Optics & Modern Physics — Physics

Reflection — Spherical Mirrors

Sign convention: distances measured from pole; along incident ray direction = +ve.

Mirror formula: \(\dfrac{1}{v} + \dfrac{1}{u} = \dfrac{1}{f} = \dfrac{2}{R}\)

Magnification: \(m = -\dfrac{v}{u} = h_i/h_o\)

Object position (concave)	Image nature	Image position
Beyond C (u > 2f)	Real, inverted, diminished	Between f and C
At C (u = 2f)	Real, inverted, same size	At C
Between C and f	Real, inverted, magnified	Beyond C
At f	Real, at infinity	∞
Inside f (u < f)	Virtual, erect, magnified	Behind mirror

Refraction — Snell's Law & TIR

Snell's law: \(n_1 \sin\theta_1 = n_2 \sin\theta_2\)

Total Internal Reflection occurs when: (1) light travels from denser to rarer medium, and (2) angle of incidence > critical angle C.

Critical angle: \(\sin C = \dfrac{n_2}{n_1}\) (\(n_1 > n_2\)). For glass–air: \(\sin C = \dfrac{1}{n_{glass}}\).

Optical fibre: uses TIR; light propagates with negligible loss through total internal reflection at core–cladding interface (\(n_{core} > n_{cladding}\)).

Refraction at Spherical Surface

$$ \frac{n_{2}}{v} - \frac{n_{1}}{u} = \frac{n_{2} - n_{1}}{R} $$

Here \(u\) = object distance, \(v\) = image distance, \(R\) = radius of curvature (sign convention applies).

Thin Lenses

Lens formula: \(\dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{f}\)

Lens maker's equation: \(\dfrac{1}{f} = (n-1)[\dfrac{1}{R_1} - \dfrac{1}{R_2}]\)

Power: \(P = \dfrac{1}{f}\) (in metres) — unit is dioptre (D)

Combination: \(P = P_1 + P_2\) (lenses in contact); \(\dfrac{1}{f} = \dfrac{1}{f_1} + \dfrac{1}{f_2}\)

Magnification: \(m = \dfrac{v}{u}\)

Prism

Angle of deviation \(\delta = i + e - A\) (\(A\) = prism angle, \(i\) = incidence angle, \(e\) = exit angle)

At minimum deviation: \(r_1 = r_2 = \dfrac{A}{2}\); \(i = e\)

\(n = \sin[\dfrac{A + \delta_{min}}{2}] / \sin(\dfrac{A}{2})\)

Dispersion: angular dispersion = \(\delta_V - \delta_R\); dispersive power \(\omega = \dfrac{n_V - n_R}{n_Y - 1}\)

Optical Instruments

Instrument	Magnification
Simple microscope	\(m = 1 + \dfrac{D}{f}\) (normal adjustment); \(m = \dfrac{D}{f}\) (far point)
Compound microscope	\(M = m_o \times m_e = (\dfrac{L}{f_o})(1 + \dfrac{D}{f_e})\)
Astronomical telescope	\(M = -\dfrac{f_o}{f_e}\) (normal adjustment)
Near-sighted correction	Concave (diverging) lens, \(P = -\dfrac{1}{d}\) (in m)
Far-sighted correction	Convex (converging) lens

Huygens Principle

Every point on a wavefront acts as a source of secondary spherical wavelets. The new wavefront is the tangent envelope of these secondary wavelets. Explains reflection, refraction, diffraction, and interference.

Young's Double Slit Experiment (YDSE)

Two coherent sources S₁ and S₂ (slit separation d) illuminate a screen at distance D.

Path difference\(: \Delta = d \sin\theta \approx \dfrac{dy}{D}\) (for small θ, where y = distance from centre)

Condition	Fringe type
\(\Delta = n\lambda (n = 0,\pm1,\pm2\ldots)\)	Bright fringe (constructive)
\(\Delta = (2n+1)\dfrac{\lambda}{2}\)	Dark fringe (destructive)

Fringe width\(: \beta = \lambda \dfrac{D}{d}\) (same for bright and dark fringes)

Intensity\(: I = 4I_{0} \cos^{2}(\dfrac{\delta}{2}),\) where \(\delta = 2\pi\dfrac{\Delta}{\lambda}\) is the phase difference

Maximum intensity\(: I_{max} = 4I_{0}\) (constructive) Minimum: I_min = 0 (for equal amplitudes)

If intensities differ: \(I_{max} = (\sqrt I_{1} + \sqrt I_{2})^{2}, I_{min} = (\sqrt I_{1} - \sqrt I_{2})^{2}\)

Diffraction

Single slit diffraction: slit width a, wavelength λ

• Minima at: \(a\sin\theta = n\lambda\) (\(n = \pm 1, \pm 2, \ldots\))
• Central maximum width = \(2\dfrac{\lambda}{a}\) (angular) or \(2\lambda \dfrac{D}{a}\) (linear on screen)
• Secondary maxima at: \(a\sin\theta = (2n+1)\dfrac{\lambda}{2}\) — much weaker than central maximum

Rayleigh criterion (resolving power): two objects just resolved when central max of one falls on first minimum of the other. \(\theta_{min} = 1.22\dfrac{\lambda}{D}\) (\(D\) = aperture diameter)

Polarisation

Light is a transverse wave — oscillations are perpendicular to propagation direction.

• Unpolarised light: random orientation of electric field vectors
• Plane polarised light: E-field oscillates in one plane only
• Malus's law: \(I = I_0 \cos^2\theta\) (\(\theta\) = angle between polariser and analyser axes)
• Brewster's law: \(\tan\theta_B = n\) (\(\theta_B\) = Brewster's angle; reflected ray is fully polarised when refracted and reflected rays are perpendicular)
• After passing through ideal polariser, intensity of unpolarised light becomes \(\dfrac{I_0}{2}\)

Photoelectric Effect

When light of frequency ν (above threshold ν₀) falls on a metal surface, electrons are emitted.

Einstein's equation\(: KE_{max} = h\nu - \phi = h(\nu - \nu_{0})\)

• Work function: \(\phi = h\nu_{0}\) (minimum energy to eject electron)
• Threshold wavelength: \(\lambda_{0} = \dfrac{hc}{\phi}\)
• Stopping potential: \(eV_{0} = KE_{max} \to V_{0} = \dfrac{h(\nu-\nu_{0})}{e}\)
• Number of photoelectrons ∝ intensity of light (not frequency)
• \(KE_{max}\) depends on frequency only, not intensity

de Broglie Wavelength

Every particle has an associated matter wave. \(\lambda = \dfrac{h}{p} = \dfrac{h}{mv}\)

• For particle accelerated through potential V: \(\lambda = \dfrac{h}{\sqrt{2mqV}}\)
• For electron: \(\lambda = \dfrac{12.27}{\sqrt V} Å\) (V in volts)
• Thermal neutrons at temperature T: \(\lambda = \dfrac{h}{\sqrt{3mkT}}\)
• Larger momentum → shorter wavelength

Heisenberg Uncertainty Principle

Δx · Δp ≥ h/4π    and    ΔE · Δt ≥ h/4π

It is impossible to simultaneously determine exact position and momentum of a particle. This is not a measurement limitation but a fundamental property of quantum mechanics.

X-Rays

• Production: produced when fast electrons decelerate hitting a target (bremsstrahlung) or eject inner shell electrons (characteristic X-rays)
• Moseley's law: \(\sqrt\nu = a(Z - b)\), where \(a\) and \(b\) are constants — establishes atomic number \(Z\)
• Bragg's law: \(2d \sin\theta = n\lambda\) — used for X-ray diffraction by crystal planes (d = interplanar spacing)
• Minimum wavelength of X-rays: \(\lambda_{min} = \dfrac{hc}{eV}\) (V = accelerating voltage)

Radioactivity

Decay law: \(N = N_{0} e^{-\lambda t}\)    \(A = \lambda N = A_0 e^{-\lambda t}\)

Half-life\(: T_{1}/_{2} = \dfrac{\ln2}{\lambda} = \dfrac{0.693}{\lambda}\)

Mean life\(: \tau = \dfrac{1}{\lambda} = T_{1}/_{2}/0.693\)

Decay type	Change in A	Change in Z	Change in N
α-decay (⁴He)	−4	−2	−2
β⁻-decay (e⁻)	0	+1	−1
β⁺-decay (e⁺)	0	−1	+1
γ-decay (photon)	0	0	0

Nuclear Binding Energy

Mass defect: \(\Delta m = Z m_p + N m_n - M_{nucleus}\)

Binding energy: \(BE = \Delta m\cdot c^2\) (in MeV: \(BE = \Delta m \times 931.5\) MeV/amu)

BE/A (binding energy per nucleon) peaks near \(A \approx 56\) (iron-56 most stable).

Fission (heavy nuclei split → increase in BE/A) and fusion (light nuclei merge → increase in BE/A) both release energy because products are more tightly bound.

Rutherford's Model

Gold foil experiment (1911): most α particles passed through; a few deflected at large angles; rare ones bounced back. Conclusion: atom has a tiny, dense, positively charged nucleus.

Limitations: orbiting electrons should radiate (Maxwell) → lose energy → spiral into nucleus. Cannot explain discrete spectral lines.

Bohr's Model of Hydrogen

Postulates: (1) Electrons move in fixed circular orbits without radiating. (2) Angular momentum is quantised: mvr = nh/2π. (3) Photon emitted/absorbed when electron transitions: hν = E_i − E_f.

Quantity	Formula	n=1 (H)
Radius	\(r_n = \dfrac{0.529n^{2}}{Z} Å\)	0.529 Å (Bohr radius a₀)
Energy	\(E_n = -\dfrac{13.6Z^{2}}{n^{2}} eV\)	−13.6 eV
Velocity	\(v_n = 2.18\times10^{6} \dfrac{Z}{n} m/s\)	2.18×10⁶ m/s

For hydrogen-like ions (He⁺, Li²⁺…): replace Z accordingly. E.g., He⁺ (Z=2): E₁ = −13.6×4 = −54.4 eV.

Hydrogen Spectral Series

Wave number: \(\dfrac{1}{\lambda} = RZ^{2}(\dfrac{1}{n_{1}^{2}} - \dfrac{1}{n_{2}^{2}})\), where \(R = 1.097\times10^{7} m^{-1}\) (Rydberg constant)

Series	n₁	n₂	Region
Lyman	1	2,3,4…	Ultraviolet
Balmer	2	3,4,5…	Visible
Paschen	3	4,5,6…	Infrared
Brackett	4	5,6,7…	Far IR
Pfund	5	6,7,8…	Far IR

Number of spectral lines from level n to ground: n(n−1)/2

Nuclear Structure

• Nucleus contains protons (Z) and neutrons (N); mass number A = Z + N
• Nuclear radius: \(R = R_0 A^{1/3}\), \(R_{0} = 1.2\ \text{fm}\) (1 fm = 10⁻¹⁵ m)
• Nuclear density is approximately constant: ρ ≈ 2.3×10¹⁷ kg/m³
• Nuclear forces: strong, short-range (~1 fm), charge-independent, saturating
• Isotopes: same Z, different A; Isobars: same A; Isotones: same N

Energy Bands

Material	Band gap E_g	Conductivity
Conductors	0 (overlap)	High (~10⁷ S/m)
Semiconductors	~0.1–2 eV (Si: 1.1 eV, Ge: 0.7 eV)	Intermediate (increases with T)
Insulators	>3 eV (diamond: 5.4 eV)	Very low

Intrinsic & Extrinsic Semiconductors

Intrinsic: pure Si or Ge; electrons thermally excited across band gap; n_e = n_h = n_i

n-type: doped with pentavalent donors (P, As, Sb); extra electron → majority carriers = electrons, minority = holes

p-type: doped with trivalent acceptors (B, Al, In); creates holes → majority carriers = holes, minority = electrons

Mass action law\(: n_e \cdot n_h = n_i^{2}\) (at equilibrium, regardless of doping)

p-n Junction

• Depletion region: forms at junction; depleted of mobile carriers; built-in potential barrier V₀ (Si: ~0.7 V, Ge: ~0.3 V)
• Forward bias: +ve to p-side; reduces barrier; current flows; diode ≈ short circuit above threshold (~0.7 V for Si)
• Reverse bias: +ve to n-side; increases barrier; only tiny leakage current flows; diode ≈ open circuit
• Breakdown: at large reverse voltage, avalanche or Zener breakdown — current increases sharply

Special Diodes

• Zener diode: operated in reverse breakdown; maintains constant voltage V_Z → used as voltage regulator
• LED: forward biased; recombination emits photons; E_photon ≈ E_g
• Photodiode: reverse biased; photons generate electron-hole pairs → current; used as detector
• Solar cell: no bias; photovoltaic effect — generates EMF from light

Transistors (BJT)

Three regions: Emitter (heavily doped), Base (thin, lightly doped), Collector (moderately doped). Types: npn and pnp.

CE configuration (most common for amplification):

• Current gain \(\beta = \dfrac{I_C}{I_B} = \dfrac{\alpha}{1-\alpha}\)
• α = I_C/I_E (always <1, typically 0.95–0.99)
• I_E = I_B + I_C
• Voltage gain A_v = β × R_C/R_in

Logic Gates

Gate	Boolean	00	01	10	11
AND	Y = A·B	0	0	0	1
OR	Y = A+B	0	1	1	1
NAND	Y = (A·B)̄	1	1	1	0
NOR	Y = (A+B)̄	1	0	0	0
XOR	Y = A⊕B	0	1	1	0

NAND and NOR are universal gates — any logic circuit can be built from NAND or NOR alone.