Modern Physics

1. Dual Nature of Radiation & Matter

Photoelectric Effect

The photoelectric effect refers to the emission of electrons from a metal surface when electromagnetic radiation of a suitable frequency falls upon it. Heinrich Hertz first observed the phenomenon in 1887, but Albert Einstein provided the correct theoretical explanation in 1905 using Max Planck's quantum theory of radiation. For this work, Einstein was awarded the Nobel Prize in Physics in 1921.

According to the quantum theory, light consists of discrete packets of energy called photons. Each photon carries an energy given by E = hf, where h = 6.63 × 10^-34 Js (Planck's constant) and f is the frequency of the radiation. The intensity of light corresponds to the number of photons per unit area per unit time, while the energy of individual photons determines their ability to eject electrons.

When a photon strikes a metal surface, its energy is transferred to an electron within the metal. The electron must overcome the attractive force binding it to the metal, called the work function φ. If the photon energy exceeds the work function, the electron is ejected with surplus energy appearing as kinetic energy.

Experimental Setup: The photoelectric effect is studied using an evacuated glass tube with two electrodes. A photosensitive metal plate (cathode) is irradiated with monochromatic light. Emitted electrons are collected by the anode, and the resulting photocurrent is measured with a microammeter. The stopping potential is determined by applying a reverse voltage until the photocurrent drops to zero.

The graph between photocurrent and applied voltage shows that for a fixed frequency and intensity, the current saturates at higher voltages. The stopping potential is independent of intensity but varies linearly with frequency. The slope of the V₀ vs f graph gives h/e, from which Planck's constant can be determined experimentally.

Einstein's Photoelectric Equation

The governing equation is:

hf = φ + KE_max

Where:

• hf = energy of the incident photon
• φ = work function of the metal (minimum energy required to eject an electron)
• KE_max = maximum kinetic energy of the emitted photoelectron

The maximum kinetic energy is directly measured using the stopping potential V₀:

KE_max = eV₀

The threshold frequency f₀ is the minimum frequency required to eject electrons:

f₀ = φ/h

Similarly, the threshold wavelength λ₀ = hc/φ. No photoelectric emission occurs when the incident wavelength is longer than the threshold wavelength.

Graphical Analysis: The photoelectric effect can be understood through several key graphs:

• Photocurrent vs Voltage: For a fixed frequency and intensity, the photocurrent increases with applied voltage and saturates when all emitted electrons reach the anode. The saturation current is proportional to light intensity. The stopping potential V₀ is the voltage at which the current becomes zero.
• Stopping Potential vs Frequency: This is a straight line with slope h/e and intercept -φ/e on the V₀ axis. The intercept on the frequency axis gives the threshold frequency f₀. This graph is the most important for NEET numerical problems.
• Kinetic Energy vs Frequency: A straight line with slope h and intercept -φ on the KE axis. KE_max increases linearly with frequency.
• Photocurrent vs Intensity: A linear relationship — current is directly proportional to intensity for a fixed frequency above threshold.

Laws of Photoelectric Effect:

• The number of photoelectrons emitted per second is directly proportional to the intensity of incident light, provided f > f₀.
• The maximum kinetic energy of photoelectrons increases linearly with the frequency of incident light and is independent of its intensity.
• There is no measurable time lag between the incidence of light and the emission of photoelectrons (less than 10^-9 s).
• Photoemission occurs only when the frequency of incident light exceeds the threshold frequency f₀.

Table 1.1: Photon Energy Across the Electromagnetic Spectrum

de Broglie's Hypothesis

In 1924, Louis de Broglie proposed that just as light exhibits both wave and particle nature, matter particles such as electrons, protons, and neutrons also possess wave-like properties. This revolutionary idea extended the wave-particle duality from radiation to all matter.

According to de Broglie, the wavelength associated with a material particle is inversely proportional to its momentum:

λ = h/p = h/(mv)

where p is the linear momentum, m is the mass, and v is the velocity of the particle. The wavelength is called the de Broglie wavelength or matter-wave wavelength.

For an electron accelerated through a potential difference V volts:

λ = h / √(2meV) = 12.27 / √V   Å

This is a highly useful formula for NEET problems. For an electron accelerated through 100 V, the de Broglie wavelength is about 1.227 Å.

Properties of Matter Waves:

• Matter waves are not electromagnetic waves; they are probability waves.
• The wavelength decreases as the momentum of the particle increases.
• Heavier particles have shorter wavelengths, making their wave nature difficult to observe.
• Matter waves travel at speeds different from the particle's speed (phase velocity vs group velocity).

Table 1.2: de Broglie Wavelengths of Common Particles

Davisson-Germer Experiment

The Davisson-Germer experiment (1927) provided the first experimental confirmation of de Broglie's hypothesis by demonstrating electron diffraction from a nickel crystal. C. J. Davisson and L. H. Germer directed a beam of electrons at a nickel crystal and observed the diffraction pattern, which could only be explained if electrons behaved as waves.

The experiment used an electron gun to produce a focused beam of electrons accelerated through a known potential. The beam was directed at a nickel crystal, and the intensity of scattered electrons was measured as a function of the scattering angle using a Faraday cup connected to a sensitive galvanometer. A sharp peak in the intensity was observed at a scattering angle of 50° for an accelerating voltage of 54 V.

2d sinθ = nλ

For the nickel crystal, the interatomic spacing d was known (0.215 nm for nickel). The first-order diffraction maximum (n = 1) was observed at θ = 65° (the incident angle measured from the crystal surface). Using Bragg's law, the wavelength was calculated to be about 1.65 Å. This matched the de Broglie wavelength predicted by λ = 12.27/√V = 12.27/√54 = 1.66 Å, providing excellent agreement.

The experimentally determined wavelength matched the de Broglie wavelength calculated from the accelerating voltage, thereby confirming the wave nature of electrons. Davisson and G. P. Thomson shared the Nobel Prize in Physics in 1937 for this discovery. The experiment was significant because it conclusively proved that particles of matter (electrons) exhibit wave-like behaviour, confirming the wave-particle duality of matter.

Table 1.3: Wave-Particle Duality Summary

Practical Applications of Wave-Particle Duality:

• Electron Microscopy: The short de Broglie wavelength of electrons (about 0.04 Å for 100 keV electrons) enables much higher resolution than optical microscopes. A transmission electron microscope (TEM) can achieve resolution below 0.1 nm, allowing visualisation of individual atoms.
• Neutron Diffraction: Neutrons have wavelengths comparable to atomic spacings and are used to study the structure of materials, especially light elements like hydrogen that are difficult to detect with X-rays.
• Scanning Tunneling Microscope (STM): Uses the quantum tunnelling of electrons between a sharp tip and a conducting surface to image individual atoms on surfaces.

Significance of the Experiment:

• It was the first direct experimental verification of de Broglie's hypothesis.
• It demonstrated that electrons can be diffracted, a property unique to waves.
• It established the wave nature of matter as a fundamental principle of quantum mechanics.
• Modern applications include electron microscopy, which uses the wave nature of electrons to image objects at atomic resolution.

2. Atoms

Rutherford's Model of the Atom

Ernest Rutherford's gold foil experiment (1911) revolutionized the understanding of atomic structure. He bombarded a thin gold foil with α-particles and observed their scattering pattern. The key observations were:

• Most α-particles passed through the foil without any deflection.
• A small fraction were deflected through small angles.
• A very few (about 1 in 8000) were deflected through large angles, some even bouncing back.

Based on these observations, Rutherford proposed the nuclear model of the atom:

• The atom contains a tiny, dense, positively charged nucleus at its center.
• The nucleus is about 10^-15 m in diameter, while the atom is about 10^-10 m.
• Electrons revolve around the nucleus in circular orbits, similar to planets around the Sun.
• Most of the atom is empty space.

Limitations of Rutherford's Model:

• According to Maxwell's electromagnetic theory, an accelerating charged particle (electron) must continuously radiate energy. This would cause the electron to spiral into the nucleus, making the atom unstable.
• Rutherford's model could not explain the discrete line spectra of atoms.
• The model did not specify the distribution of electrons or their orbits.

Table 2.1: Comparison of Atomic Models

Bohr's Model of the Hydrogen Atom

Niels Bohr (1913) combined Rutherford's nuclear model with Planck's quantum theory to explain atomic spectra and stability. He proposed three postulates:

First Postulate (Stationary Orbits): Electrons revolve around the nucleus only in certain permitted circular orbits called stationary orbits. In these orbits, the electron does not radiate energy despite being accelerated. This was a radical departure from classical electromagnetic theory, which predicted that accelerating charges must radiate energy.

Second Postulate (Angular Momentum Quantisation): The angular momentum of the electron in these stationary orbits is an integer multiple of h/2π:

mvr = nh/2π,   where n = 1, 2, 3, ...

The integer n is called the principal quantum number. Each value of n corresponds to a specific orbit. The quantisation of angular momentum is the key postulate that leads to discrete energy levels.

Third Postulate (Frequency of Radiation): When an electron jumps from a higher energy orbit (E_i) to a lower energy orbit (E_f), the difference in energy is emitted as a photon of frequency:

hf = E_i - E_f

Conversely, an electron can jump from a lower to a higher energy orbit by absorbing a photon of exactly the right energy. This explains why atomic spectra consist of discrete lines rather than a continuous spectrum.

Derivation of Key Results: Combining the second postulate (mvr = nh/2π) with the Coulomb force providing the centripetal acceleration (mv²/r = kZe²/r², where k = 1/4πε₀), we can eliminate v to obtain the radius, velocity, and energy expressions.

Step 1 — Velocity: From Bohr's second postulate, v = nh/2πmr. From Coulomb's law, mv²/r = kZe²/r². Substituting v gives: m(nh/2πmr)²/r = kZe²/r², which simplifies to find r.

Step 2 — Radius: Solving for r: r_n = n²h²ε₀/πmZe² = n² × (0.529 Å)/Z.

Step 3 — Velocity: Substituting r back: v_n = Ze²/2ε₀nh = (2.18 × 10⁶)Z/n m/s.

Step 4 — Energy: Total energy E = KE + PE = (1/2)mv² + (−kZe²/r) = −kZe²/(2r). Substituting r: E_n = −mZ²e⁴/8ε₀²n²h² = −13.6Z²/n² eV.

Step 5 — Frequency of Revolution: f_n = v_n/2πr_n = mZ²e⁴/4ε₀²n³h³. The frequency of emitted radiation when an electron jumps from n+1 to n is approximately equal to the orbital frequency for large n (correspondence principle).

Energy Levels, Radius, and Velocity

From Bohr's postulates and Coulomb's law, we can derive the following key formulas for hydrogen-like atoms (atomic number Z, single electron):

Radius of the n^th orbit:

r_n = n² × (0.529 Å) / Z

For hydrogen (Z = 1), the first Bohr radius r₁ = 0.529 Å.

Velocity of the electron in the n^th orbit:

v_n = (2.18 × 10⁶) × Z / n   m/s

Total energy of the electron in the n^th orbit:

E_n = -13.6 × Z² / n²   eV

For hydrogen (Z = 1): E₁ = -13.6 eV, E₂ = -3.4 eV, E₃ = -1.51 eV, etc.

Table 2.2: Bohr Orbits for Hydrogen Atom (Z = 1)

Success of Bohr's Model: The model successfully explained the discrete line spectrum of hydrogen, the Rydberg formula, and the existence of discrete energy levels. It also correctly predicted the ionisation energy of hydrogen (13.6 eV) and the wavelengths of the Lyman, Balmer, and other series.

Limitations of Bohr's Model:

• It could not explain the spectra of multi-electron atoms (helium, lithium, etc.).
• It could not explain the fine structure of spectral lines (splitting in magnetic fields).
• It violated the Heisenberg uncertainty principle by assuming definite electron paths.
• It could not explain the relative intensities of spectral lines.
• It worked only for hydrogen-like atoms (single electron systems).
• The model was a semi-classical hybrid — it used classical orbits with quantum conditions.

Spectral Series of Hydrogen

When an electron jumps from a higher orbit (n₂) to a lower orbit (n₁), the emitted photon's wavelength is given by the Rydberg formula:

1/λ = R(1/n₁² - 1/n₂²)

where R = 1.097 × 10⁷ m^-1 is the Rydberg constant. Different spectral series correspond to transitions ending at different lower orbits.

Table 2.3: Spectral Series of Hydrogen Atom

X-rays

X-rays are electromagnetic waves of very short wavelength (0.01 nm to 10 nm) produced when high-energy electrons strike a metal target. Wilhelm Röntgen discovered them in 1895.

Production: A Coolidge tube uses a heated filament (cathode) that emits electrons. These electrons are accelerated by a high voltage (typically 30-150 kV) toward a metal anode (target). When the electrons strike the target, X-rays are produced through two mechanisms:

• Characteristic X-rays: Produced when an incident electron knocks out an inner-shell electron, and an outer electron fills the vacancy. The emitted photon has an energy equal to the difference between the two energy levels.
• Continuous X-rays (Bremsstrahlung): Produced when incident electrons are decelerated by the electric field of the target nuclei. This gives a continuous spectrum with a sharp cutoff at the minimum wavelength:

λ_min = hc/eV = 12400/V   Å (V in volts)

Moseley's Law: For characteristic X-rays, the frequency is related to the atomic number of the target by:

√f = a(Z - b)

where a and b are constants. For K_α lines, a ≈ 4.97 × 10⁷ Hz^1/2 and b ≈ 1 (shielding constant). This law established the systematic arrangement of elements in the periodic table and allowed the prediction of undiscovered elements based on their X-ray spectra.

Types of X-ray Spectra:

• Continuous spectrum: Produced by bremsstrahlung (braking radiation) — electrons decelerated by the target nuclei emit photons of various energies up to a maximum determined by the applied voltage. The minimum wavelength λ_min = hc/eV.
• Characteristic spectrum: Sharp peaks superimposed on the continuous spectrum. These occur at specific wavelengths characteristic of the target material. The K_α line results from an electron transition from L-shell (n=2) to K-shell (n=1), while K_β is from M-shell (n=3) to K-shell (n=1).

Table 2.4: Characteristic X-ray Transitions

3. Nuclei

Nuclear Structure

The atomic nucleus was discovered by Rutherford in 1911. It is the tiny, dense, positively charged core of the atom that contains nearly all of its mass. The nucleus consists of:

• Protons: Positively charged particles with charge +e and mass 1.6726 × 10^-27 kg (1.00727 u).
• Neutrons: Neutral particles with mass 1.6749 × 10^-27 kg (1.00866 u), slightly heavier than protons.

Collectively, protons and neutrons are called nucleons. The number of protons is the atomic number (Z), and the total number of nucleons is the mass number (A). Nuclides are represented as ^A_ZX.

Nuclear Size and Density: The nuclear radius is given by:

R = R₀ A^1/3

where R₀ ≈ 1.2 × 10^-15 m. Nuclear density is enormous: about 2.3 × 10¹⁷ kg/m³, independent of the nucleus size. This uniform density indicates that nuclear matter is incompressible.

Nuclear Force: The strong nuclear force binds nucleons together. It is a short-range force (acts only within about 10^-15 m) that is independent of charge (acts between proton-proton, proton-neutron, and neutron-neutron). It is about 100 times stronger than the electromagnetic force at nuclear distances. The nuclear force is saturable, meaning a nucleon interacts only with its nearest neighbours, which is why the binding energy per nucleon is approximately constant for medium-mass nuclei (saturation property).

Nuclear Stability: The stability of a nucleus depends on the balance between the attractive nuclear force and the repulsive Coulomb force between protons. Light stable nuclei have roughly equal numbers of protons and neutrons (N ≈ Z). As Z increases, the Coulomb repulsion grows, requiring more neutrons for stability (N > Z). The most stable nuclei lie along the "line of stability" on the N-Z plot. Nuclei with Z > 82 (lead) are unstable and undergo radioactive decay. The "magic numbers" (2, 8, 20, 28, 50, 82, 126) correspond to particularly stable nuclei with filled nuclear shells.

Table 3.1: Comparison of Atomic and Nuclear Sizes

Mass Defect and Binding Energy: The mass of a nucleus is always less than the sum of the masses of its constituent nucleons. This difference is the mass defect (Δm):

Δm = [Zm_p + (A-Z)m_n] - M_nucleus

The energy equivalent of this mass defect is the binding energy (BE = Δm × c²). Using the conversion 1 u = 931.5 MeV/c²:

BE = Δm (in u) × 931.5   MeV

Binding energy per nucleon (BE/A) is a measure of nuclear stability. Iron-56 has the highest BE/A (~8.8 MeV), making it the most stable nucleus.

Table 3.2: Binding Energy per Nucleon for Selected Nuclides

Radioactivity

Radioactivity is the spontaneous emission of radiation from the nucleus of an unstable atom. It was discovered by Henri Becquerel in 1896. Marie and Pierre Curie isolated the radioactive elements polonium and radium.

Alpha, Beta, and Gamma Decay

There are three types of radioactive emissions:

Table 3.3: Properties of Alpha, Beta, and Gamma Radiation

Alpha Decay: ^A_ZX → ^A-4_Z-2Y + ⁴₂He. Example: ²³⁸₉₂U → ²³⁴₉₀Th + α.

Beta-minus Decay: ^A_ZX → ^A_Z+1Y + e^- + ν̅_e. Example: ¹⁴₆C → ¹⁴₇N + β^- + ν̅_e.

Beta-plus Decay: ^A_ZX → ^A_Z-1Y + e⁺ + ν_e.

Gamma Decay: ^A_ZX* → ^A_ZX + γ (excited nucleus to ground state).

Half-Life and Mean Life

The rate of radioactive decay follows first-order kinetics. The number of nuclei remaining after time t is:

N = N₀ e^-λt

where N₀ is the initial number of nuclei and λ is the decay constant. The activity (A) is A = λN, measured in becquerel (Bq) or curie (Ci).

Half-Life (T_1/2): The time required for half of the radioactive nuclei to decay:

T_1/2 = ln(2)/λ = 0.693/λ

Mean Life (τ): The average lifetime of a radioactive nucleus:

τ = 1/λ = T_1/2/0.693 = 1.44 T_1/2

After n half-lives (t = nT_1/2): N = N₀(1/2)ⁿ.

Table 3.4: Radioactive Decay After n Half-Lives

Nuclear Reactions

A nuclear reaction is a process in which two nuclei or a nucleus and a subatomic particle collide to produce different nuclei. The general form is:

^A_ZX + a → ^A'_Z'Y + b

The Q-value of a nuclear reaction is the energy released or absorbed:

Q = (initial mass - final mass) × c²

If Q > 0, energy is released (exothermic). If Q < 0, energy is absorbed (endothermic).

Nuclear Fission

Nuclear fission is the splitting of a heavy nucleus (like ²³⁵U or ²³⁹Pu) into two or more lighter nuclei, accompanied by the release of a large amount of energy and several neutrons.

²³⁵₉₂U + ¹₀n → ²³⁶₉₂U* → ¹⁴¹₅₆Ba + ⁹²₃₆Kr + 3¹₀n + ~200 MeV

Key features of fission:

• The chain reaction is sustained by the neutrons released.
• Critical mass is required for a self-sustaining chain reaction.
• Used in nuclear reactors and atomic bombs.
• 233 MW of thermal energy from 1 kg of ²³⁵U (vs. 24 MWh from 1 kg of coal).

Nuclear Fusion

Nuclear fusion is the combining of two light nuclei to form a heavier nucleus, with enormous energy release. Fusion powers the Sun and other stars.

²₁H + ³₁H → ⁴₂He + ¹₀n + 17.6 MeV

Key features of fusion:

• Requires extremely high temperatures (~10⁸ K) to overcome Coulomb repulsion.
• Produces more energy per unit mass than fission.
• Produces less radioactive waste than fission.
• Not yet commercially viable for power generation on Earth (research ongoing).

Table 3.5: Comparison of Nuclear Fission and Fusion

4. Semiconductor Electronics

Energy Bands in Solids

In solids, atomic energy levels merge to form energy bands due to the interaction between large numbers of atoms. The key bands are:

• Valence Band: The band occupied by valence electrons. At 0 K, it is completely filled.
• Conduction Band: The band where electrons are free to conduct electricity. At 0 K, it is empty.
• Forbidden Gap (E_g): The energy difference between the top of the valence band and the bottom of the conduction band.

Table 4.1: Energy Band Gap of Different Materials

Intrinsic and Extrinsic Semiconductors:

• Intrinsic: Pure semiconductor with equal number of electrons and holes (n_e = n_h).
• Extrinsic: Doped semiconductor with added impurities to increase conductivity.
• n-type: Doped with pentavalent atoms (P, As, Sb), electrons are majority carriers.
• p-type: Doped with trivalent atoms (B, Al, In), holes are majority carriers.

Table 4.2: Comparison of n-type and p-type Semiconductors

Diodes

A p-n junction diode is formed by joining p-type and n-type semiconductors. At the junction, diffusion of charge carriers creates a depletion region (no free carriers), establishing a built-in potential barrier (about 0.3 V for Ge, 0.7 V for Si).

Formation of Depletion Region: When p-type and n-type materials are joined, holes from the p-side diffuse into the n-side, and electrons from the n-side diffuse into the p-side. This diffusion leaves behind immobile charged ions: negative acceptor ions on the p-side and positive donor ions on the n-side. The region devoid of mobile charge carriers is called the depletion region or space charge region. The electric field created by these immobile ions opposes further diffusion, establishing equilibrium. The width of the depletion region is typically about 0.5 μm and depends on the doping concentration, applied voltage, and temperature.

Energy Band Diagram: In equilibrium, the Fermi level is constant throughout the p-n junction. The conduction and valence bands bend near the junction, creating an energy barrier (qV_B, where V_B is the built-in potential). The built-in potential for a silicon p-n junction at room temperature is approximately V_B = (kT/e)ln(N_AN_D/n_i²), where N_A and N_D are acceptor and donor concentrations, and n_i is the intrinsic carrier concentration.

Forward Bias: p-side connected to positive, n-side to negative. The applied voltage opposes the built-in potential, reducing the depletion width. Once the applied voltage exceeds the barrier potential, a large forward current flows. The forward current increases exponentially with voltage.

Reverse Bias: p-side connected to negative, n-side to positive. The applied voltage adds to the built-in potential, widening the depletion region. Only a small reverse saturation current (due to minority carriers) flows. At high reverse voltage, breakdown occurs through either the Zener effect (at low voltages, due to tunnelling) or avalanche breakdown (at higher voltages, due to impact ionisation).

Table 4.3: Diode Characteristics Summary

V-I Characteristics: The current-voltage relationship of a p-n junction diode is given by the diode equation:

I = I₀[exp(eV/ηkT) - 1]

where I₀ is the reverse saturation current, V is the applied voltage, k = 1.38 × 10^-23 J/K is Boltzmann's constant, T is the absolute temperature, and η is the ideality factor (η = 1 for Ge, η = 2 for Si). At room temperature (300 K), kT/e = 0.026 V. For forward bias V > 0.1 V, the exponential term dominates and I ≈ I₀exp(eV/ηkT). For reverse bias, I ≈ -I₀.

The dynamic (AC) resistance of a diode is given by r_d = ΔV/ΔI = ηkT/(eI) at a given operating current I. At room temperature for a silicon diode, r_d ≈ 0.026/I for η = 1.

Diode Applications:

• Rectifier: Converts AC to DC (discussed below).
• Clipper: Removes portions of a signal above or below a reference level.
• Clamper: Shifts the DC level of a signal.
• Zener regulator: Maintains constant output voltage using Zener breakdown.
• LED: Emits light when forward biased (electroluminescence).
• Photodiode: Conducts more when illuminated (used in light sensors).
• Solar cell: Converts light energy into electrical energy.

Rectifiers

A rectifier converts AC to DC. There are two types:

Half-Wave Rectifier: Uses a single diode. Conducts only during positive half-cycles. Efficiency = 40.6%. Ripple factor = 1.21.

Full-Wave Rectifier: Uses two diodes (center-tapped) or a bridge rectifier (4 diodes). Conducts during both half-cycles. Efficiency = 81.2%. Ripple factor = 0.48.

Table 4.4: Comparison of Half-Wave and Full-Wave Rectifiers

Transistors

A bipolar junction transistor (BJT) is a three-terminal semiconductor device consisting of two p-n junctions. There are two types: n-p-n and p-n-p. The three regions are: emitter (E — heavily doped), base (B — thin and lightly doped), and collector (C — moderately doped). In an n-p-n transistor, two n-type regions sandwich a thin p-type base. In a p-n-p transistor, two p-type regions sandwich a thin n-type base.

The transistor can be connected in three configurations, each with distinct characteristics:

Table 4.5: Comparison of Transistor Configurations

Common Emitter Amplifier

The CE configuration is the most widely used amplifier circuit. It provides both high current gain and high voltage gain, making it suitable for most amplification applications.

Current gain (β): β = I_C/I_B (typically 20-200). Also known as the common emitter DC current gain.

Input resistance (r_i): r_i = ΔV_BE/ΔI_B (low, typically ~1 kΩ). The low input resistance means the CE amplifier draws significant current from the source.

Output resistance (r_o): r_o = ΔV_CE/ΔI_C (high, typically ~40 kΩ). The high output resistance allows the amplifier to drive moderate loads.

Voltage gain (A_V): A_V = V_out/V_in = -β × R_C/r_i. The negative sign indicates a 180° phase shift between input and output.

Power gain (A_P): A_P = A_V × β. The CE amplifier provides the highest power gain among all three configurations.

Operating Point (Q-point): For faithful amplification, the transistor must be biased in the active region. The DC load line is drawn on the output characteristics, and the Q-point (quiescent point) is set at the middle of the load line to allow maximum undistorted output swing. Proper biasing is achieved through voltage divider bias or self-bias circuits.

Coupling and Bypass Capacitors: In a practical CE amplifier, coupling capacitors are used at the input and output to block DC while allowing AC signals to pass. A bypass capacitor across the emitter resistor increases the AC voltage gain by providing a low-impedance path for AC signals.

Input Characteristics: The input characteristics of a CE transistor show the relationship between I_B (base current) and V_BE (base-emitter voltage) for a fixed V_CE (collector-emitter voltage). The curve is similar to a forward-biased diode characteristic with a knee voltage of about 0.7 V for silicon and 0.3 V for germanium.

Output Characteristics: The output characteristics show the relationship between I_C (collector current) and V_CE for different fixed values of I_B. Three regions are clearly visible: the cut-off region (both junctions reverse biased, I_C ≈ 0), the active region (emitter junction forward biased, collector junction reverse biased, I_C = βI_B), and the saturation region (both junctions forward biased, I_C is maximum).

Logic Gates

Logic gates are the basic building blocks of digital circuits. They perform Boolean operations on one or more binary inputs and produce a single binary output (0 or 1, corresponding to LOW and HIGH voltage levels).

Table 4.6: Truth Tables of Basic Logic Gates

Universal Gates: NAND and NOR gates are called universal gates because any logic circuit can be implemented using only NAND gates or only NOR gates.

Chapter Summary

Table 5.1: Quick Revision — Key Formulae in Modern Physics

Timeline of Key Discoveries in Modern Physics

• 1895: Röntgen discovers X-rays.
• 1896: Becquerel discovers radioactivity.
• 1897: J.J. Thomson discovers the electron.
• 1900: Planck proposes quantum theory of radiation.
• 1905: Einstein explains the photoelectric effect.
• 1911: Rutherford proposes nuclear model of atom.
• 1913: Bohr proposes quantum model of hydrogen atom.
• 1924: de Broglie proposes matter waves.
• 1925: Heisenberg develops matrix mechanics.
• 1926: Schrödinger develops wave mechanics.
• 1927: Davisson-Germer confirm electron diffraction; Heisenberg uncertainty principle.
• 1932: Chadwick discovers the neutron.
• 1939: Hahn and Strassmann discover nuclear fission.
• 1947: Bardeen, Brattain, and Shockley invent the transistor.

📋 Formula Sheet

▶ 1. Dual Nature of Radiation & Matter

Photoelectric effect: K_max = hf − φ = eV₀

h = 6.63×10⁻³⁴ J·s (Planck's constant)

φ = work function (minimum energy to eject electron)

V₀ = hf/e − φ/e (stopping potential vs frequency: linear, slope = h/e)

Threshold frequency: f₀ = φ/h

Einstein's equation: hf = φ + ½mv²_max

de Broglie wavelength: λ = h/p = h/mv

For electron: λ = h/√(2mE) = h/√(2meV) ≈ 12.27/√V Å (V in volts)

Davisson-Germer experiment: confirmed wave nature of electrons

Compton effect: Δλ = h(1−cosθ)/m₀c (wavelength shift in X-ray scattering)

NEET Tip: Photoelectric effect is instantaneous (10⁻⁹ s). Intensity of light affects NUMBER of photoelectrons (saturation current), not their kinetic energy. Frequency determines KE of electrons.

▶ 2. Atomic Structure

Bohr's postulates: mvr = nh/2π (angular momentum quantization)

Radius of nth orbit: r_n = n²h²ε₀/πmZe² = 0.529·n²/Z Å

Velocity in nth orbit: v_n = Ze²/2ε₀nh = (c/137)·Z/n

Energy of nth orbit: E_n = −mZ²e⁴/8ε₀²n²h² = −13.6·Z²/n² eV

For hydrogen (Z=1): E_n = −13.6/n² eV

Wavelength of emitted photon: 1/λ = R(1/n₁² − 1/n₂²)

Rydberg constant: R = me⁴/8ε₀²h³c = 1.097×10⁷ m⁻¹

Spectral series for Hydrogen:

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...	Infrared
Pfund	5	6,7,8...	Infrared

Ionization energy: E_∞ − E₁ = 13.6Z² eV (energy to remove electron)

NEET Tip: Balmer series (visible) is most commonly tested. First line of Balmer = H_α (n=3→2, 656.3 nm). Shortest wavelength of a series corresponds to n₂ = ∞.

▶ 3. Nuclear Physics

Nuclear composition: A = Z + N (mass number = protons + neutrons)

Nuclear radius: R = R₀A^⅓, R₀ = 1.2×10⁻¹⁵ m

Nuclear density: ~2.3×10¹⁷ kg/m³ (constant for all nuclei)

Mass defect: Δm = [Zm_p + (A−Z)m_n] − M_nucleus

Binding energy: BE = Δm·c² (in Joules) = Δm·931.5 MeV/amu

Binding energy per nucleon: BE/A — peaks at iron (Fe-56, ~8.8 MeV)

Radioactive decay law: N = N₀e^−λt

Half-life: T_½ = ln2/λ = 0.693/λ

Mean life: τ = 1/λ = T_½/0.693

Activity: A = λN = A₀e^−λt

Types of decay:

Decay	Emitted	A change	Z change
α	²He⁴	−4	−2
β⁻	e⁻ + ν̄	0	+1
β⁺	e⁺ + ν	0	−1
γ	photon	0	0

Nuclear fission: heavy nucleus splits into lighter ones + neutrons + energy. Chain reaction.

Nuclear fusion: light nuclei combine to form heavier nucleus + energy. Requires high T (plasma).

NEET Tip: After n half-lives, fraction remaining = (½)ⁿ. After 7 half-lives, ~0.8% remains.

▶ 4. Semiconductors & Electronic Devices

Energy bands: Valence band (VB) and Conduction band (CB), gap = E_g

Intrinsic (pure Si/Ge): nₑ = n_h (equal electrons and holes)

Extrinsic: n-type (donor, pentavalent), p-type (acceptor, trivalent)

Fermi level: In n-type → near CB, In p-type → near VB

p-n junction: Depletion region forms. Built-in potential ~0.7V (Si), ~0.3V (Ge)

Forward bias: V > 0, current flows (V > threshold)

Reverse bias: V < 0, very small current (leakage)

Diode equation: I = I₀(e^eV/kT − 1)

Zener diode: operates in reverse breakdown (constant voltage regulation)

Rectifiers:

Half-wave: η = 40.6%, ripple factor = 1.21

Full-wave: η = 81.2%, ripple factor = 0.48

Logic Gates:

AND: Y = A·B, OR: Y = A+B, NOT: Y = Ā

NAND: Y = A·B (universal gate), NOR: Y = A+B (universal gate)

XOR: Y = A⊕B = ĀB + AB̄

Transistor (BJT):

Common emitter: β = I_C/I_B (current gain), α = I_C/I_E

Relation: β = α/(1−α), I_E = I_B + I_C

Transistor as amplifier (CE): A_V = βR_C/r_be

Transistor as switch: Cutoff (off), Saturation (on)

NEET Tip: NAND and NOR are universal gates — any logic gate can be made using only NAND or only NOR gates.

⚠️ NEET Exam Tips & Common Mistakes

▶ Frequently Made Mistakes

❌ Photoelectric effect: Intensity vs Frequency: Intensity → number of electrons (current). Frequency → kinetic energy (stopping potential). This is the #1 NEET question on this topic.

❌ Stopping potential graph: V₀ vs f is a straight line. Slope = h/e (universal, same for all metals). Intercept = −φ/e (depends on metal work function).

❌ Bohr model limitations: Works only for single-electron atoms/ions (H, He⁺, Li²⁺). Fails for multi-electron atoms.

❌ Orbit vs Orbital: Bohr orbits are fixed circular paths (old model). Quantum orbitals are probability distributions — don't confuse.

❌ Nuclear binding energy: Higher BE/nucleon = more stable nucleus. Iron (Fe-56) is most stable. Fusion of lighter elements and fission of heavier elements both release energy.

❌ Half-life confusion: After 2 half-lives, fraction = ¼ (not 0). After 3, fraction = ⅛. Never zero — decay is exponential, asymptotic to zero.

❌ pn junction forward bias: Current flows only when applied voltage > knee voltage (0.7V for Si). Not immediate at V = 0+.

❌ Logic gate universality: NAND + NAND can make AND, OR, NOT. Remember: NAND is NOT-AND.

▶ Shortcut Formulas for NEET

⚡ de Broglie λ for electron: λ = 12.27/√V Å (V = accelerating potential in volts)

⚡ For hydrogen, energy of nth orbit = −13.6/n² eV. Radius = 0.529·n² Å.

⚡ Speed of electron in nth Bohr orbit: v_n = c/137 · Z/n (α = 1/137 = fine structure constant)

⚡ Wavelength of emitted photon: 1/λ = R(1/n₁² − 1/n₂²). Longest λ in a series = n₁→n₁+1 transition.

⚡ Number of spectral lines from n₂ to n₁: N = (n₂−n₁)(n₂−n₁+1)/2

⚡ Activity after n half-lives: A = A₀/2ⁿ. Mass remaining = M₀/2ⁿ.

⚡ 1 amu = 931.5 MeV/c². To convert mass defect to energy: E(MeV) = Δm(amu) × 931.5

⚡ For fastest NEET: memorize that mass of proton ≈ mass of neutron ≈ 1.67×10⁻²⁷ kg ≈ 1 amu

Practice Questions