Formula Sheet
Every key formula of the course on one page, grouped by chapter, with a link back to the section that develops it.
Chapter 1 · Light – Reflection and Refraction
Laws of reflection§ 1.1 →
$\angle i=\angle r$, angles measured from the normal; incident ray, normal and reflected ray lie in one plane
Plane-mirror image§ 1.1 →
Image distance behind = object distance in front; virtual, erect, same size, laterally inverted
Radius of curvature and focal length§ 1.1 →
$R=2f$ (spherical mirror of small aperture)
Rotating a plane mirror (extension)§ 1.1 →
Mirror turns through $\theta$ ⟹ reflected ray turns through $2\theta$ (incident ray fixed)
New Cartesian sign convention (mirrors)§ 1.2 →
Distances from $P$; along the incident light $+$, against it $-$; heights up $+$. $u\lt0$; $f\lt0$ concave, $f\gt0$ convex; real image $v\lt0$, virtual image $v\gt0$
Mirror formula§ 1.2 →
$\dfrac1v+\dfrac1u=\dfrac1f$ (New Cartesian convention; small aperture)
Magnification by a mirror§ 1.2 →
$m=\dfrac{h'}{h}=-\dfrac vu$; $m\lt0$ real and inverted, $m\gt0$ virtual and erect, $|m|\gt1$ enlarged
Concave mirror: where the image goes§ 1.2 →
Beyond $C$ → between $F$ and $C$ (diminished); at $C$ → at $C$ (same size); $C$–$F$ → beyond $C$ (enlarged); at $F$ → infinity; inside $F$ → behind, virtual, erect, enlarged. Convex mirror: always behind, between $P$ and $F$, virtual, erect, diminished
Snell's law of refraction§ 1.3 →
$\dfrac{\sin i}{\sin r}=n_{21}$, i.e. $n_1\sin i=n_2\sin r$ (angles from the normal; given colour and pair of media)
Absolute refractive index§ 1.3 →
$n=\dfrac cv$, $c=3.0\times10^{8}\ \text{m/s}$; $n\ge1$, no unit
Relative refractive index§ 1.3 →
$n_{21}=\dfrac{v_1}{v_2}=\dfrac{n_2}{n_1}$, $\ n_{12}=\dfrac{1}{n_{21}}$
Which way does light bend?§ 1.3 →
Rarer → denser: towards the normal ($r\lt i$); denser → rarer: away from it ($r\gt i$); along the normal: no bending. Glass slab with parallel faces: $e=i$, emergent ray parallel to incident ray, with a lateral shift
Lens formula§ 1.4 →
$\dfrac1v-\dfrac1u=\dfrac1f$ (thin lens, small aperture; New Cartesian convention from $O$; $f\gt0$ convex, $f\lt0$ concave)
Magnification by a lens§ 1.4 →
$m=\dfrac{h'}{h}=\dfrac vu$; real image: $v\gt0$, $m\lt0$ (inverted); virtual image: $v\lt0$, $m\gt0$ (erect)
Power of a lens§ 1.4 →
$P=\dfrac1f$ with $f$ in metres; unit dioptre, $1\ \text{D}=1\ \text{m}^{-1}$; convex $P\gt0$, concave $P\lt0$
Thin lenses in contact§ 1.4 →
$P=P_1+P_2+P_3+\cdots$, then $f=\dfrac1P$
Convex lens: where the image goes§ 1.4 →
Beyond $2F_1$ → between $F_2$ and $2F_2$ (diminished); at $2F_1$ → at $2F_2$ (same size); $F_1$–$2F_1$ → beyond $2F_2$ (enlarged); at $F_1$ → infinity; inside $F_1$ → same side, virtual, erect, enlarged. Concave lens: always between $F_1$ and $O$, virtual, erect, diminished
Chapter 2 · The Human Eye and the Colourful World
Range of vision of a normal eye§ 2.1 →
Near point $\approx25\ \text{cm}$ (least distance of distinct vision); far point at infinity
Correcting myopia§ 2.1 →
Concave lens, $f=-(\text{far point})$, $P=\dfrac1f$ ($f$ in m) — negative power; lens taken close to the eye
Correcting hypermetropia / presbyopia§ 2.1 →
Convex lens: $u=-25\ \text{cm}$, $v=-N$ (near point), $\dfrac1f=\dfrac1{25\ \text{cm}}-\dfrac1N$; in dioptres $P=\dfrac{100}{25}-\dfrac{100}{N}$ with $N$ in cm
Bifocal lens§ 2.1 →
Upper part concave (distance, from the far point); lower part convex (reading, from the near point)
Refraction through a prism§ 2.2 →
Snell's law at each face: towards the normal on entering, away on leaving; the emergent ray is deviated by $\delta$ towards the base (not parallel to the incident ray)
Prism at normal incidence§ 2.2 →
Angle inside at the second face $=A$; $\sin e=n\sin A$, $\ \delta=e-A$
Dispersion§ 2.2 →
$n_{\text{violet}}\gt n_{\text{red}}$: violet deviated most, red least; spectrum VIBGYOR
Recombination§ 2.2 →
Identical prism, inverted, after the first: colours emerge parallel and mix to white
Rainbow§ 2.2 →
Refraction + dispersion, internal reflection, refraction; red at $\approx42.4^\circ$, violet at $\approx40.6^\circ$ from the antisolar point; top of a band is (rainbow angle − Sun's height) above the horizon
Atmospheric refraction§ 2.3 →
Stars appear higher than they are; stars (point sources) twinkle, planets (discs) do not
Advance sunrise, delayed sunset§ 2.3 →
Earth turns $0.25^\circ$ per minute; lift $\approx0.5^\circ$ ⟹ Sun seen $\approx2\ \text{min}$ early and $\approx2\ \text{min}$ late; day $\approx4\ \text{min}$ longer
Scattering and particle size§ 2.3 →
Very fine particles: blue scattered most (blue sky, Tyndall glow); large particles: all colours alike (white clouds); red scattered least (danger signals)
Path through the air from the horizon§ 2.3 →
Star or Sun on the horizon: $L=\sqrt{(R+H)^2-R^2}$ (about $40$ times $H$ for $H=8\ \text{km}$)
Chapter 3 · Electricity
Electric current§ 3.1 →
$I=\dfrac Qt$; $1\ \text{A}=1\ \text{C/s}$ ($Q$ in C, $t$ in s)
Charge and electrons§ 3.1 →
$Q=ne$, $e=1.6\times10^{-19}\ \text{C}$; $1\ \text{C}=6.25\times10^{18}$ electrons
Potential difference§ 3.1 →
$V=\dfrac WQ$, so $W=VQ$; $1\ \text{V}=1\ \text{J/C}$
Connecting and reading meters§ 3.1 →
Ammeter (very low resistance) in series; voltmeter (very high resistance) in parallel. Least count $=\dfrac{\text{range}}{\text{number of divisions}}$ for a uniform scale from zero.
Ohm's law§ 3.2 →
$V=IR$, $I=\dfrac VR$, $R=\dfrac VI$ (metallic conductor at constant temperature); $1\ \Omega=1\ \text{V/A}$
V–I graph§ 3.2 →
$V$ up, $I$ across: slope $=R$; $I$ up, $V$ across: slope $=\dfrac1R$
Resistance of a wire§ 3.2 →
$R=\rho\dfrac lA$, $A=\pi r^2=\dfrac{\pi d^2}{4}$ (uniform wire, given temperature)
Resistivity§ 3.2 →
$\rho=\dfrac{RA}{l}$, unit $\Omega\ \text{m}$; depends on the material only (and temperature)
Cutting and stretching a wire§ 3.2 →
Cut into $n$ equal pieces: each $\dfrac Rn$. Stretched to $n$ times its length (volume constant): $n^2R$. $\rho$ unchanged in both.
Resistors in series§ 3.3 →
$R_s=R_1+R_2+R_3$; same current $I$ in each; $V=V_1+V_2+V_3$
Resistors in parallel§ 3.3 →
$\dfrac1{R_p}=\dfrac1{R_1}+\dfrac1{R_2}+\dfrac1{R_3}$; same $V$ across each; $I=I_1+I_2+I_3$
Two resistors in parallel§ 3.3 →
$R_p=\dfrac{R_1R_2}{R_1+R_2}$ (two resistors only); $\dfrac{I_1}{I_2}=\dfrac{R_2}{R_1}$
$n$ identical resistors§ 3.3 →
Series: $nR$. Parallel: $\dfrac Rn$. Ratio $n^2$.
Sharing the voltage in series§ 3.3 →
$V_1=\dfrac{R_1}{R_1+R_2+R_3}\,V$ — the largest resistor takes the largest share
Joule's law of heating§ 3.4 →
$H=VIt=I^2Rt=\dfrac{V^2}{R}\,t$ (heat in joules, steady current)
Electric power§ 3.4 →
$P=\dfrac Wt=VI=I^2R=\dfrac{V^2}{R}$; $1\ \text{W}=1\ \text{J/s}=1\ \text{V A}$
From an appliance's rating§ 3.4 →
$I=\dfrac PV$, $R=\dfrac{V^2}{P}$; on another voltage $P'=\dfrac{V'^2}{R}$
The kilowatt-hour§ 3.4 →
$1\ \text{kWh}=3.6\times10^6\ \text{J}$; energy (kWh) $=P$ (kW) $\times t$ (h)
Bill and fuse§ 3.4 →
Cost $=$ units (kWh) $\times$ price per unit. Fuse rating just above the normal current $\dfrac PV$.
Chapter 4 · Magnetic Effects of Electric Current
Rules of magnetic poles§ 4.1 →
Like poles repel, unlike poles attract; poles come in pairs — $n$ pieces of a magnet have $2n$ poles. Only repulsion proves a bar is a magnet.
Direction of a magnetic field§ 4.1 →
The way the north pole of a compass needle points; along the tangent to the field line.
Properties of magnetic field lines§ 4.1 →
N $\to$ S outside, S $\to$ N inside (closed curves); never intersect; crowded where the field is strong; straight, parallel and equally spaced in a uniform field.
Strength from a field-line diagram§ 4.1 →
$\dfrac{B_A}{B_B}=\dfrac{\text{lines per unit width at A}}{\text{lines per unit width at B}}$
Field of a straight wire§ 4.2 →
Concentric circles; $B\propto\dfrac Ir$, so $\dfrac{B_2}{B_1}=\dfrac{I_2}{I_1}\cdot\dfrac{r_1}{r_2}$
Right-hand thumb rule§ 4.2 →
Thumb along the current, fingers curl with the field. Current out of the page ⊙: anticlockwise; into the page ⊗: clockwise.
Field at the centre of a coil§ 4.2 →
$B\propto\dfrac{nI}{r}$; face with anticlockwise current = N, clockwise = S
Solenoid and electromagnet§ 4.2 →
Uniform field inside; $B$ grows with $I$, turns per length, soft-iron core; reversing $I$ swaps N and S.
The formulas (JEE/NEET extension)§ 4.2 →
$B=\dfrac{\mu_0I}{2\pi r}$ (wire), $B=\dfrac{\mu_0nI}{2r}$ (coil centre), $B=\mu_0n'I$ (solenoid, $n'$ turns per metre); $\mu_0=4\pi\times10^{-7}\ \text{T m/A}$
Force on a current-carrying conductor§ 4.3 →
At right angles to the field, $F\propto B\,I\,l$; $F=0$ if the conductor is parallel to the field. Reversing $I$ or $B$ (not both) reverses $F$.
Fleming's left-hand rule§ 4.3 →
Left hand, fingers mutually perpendicular: forefinger = field, middle finger = current, thumb = force.
Domestic wiring§ 4.3 →
Live (red, $220\ \text{V}$), neutral (black, $\approx0\ \text{V}$), earth (green); appliances in parallel; switch and fuse/MCB in the live wire.
AC and DC§ 4.3 →
Mains AC at $50\ \text{Hz}$ changes direction $100$ times a second (every $0.01\ \text{s}$); advantage over DC: transmitted over long distances without much energy loss.
Current and circuit rating§ 4.3 →
$I=\dfrac PV$ ($V=220\ \text{V}$); currents of appliances in parallel add; total $\lt$ MCB/fuse rating. Short circuit: $I=\dfrac VR$ very large.
The formula (JEE/NEET extension)§ 4.3 →
$F=BIl\sin\theta$, with $\theta$ the angle between the conductor and the field; $B$ in tesla ($1\ \text{T}=1\ \text{N A}^{-1}\text{m}^{-1}$)