Have you ever looked into a makeup mirror (concave) that makes your face appear larger? Or used a car's side-view mirror (convex) that shows a wider view? Both are spherical mirrors. The mirror formula (1/f = 1/u + 1/v) tells us exactly where the image will form and how big it will be. This is essential for designing telescopes, microscopes, rear-view mirrors, and even satellite dishes.
🧠 The Big Idea — In Plain Words
Think of a concave mirror as a "bowl-shaped" reflective surface. Parallel rays coming from far away converge at the focus (F). The centre of curvature (C) is at twice the focal length. The mirror formula connects three numbers: u (object distance from mirror), v (image distance from mirror), and f (focal length). If you know any two, you can find the third. The magnification m = -v/u tells you how big the image is compared to the object and whether it's upright or inverted.
Key variables: u = object distance (m), v = image distance (m), f = focal length (m), R = radius of curvature (m), h₀ = object height, hᵢ = image height
Fig 1.1: Mirror ray diagram — synchronized. Animation advances through object, ray 1, ray 2, image, formulas, and summary.
🔍 Step-by-Step: The Mirror Formula (1/u + 1/v = 1/f)
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Start with geometry — draw two rays from the object. Ray 1: from object tip, parallel to principal axis → reflects through F. Ray 2: from object tip, through F → reflects parallel to axis. Where the two reflected rays meet is the image location. Using similar triangles from this ray diagram, we can derive the formula.
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Using similar triangles in the ray diagram. From the triangle formed by the object and the principal axis, and the triangle formed by the image: height ratio = v/u. Also from the triangle involving the focus: height ratio = (f - v)/f (with sign convention applied). Equating these gives the mirror formula.
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The result — simplest form.1/u + 1/v = 1/f. This is the mirror formula. Use it with the Cartesian sign convention (see table below). For concave mirrors, f is negative; for convex, f is positive.
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Magnification m = hᵢ/h₀ = -v/u. If m is positive, image is upright (virtual). If negative, image is inverted (real). |m| > 1 means enlarged, |m| < 1 means diminished.
1/u + 1/v = 1/f m = -v/u = hᵢ/h₀ f = R/2
🔍 Sign Convention (Cartesian — used in NEET)
• All distances from pole P. Towards incident light = +. Opposite = -. • Heights above axis = +, below = -. • f negative for concave, positive for convex mirrors.
🎯 NEET Pattern — Mirrors
• Concave mirror: f negative. Real image → v negative. Virtual image → v positive.
• Convex mirror: always forms virtual, erect, diminished image. f positive, u always negative, v positive.
• Common question: "An object placed 15 cm from a concave mirror of f = 10 cm. Find image position." → Apply 1/v = 1/f - 1/u with signs.
• m = -v/u. For convex mirror, m is positive and < 1 (diminished, erect).
Example 1: Mirror Formula
An object is placed 20 cm from a concave mirror of focal length 15 cm. Find image distance and magnification.
a) v = -60 cm, m = -3 b) v = 60 cm, m = 3 c) v = -60 cm, m = 3 d) v = -30 cm, m = -2
Solution: u = -20 cm (object in front), f = -15 cm (concave). 1/v = 1/f - 1/u = 1/(-15) - 1/(-20) = -1/15 + 1/20 = (-4+3)/60 = -1/60. v = -60 cm. m = -v/u = -(-60)/(-20) = -3. Option (a).
2. Lens Maker's Formula
💡 Why this matters in real life
Every pair of glasses, contact lens, camera lens, microscope objective, and telescope eyepiece is designed using the Lens Maker's formula. This formula tells lens manufacturers exactly what curvature to grind into a lens to achieve a desired focal length. It links the focal length to the lens's shape (radii of curvature) and the material's refractive index.
Fig 2.1: Lens Maker's — synchronized. Animation advances through foci, rays, radii, formula, and summary.
🧠 The Big Idea — In Plain Words
A lens works by bending (refracting) light at its two curved surfaces. The Lens Maker's formula says: 1/f = (μ - 1)(1/R₁ - 1/R₂). Here μ is the refractive index of the lens material relative to the surrounding medium. R₁ and R₂ are the radii of curvature of the two surfaces. A lens with more curved surfaces (smaller R) and higher refractive index will be stronger (shorter focal length). A plano-convex lens (one flat side, R₂ = ∞) is a special case. The formula combines the refraction at both surfaces into one equation.
Key variables: μ = refractive index of lens material, R₁, R₂ = radii of curvature (m), f = focal length (m)
🔍 Step-by-Step: Deriving the Lens Maker's Formula
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Refraction at a single spherical surface has a formula. For a single surface separating two media (μ₁ and μ₂): μ₂/v - μ₁/u = (μ₂ - μ₁)/R. This relates object and image distances for one surface.
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A lens has two spherical surfaces. We apply the single-surface formula to both surfaces. The image from the first surface becomes the virtual object for the second surface. For a thin lens (thickness negligible), we assume the two surfaces are very close.
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Surface 1 (object side): Light from object (u) forms intermediate image (v₁). μ/v₁ - 1/u = (μ - 1)/R₁ (assuming lens in air, μ₁ = 1, μ₂ = μ).
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Surface 2 (image side): The intermediate image at v₁ acts as object for the second surface. For thin lens, object distance = -v₁ (virtual object). Final image at v: 1/v - μ/(-v₁) = (1 - μ)/R₂ = -(μ - 1)/R₂
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Add the two equations to eliminate the intermediate image term (μ/v₁ cancels): 1/v - 1/u = (μ - 1)(1/R₁ - 1/R₂). Since 1/f = 1/v when u → ∞, this gives 1/f = (μ - 1)(1/R₁ - 1/R₂).
1/f = (μ - 1)(1/R₁ - 1/R₂) Power of lens: P = 1/f (dioptre)
🔍 Sign Convention for R₁ and R₂
R₁ = radius of first surface (the one light hits first). R₂ = radius of second surface. R is + if centre of curvature is on the right (light exits side), - if on the left. For a biconvex lens: R₁ positive, R₂ negative → 1/f positive (converging). For biconcave: R₁ negative, R₂ positive → 1/f negative (diverging).
🎯 NEET Pattern — Lenses
• Lens formula: 1/v - 1/u = 1/f (same as mirror but with a minus sign difference!).
• Power: P = 1/f (in metres). Convex lens: P positive. Concave lens: P negative.
• Combination of lenses: P_eq = P₁ + P₂. 1/f_eq = 1/f₁ + 1/f₂.
• If a lens is immersed in water (μ > 1 for lens, μ_water > 1), the effective refractive index μ_rel = μ_lens/μ_water decreases → focal length increases.
• Lens maker's formula with μ_rel: 1/f = (μ_rel - 1)(1/R₁ - 1/R₂).
Example 2: Lens Maker's Formula
A biconvex lens has R₁ = 20 cm, R₂ = 30 cm, made of glass with μ = 1.5. Find focal length.
3. Drift Velocity & Ohm's Law (Microscopic Derivation)
💡 Why this matters in real life
When you flip a switch, the light turns on almost instantly — but the individual electrons barely move! This is the key insight of drift velocity. Understanding how current flows at the atomic level helps us design better wires, understand why wires heat up (I²R loss), and explains why some materials conduct electricity better than others.
Fig 3.1: Drift velocity — synchronized. Animation advances through atoms, free electrons, net drift, formulas, and summary.
🧠 The Big Idea — In Plain Words
Electrons in a metal move randomly in all directions at huge speeds (~10⁶ m/s) due to thermal energy. When you apply a voltage, it creates an electric field that drifts the electrons slowly (about 1 mm/s!) in one direction. This drift velocity v_d is very small. But because there are so many electrons, the total current is substantial. The formula I = nAev_d connects the current to the number of electrons (n), wire area (A), and drift velocity (v_d).
Ohm's law (V = IR) is a macroscopic law. The microscopic version shows that resistivity ρ depends on the electron density, charge, and how often they collide with atoms (relaxation time τ). This explains why conductors have low resistivity and insulators high resistivity.
Key variables: n = free electron density (m⁻³), A = cross-section (m²), e = 1.6×10⁻¹⁹ C, v_d = drift velocity (m/s), τ = relaxation time (s), σ = conductivity (Ω⁻¹m⁻¹), ρ = resistivity (Ω·m)
🔍 Step-by-Step: Current in terms of Drift Velocity
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Picture a wire with cross-section A. When an electric field E is applied, electrons drift with velocity v_d along the wire. In time Δt, an electron moves a distance v_d·Δt. So the volume of wire from which electrons exit in Δt = A × v_d·Δt.
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Number of electrons in this volume = n × Volume = n·A·v_d·Δt. Each electron carries charge e. Total charge flowing in Δt: Q = n·A·v_d·Δt·e
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Current I = Q/Δt = n·A·v_d·e.I = nAev_d. This tells us: for a given current, if the wire is thicker (A↑) or has more free electrons (n↑), the drift velocity is smaller.
I = nAev_d Drift velocity: v_d = I/(nAe)
🔍 Step-by-Step: Deriving Ohm's Law Microscopically
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An electron in an electric field E experiences force F = eE. This accelerates it: a = eE/m. But before it speeds up too much, it crashes into an atom (collision). The relaxation time τ is the average time between collisions.
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Drift velocity = acceleration × average time between collisions. The electron accelerates from 0 after each collision. Average velocity during one free run = ½·a·τ. But more precisely: v_d = (eE/m)·τ (the factor of ½ is absorbed into a more detailed calculation).
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Substitute v_d into I = nAev_d.I = nAe·(eEτ/m) = (nAe²τ/m)·E
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Express E in terms of V and L: E = V/L (potential difference divided by length).I = (nAe²τ/m)·(V/L) → V = I·(mL)/(nAe²τ)
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Compare with V = IR to identify resistance R.R = mL/(nAe²τ) = ρ·L/A, where ρ = m/(ne²τ) is the resistivity. This microscopic model shows that resistivity depends on the electron density n and the relaxation time τ (which decreases with temperature → ρ increases with temperature in conductors).
R = ρL/A ρ = m/(ne²τ) σ = 1/ρ = ne²τ/m
🔍 Common Mistake
Students think drift velocity is the speed of electrons. It's not! Electrons move randomly at ~10⁶ m/s, but their net drift is only ~10⁻⁴ m/s. Also: Ohm's law (V = IR) is not a universal law — it only holds for ohmic conductors at constant temperature. Semiconductors, electrolytes, and gases do NOT follow Ohm's law.
🎯 NEET Pattern — Drift Velocity
• "A wire of length L, area A has n free electrons per m³. Find drift velocity for current I." → v_d = I/(nAe).
• As temperature increases, τ decreases (more collisions) → ρ increases → R increases for conductors.
• For insulators: n is very small (few free electrons) → ρ is huge.
• NEET common: "Two wires of same material, one twice as thick. Same current → drift velocity in thinner wire is ___" → 4 times larger (since v_d ∝ 1/A ∝ 1/r²).
Example 3: Drift Velocity
A copper wire of cross-section 1 mm² carries a current of 1 A. If the free electron density in copper is 8.5×10²⁸ m⁻³, find the drift velocity. (e = 1.6×10⁻¹⁹ C)
The Biot-Savart law tells us how electric currents create magnetic fields. This is the physics behind electromagnets (used in scrap yards to lift cars), MRI machines (which use strong magnetic fields to image your body), electric motors, and even the Earth's magnetic field (generated by currents in the molten core).
Fig 4.1: Biot-Savart — synchronized. Animation advances through current, field lines, right-hand rule, formula, and summary.
🧠 The Big Idea — In Plain Words
A moving charge (current) creates a magnetic field around it. The Biot-Savart law gives the magnetic field dB produced by a tiny segment of current-carrying wire. The field dB is: (1) proportional to the current I and the length of the segment dl, (2) inversely proportional to the square of the distance r², and (3) depends on the sine of the angle between dl and the direction to the point. The right-hand thumb rule gives the direction: point your thumb along current, fingers curl in direction of B.
Key variables: I = current (A), dl = tiny length element (m), r = distance from element (m), θ = angle between dl and r, μ₀ = 4π×10⁻⁷ T·m/A (permeability of free space)
🔍 Step-by-Step: Biot-Savart Law
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The law in its general form. A current element Idl (current I flowing through length dl) produces a magnetic field dB at a point at distance r: dB = (μ₀/4π)·(Idl × r̂)/r² = (μ₀/4π)·(I·dl·sinθ)/r²
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Application 1: Magnetic field at centre of a circular current loop. For a circular loop of radius R, every dl is perpendicular to r (θ = 90°, sinθ = 1), and r = R constant. dB from each element points in the same direction (out of plane, by right-hand rule): B = ∫dB = (μ₀/4π)·(I/R²)∫dl = (μ₀/4π)·(I/R²)·(2πR) = μ₀I/(2R)
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Application 2: Field due to a long straight wire. For a straight wire of length L, at a perpendicular distance a from the wire, we integrate along the wire. Each element dl at distance r = a/sinθ from P: B = (μ₀I)/(4πa)[sinφ₁ + sinφ₂]. For an infinitely long wire (φ₁ = φ₂ = 90°): B = μ₀I/(2πa)
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Application 3: Field on the axis of a circular loop. At distance x from centre on the axis, only the axial component adds up (perpendicular components cancel by symmetry): B = (μ₀IR²)/(2(R² + x²)^(3/2)). At centre (x = 0): B = μ₀I/(2R). Far away (x >> R): B ≈ μ₀IR²/(2x³) — like a magnetic dipole.
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Application 4: Field inside a solenoid. A solenoid is many circular loops stacked together. Using the axial field formula and integrating along the length: B = μ₀nI (inside a long solenoid), where n = N/L is the number of turns per unit length. The field is nearly uniform inside and near zero outside.
Straight wire: B = μ₀I/(2πa) Loop centre: B = μ₀I/(2R) Solenoid: B = μ₀nI
🔍 Common Mistake
Students confuse the Biot-Savart law (current → magnetic field) with Faraday's law (changing magnetic field → EMF). Also: the Biot-Savart law gives the field due to a steady current. For time-varying currents, the full Maxwell-Ampere law is needed (which includes displacement current).
🎯 NEET Pattern — Biot-Savart
• Straight wire: B ∝ I/a. Double the current → double B. Double the distance → half B.
• Circular loop: B at centre ∝ I/R. Half the radius → double B.
• Solenoid: B = μ₀nI = μ₀NI/L. Independent of cross-sectional area!
• Ampere's circuital law (alternative way): ∮B·dl = μ₀I_enclosed. Useful for symmetric situations (infinite wire, solenoid, toroid).
• NEET 2023: "Two parallel wires carrying currents in same direction ___" → attract each other.
Example 4: Biot-Savart Law
A long straight wire carries 5 A current. Find the magnetic field at a distance of 2 cm from the wire. (μ₀ = 4π×10⁻⁷)
a) 5×10⁻⁵ T b) 2.5×10⁻⁵ T c) 1.25×10⁻⁵ T d) 10⁻⁴ T
Solution: B = μ₀I/(2πa) = (4π×10⁻⁷×5)/(2π×0.02) = (20π×10⁻⁷)/(0.04π) = 500×10⁻⁷ = 5×10⁻⁵ T. Option (a).
5. Bohr's Model of Hydrogen Atom
💡 Why this matters in real life
Bohr's model was the first successful explanation of why atoms emit light at specific colours (like the red glow of a neon sign or the yellow of sodium street lamps). It introduced the revolutionary idea that electrons can only exist in certain allowed orbits — quantisation of angular momentum. This was the first step toward quantum mechanics, which underlies all of modern electronics, lasers, and even how we understand the chemical properties of elements.
Fig 5.1: Bohr model — synchronized. Animation advances through orbits, electrons, energy levels, formulas, and summary.
🧠 The Big Idea — In Plain Words
Bohr proposed three postulates: (1) Electrons orbit the nucleus in stationary orbits without radiating energy — something classical physics couldn't explain. (2) The angular momentum of an electron is quantised: mvr = nh/(2π), where n = 1, 2, 3... (the principal quantum number). (3) Electrons can "jump" between orbits by absorbing or emitting a photon of energy equal to the difference in orbit energies: ΔE = hf.
From these postulates, we can derive the radius, velocity, and energy of the nth orbit. The radius increases as n², the velocity decreases as 1/n, and the energy is -13.6/n² eV (negative means the electron is bound).
Key variables: n = principal quantum number, r_n = radius of nth orbit, v_n = speed, E_n = total energy, h = Planck's constant
🔍 Step-by-Step: Radius of nth Bohr Orbit
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Coulomb force provides the centripetal force. The electrostatic attraction between the nucleus (+Ze) and electron (-e) keeps the electron in circular motion. For hydrogen, Z = 1: ke²/r² = mv²/r → ke²/r = mv²
Substitute v into the centripetal equation. From step 1: mv² = ke²/r. Substitute v² = n²h²/(4π²m²r²): m·[n²h²/(4π²m²r²)] = ke²/r → n²h²/(4π²mr²) = ke²/r → r = n²h²/(4π²mke²)
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Plug in the constants. h = 6.63×10⁻³⁴, m = 9.1×10⁻³¹, k = 9×10⁹, e = 1.6×10⁻¹⁹: r_n = n² × 0.529 Å = n²a₀, where a₀ = 0.529 Å is the Bohr radius (radius of the smallest orbit, n=1). The radius grows as n² — higher orbits are much larger.
r_n = n²a₀ a₀ = 0.529 Å = 5.29×10⁻¹¹ m (Bohr radius)
🔍 Step-by-Step: Velocity and Energy of nth Orbit
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Velocity from quantisation: v = nh/(2πmr). Using r = n²a₀: v_n = h/(2πma₀)·(1/n) = v₁/n. The velocity decreases as 1/n. v₁ ≈ 2.2×10⁶ m/s (about 1/137 of speed of light — the fine structure constant α = v₁/c = 1/137).
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Total energy = Kinetic energy + Potential energy. KE = ½mv². From the force equation, mv² = ke²/r, so KE = ke²/(2r). PE = -ke²/r (attractive potential, negative). Total: E = KE + PE = ke²/(2r) - ke²/r = -ke²/(2r)
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Substitute r = n²a₀:E_n = -ke²/(2n²a₀) = -13.6/n² eV. For n=1 (ground state): -13.6 eV — this is the ionisation energy of hydrogen. For n=2: -3.4 eV. For n=3: -1.51 eV. As n → ∞, E → 0 (electron is free).
E_n = -13.6/n² eV v_n = v₁/n r_n = n²a₀
🔍 Common Mistake
Students often forget that the total energy of an electron in orbit is negative. A negative energy means the electron is bound to the nucleus — you need to give it +13.6 eV to free it. Also: the Bohr model only works for single-electron atoms (H, He⁺, Li²⁺). For multi-electron atoms, the full Schrödinger equation is needed.
🎯 NEET Pattern — Bohr's Model
• Energy of photon emitted when electron jumps from n₂ to n₁: ΔE = 13.6(1/n₁² - 1/n₂²) eV.
• Wavelength of emitted photon: hc/λ = ΔE. For Lyman series (n₁ = 1): UV. Balmer (n₁ = 2): visible. Paschen (n₁ = 3): infrared.
• Spectral lines in a transition from n₂ to n₁: number of lines = n₂(n₂-1)/2.
• For hydrogen-like atoms (Z > 1): E_n = -13.6 Z²/n² eV, r_n = n²a₀/Z.
Example 5: Bohr's Model
Find the energy required to excite an electron in hydrogen from n=1 to n=3.
The photoelectric effect is how solar panels generate electricity! When light hits a metal surface, it can knock out electrons. More importantly, this experiment proved that light behaves as particles (photons) — not just waves — and won Einstein the Nobel Prize. It's the foundation of quantum mechanics and is used in light sensors, automatic doors, digital cameras, and solar cells.
Fig 6.1: Photoelectric effect — synchronized. Animation advances through photons, ejected electrons, I-V graph, formula, and summary.
🧠 The Big Idea — In Plain Words
Classical physics predicted that shining brighter light (higher intensity) should knock out electrons with more energy. But experiments showed something strange: brighter light knocked out more electrons, not more energetic ones. And if the light's frequency was below a certain threshold, no electrons came out at all, no matter how bright the light!
Einstein explained this by proposing that light comes in packets (photons) each with energy E = hf. One photon gives ALL its energy to ONE electron. The electron uses some energy to escape the metal (work function φ) and the rest becomes kinetic energy: hf = φ + ½mv². If hf < φ, no electron can escape (no matter how many photons). This instantly explained the threshold frequency and why intensity only affects the number of electrons, not their energy.
Key variables: h = Planck's constant, f = frequency (Hz), λ = wavelength (m), φ = work function (eV or J), K_max = maximum KE of photoelectron (eV or J), V₀ = stopping potential (V)
🔍 Step-by-Step: Einstein's Photoelectric Equation
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Energy of the incident photon: E = hf = hc/λ. Planck had already shown that light energy is quantised as E = hf. Einstein took the next step: this photon behaves like a particle that can transfer all its energy to a single electron.
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The work function φ: the minimum energy needed to free an electron from the metal. Different metals have different φ. Sodium has φ ≈ 2.3 eV (easy to knock out), while platinum has φ ≈ 6.3 eV (harder). The energy needed depends on how tightly the metal holds its electrons.
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Einstein's photoelectric equation (energy conservation). The photon's energy goes into two things: (a) freeing the electron from the metal (φ), and (b) giving it kinetic energy: hf = φ + K_max, where K_max = ½mv_max²
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Stopping potential V₀: the voltage that stops the fastest electrons. If you apply a reverse voltage, the fastest electrons are stopped when eV₀ = K_max: eV₀ = hf - φ → V₀ = (h/e)f - φ/e. A graph of V₀ vs f is a straight line with slope = h/e. This is how Millikan experimentally verified Einstein's equation and measured Planck's constant!
Students confuse intensity and frequency. Intensity = number of photons per second (affects number of photoelectrons, i.e., current). Frequency = energy per photon (affects K_max of electrons). Also: photoelectric emission is instantaneous — the electron absorbs the photon's energy immediately (no time delay as wave theory predicted!).
🎯 NEET Pattern — Photoelectric Effect
• Threshold frequency f₀ = φ/h. Below f₀: no emission regardless of intensity.
• Above threshold: K_max = h(f - f₀). Increasing intensity → more electrons (higher photocurrent), same K_max.
• Stopping potential V₀ = (h/e)(f - f₀). Independent of intensity.
• NEET classic: "Light of wavelength λ falls on a metal with work function φ. Find stopping potential." → V₀ = (hc/λ - φ)/e.
• Factors affecting photocurrent: intensity of light (↑ intensity → ↑ current), potential difference (saturates at high V).
• Davis and Goucher experiment showed that wave theory cannot explain photoemission.
Example 6: Photoelectric Effect
Light of wavelength 200 nm falls on a metal surface with work function 4.0 eV. Find the stopping potential. (h = 6.63×10⁻³⁴, c = 3×10⁸, e = 1.6×10⁻¹⁹)