfaces (with sign) · $n_{21}$: Refractive index of lens material relative to the surrounding medium

n₂₁ is the refractive index of the lens material relative to the surrounding medium. For an equiconvex lens R₁ = +R and R₂ = −R. A convex lens placed in a liquid denser than the glass (n₂₁ < 1) diverges light.

Distances from the optical centre. Convex lens: f > 0; concave lens: f < 0. A real object has u < 0.

No minus sign for a lens, unlike a mirror. m negative: real, inverted image. m positive: virtual, erect image.

$P$: Power (dioptre, D = m⁻¹) · $f$: Focal length (m)

With f in metres, P is in dioptre (D). Converging (convex) lens: P > 0; diverging (concave) lens: P < 0.

Keep the sign of each focal length. For any system of lenses the total magnification is the product of the individual magnifications.

$A$: Angle of the prism · $\delta$: Angle of deviation · $i,e$: Angles of incidence and emergence

A is the angle of the prism, δ the angle of deviation, i and e the angles of incidence and emergence. At minimum deviation i = e and r₁ = r₂ = A/2.

$D_m$: Angle of minimum deviation

D_m is the angle of minimum deviation. This is the standard way of measuring the refractive index of a prism material.

Valid only for a thin prism (A of a few degrees), where sines can be replaced by the angles themselves. Do not use it for a 60° prism.

$D$: Least distance of distinct vision (about 25 cm) · $f$: Focal length of the convex lens

D ≈ 25 cm is the least distance of distinct vision. The image at the near point gives the larger magnification; the image at infinity is more relaxing for the eye.

$f_o,f_e$: Focal lengths of objective and eyepiece · $L$: Tube length

Final image at infinity. L is the tube length: the distance between the second focal point of the objective and the first focal point of the eyepiece. If the final image is at the near point, m_e = 1 + D/f_e instead.

$f_o$: Focal length of the objective · $f_e$: Focal length of the eyepiece · $L$: Length of the telescope tube

Normal adjustment (final image at infinity). A long-focus objective and a short-focus eyepiece give high magnifying power; a large objective diameter gathers more light and resolves better.

Envelope of secondary wavelets after time t gives the new wavefront. Explains reflection, refraction, diffraction.

$\Delta$: Path difference (m) · $\Delta \varphi$: Phase difference (rad)

Phase difference = (2\pi/\lambda) × path difference. Full wavelength = 2\pi radians.

Waves arrive in phase, amplitudes add. Intensity is 4I_0 for two equal sources.

Waves arrive out of phase by \pi, amplitudes cancel. Intensity is zero for equal amplitudes.

I_max = (\sqrt{I_1}+\sqrt{I_2})²; I_min = (\sqrt{I_1}-\sqrt{I_2})². Ratio predicts fringe visibility.

$\beta$: Fringe width (m) · $\lambda$: Wavelength (m) · $D$: Slit-to-screen distance (m) · $d$: Slit separation (m)

Distance between two consecutive bright (or dark) fringes.

Central maximum at y = 0. Fringes are equally spaced.

First dark fringe at y = \lambda D/(2d).

Position of minima in single-slit diffraction pattern. n = 0 is the central maximum, NOT a minimum.

Twice the width of any secondary maximum. Wider slit ⇒ narrower central peak.

Minimum angular separation of two point objects to be resolved by a circular aperture of diameter D.

At the polarising angle \theta_B, the reflected ray is 100% plane-polarised (perpendicular to plane of incidence).

Intensity of polarised light after passing through a polariser at angle \theta to its polarisation axis.

$h$: Planck's constant = 6.626×10⁻³⁴ J·s · $\nu$: Frequency (Hz) · $\lambda$: Wavelength (m) · $c$: Speed of light = 3×10⁸ m/s

Each photon carries a fixed quantum of energy proportional to its frequency.

$\varphi$: Work function of metal (J or eV) · ${\nu }_0$: Threshold frequency (Hz)

Photon energy = work function + kinetic energy of ejected electron. Explains threshold frequency.

Reverse potential that just stops the fastest photoelectrons. Independent of intensity.

Every moving particle has a matter wave. Wavelength inversely proportional to momentum.

Convenient form for electron accelerated through potential V (in volts).

Angular momentum in nth orbit is an integer multiple of \hbar. Bohr's second postulate.

First Bohr radius (n = 1) ≈ 0.529 Å = 5.29×10⁻¹¹ m.

Negative because electron is bound. Ground state (n = 1) has E = −13.6 eV.

Emission when n_2 > n_1; absorption otherwise.

Wavelength of spectral line in hydrogen. R = 1.097×10⁷ m⁻¹. Lyman: n_1=1, Balmer: n_1=2, Paschen: n_1=3.

Rest-mass energy. 1 u ≈ 931.5 MeV.

Number of surviving nuclei decreases exponentially. \lambda is decay constant, not wavelength.

T_{1/2} < \tau. Ratio: T_{1/2} = 0.693 \tau.

Number of disintegrations per second. SI unit: becquerel (Bq). 1 Ci = 3.7×10¹⁰ Bq.

$E_g$: Energy band gap (eV) · $E_C$: Energy at the bottom of the conduction band (eV) · $E_V$: Energy at the top of the valence band (eV)

E_C is the bottom of the conduction band and E_V the top of the valence band. Conductor: bands overlap (no gap). Insulator: E_g > 3 eV. Semiconductor: E_g < 3 eV, e.g. Si ≈ 1.1 eV, Ge ≈ 0.7 eV.

$n_e$: Free-electron density (m⁻³) · $n_h$: Hole density (m⁻³) · $n_i$: Intrinsic carrier concentration (m⁻³)

In a pure semiconductor each electron freed by thermal energy leaves behind one hole, so the two densities are equal. n_i increases rapidly with temperature.

$I_e$: Current due to free electrons (A) · $I_h$: Current due to holes (A)

Electrons and holes move in opposite directions under an applied field, but both give current in the same direction, so the two currents add.

Holds in thermal equilibrium for both intrinsic and doped semiconductors. Raising one carrier density by doping lowers the other.

$N_D$: Donor atom concentration (m⁻³)

Doped with pentavalent donor atoms (As, Sb, P). Electrons are the majority carriers. The donor level lies just below the bottom of the conduction band.

$N_A$: Acceptor atom concentration (m⁻³)

Doped with trivalent acceptor atoms (In, B, Al). Holes are the majority carriers. The acceptor level lies just above the top of the valence band.

The acceptors take up electrons released by the donors, so the two dopings partly cancel. If N_A > N_D the material is p-type with n_h ≈ N_A − N_D. Valid when the difference is much larger than n_i.

$V_0$: Barrier potential at equilibrium (V) · $V$: Applied bias voltage (V)

V₀ is the barrier potential with no bias and V is the applied voltage. Forward bias lowers the barrier and narrows the depletion layer; reverse bias raises the barrier and widens the depletion layer.

In forward bias the current stays very small until the applied voltage reaches the threshold value; beyond it the current rises steeply (exponentially). These are the NCERT values; some other books quote about 0.3 V for Ge.

$r_d$: Dynamic resistance (Ω) · $\Delta V$: Small change in voltage (V) · $\Delta I$: Corresponding change in current (A)

The ratio of a small change in voltage to the resulting small change in current, read from the V–I curve at the operating point. Low (ohms to tens of ohms) in forward bias above threshold; very high in reverse bias.

$V$: Battery voltage (V) · $R$: Series resistance (Ω)

Kirchhoff's loop rule with the diode treated as a fixed forward drop V_th (≈ 0.7 V for Si). In reverse bias only a tiny reverse saturation current flows, so I ≈ 0.

A single diode conducts only during the half-cycle in which it is forward biased, so there is one output pulse per input cycle.

Two diodes with a centre-tapped transformer conduct on alternate half-cycles, giving two output pulses per input cycle: 100 Hz output from 50 Hz mains.