Definitions & Solutions

Electric Charge: It is an intrinsic property of elementary particles of matter which gives rise to electric force between various objects.

Quantization of Charge: The total charge on a body is always an integral multiple of a basic quantum of charge (\(e\)).
Formula: \( q = \pm ne \) (where \(n = 1, 2, 3...\) and \(e = 1.6 \times 10^{-19} C\)).

Conservation of Charge: For an isolated system, the total charge remains constant. Charge can neither be created nor destroyed, only transferred from one body to another.

Coulomb's Law: The force of attraction or repulsion between two stationary point charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them.

Formula: \( F = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r^2} \)
Vector Form: \( \vec{F}_{12} = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r^2} \hat{r}_{21} \)

Relative Permittivity (\(\epsilon_r\) or \(K\)): It is the ratio of the permittivity of the medium (\(\epsilon\)) to the permittivity of free space (\(\epsilon_0\)). \( \epsilon_r = \frac{\epsilon}{\epsilon_0} = \frac{F_{vacuum}}{F_{medium}} \).

Electric Field Intensity: It is defined as the electrostatic force experienced per unit positive test charge placed at that point.

Formula: \( \vec{E} = \lim_{q_0 \to 0} \frac{\vec{F}}{q_0} \)

SI Unit: Newton/Coulomb (N/C) or Volt/meter (V/m).

Dimensional Formula: \( [M^1 L^1 T^{-3} A^{-1}] \)

Electric Dipole: A system of two equal and opposite point charges separated by a small distance.

Electric Dipole Moment (\(\vec{p}\)): It is the product of the magnitude of either charge and the vector distance between them.

Formula: \( \vec{p} = q \times 2\vec{a} \)

SI Unit: Coulomb-meter (C·m).

Direction: By convention, it is directed from the negative charge to the positive charge.

Electric Flux (\(\Phi_E\)): It is the total number of electric field lines passing normally through a given surface area.

Formula: \( \Phi_E = \oint \vec{E} \cdot d\vec{S} = E S \cos\theta \)
SI Unit: \( N \cdot m^2 / C \) or \( V \cdot m \).

Gauss's Law: The total electric flux through any closed surface in free space is equal to \(1/\epsilon_0\) times the net charge enclosed by the surface. \( \Phi_E = \frac{q_{enclosed}}{\epsilon_0} \)

• Linear Charge Density (\(\lambda\)): Charge per unit length. \( \lambda = \frac{q}{L} \). SI Unit: \( C/m \). Dimension: \( [L^{-1}TA] \).
• Surface Charge Density (\(\sigma\)): Charge per unit area. \( \sigma = \frac{q}{A} \). SI Unit: \( C/m^2 \). Dimension: \( [L^{-2}TA] \).
• Volume Charge Density (\(\rho\)): Charge per unit volume. \( \rho = \frac{q}{V} \). SI Unit: \( C/m^3 \). Dimension: \( [L^{-3}TA] \).

Electrostatic Potential (\(V\)): The amount of work done in bringing a unit positive test charge from infinity to a point in the electric field, against the electrostatic force.

Potential Difference (\(\Delta V\)): The work done in moving a unit positive charge from one point to another in an electric field. \( \Delta V = V_B - V_A = \frac{W_{AB}}{q_0} \).

Formula: \( V = \frac{W}{q_0} \) | SI Unit: Volt (V) or Joule/Coulomb (J/C).

Dimension: \( [M^1 L^2 T^{-3} A^{-1}] \). It is a Scalar quantity.

Equipotential Surface: Any surface that has the same electric potential at every point on it.

Properties:

• No work is done in moving a test charge from one point to another on an equipotential surface (\( W = q\Delta V = 0 \)).
• The electric field is always perpendicular to the equipotential surface at every point.
• Two equipotential surfaces can never intersect each other.

Electrostatic Potential Energy (\(U\)): It is the total work done by an external agent in assembling a configuration of charges by bringing them from infinity to their present locations.

Formula for two point charges: \( U = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r} \) (where \(r\) is the distance between \(q_1\) and \(q_2\)).

Capacitance (\(C\)): The ability of a conductor to store electric charge and electric energy. It is the ratio of charge given to the conductor to the rise in its potential.

Formula: \( C = \frac{Q}{V} \) | SI Unit: Farad (F). | Dimension: \( [M^{-1} L^{-2} T^4 A^2] \).

Factors affecting Parallel Plate Capacitor (\(C = \frac{\epsilon_0 A}{d}\)):

• Area of the plates (\(A\)) [Directly proportional]
• Distance between plates (\(d\)) [Inversely proportional]
• Nature of the dielectric medium between the plates.

Dielectric Strength: The maximum electric field that a dielectric medium can withstand without breaking down (losing its insulating property).

Polarization Vector (\(\vec{P}\)): It is defined as the induced dipole moment per unit volume of the dielectric material when placed in an external electric field.

Energy Stored (\(U\)): The work done in charging a capacitor is stored as electrostatic potential energy.
\( U = \frac{1}{2} C V^2 = \frac{Q^2}{2C} = \frac{1}{2} Q V \)

Energy Density (\(u\)): The energy stored per unit volume in the electric field between the plates.
\( u = \frac{1}{2} \epsilon_0 E^2 \)

Electric Current (\(I\)): The rate of flow of electric charge through any cross-section of a conductor. \( I = \frac{dq}{dt} \). SI Unit: Ampere (A).

Drift Velocity (\(v_d\)): The average velocity with which free electrons get drifted towards the positive end of a conductor under the influence of an external electric field.
Formula: \( v_d = -\frac{eE}{m}\tau \) (where \(\tau\) is relaxation time). SI Unit: \(m/s\).

Mobility (\(\mu\)): The magnitude of drift velocity acquired per unit applied electric field.
Formula: \( \mu = \frac{|v_d|}{E} \). SI Unit: \( m^2 V^{-1} s^{-1} \).

Ohm's Law: At constant physical conditions (like temperature, pressure), the current flowing through a conductor is directly proportional to the potential difference applied across its ends. \( V \propto I \) or \( V = IR \).

Electrical Resistance (\(R\)): The opposition offered by a conductor to the flow of electric current through it. \( R = \frac{V}{I} \).

SI Unit: Ohm (\(\Omega\)). | Dimension: \( [M^1 L^2 T^{-3} A^{-2}] \).

Resistivity (\(\rho\)): It is equal to the resistance of a conductor of unit length and unit cross-sectional area. It depends only on the material and temperature.
Formula: \( \rho = \frac{RA}{l} = \frac{m}{ne^2\tau} \). | SI Unit: \( \Omega \cdot m \). | Dimension: \( [M^1 L^3 T^{-3} A^{-2}] \).

Conductivity (\(\sigma\)): The reciprocal of resistivity is called conductivity.
Formula: \( \sigma = \frac{1}{\rho} \). | SI Unit: \( \Omega^{-1} m^{-1} \) or \( S/m \). | Dimension: \( [M^{-1} L^{-3} T^3 A^2] \).

Current Density (\(\vec{J}\)): It is the amount of current flowing per unit area placed normal to the direction of current flow. It is a vector quantity.
\( \vec{J} = \frac{I}{A} \) (SI Unit: \(A/m^2\)).

Vector form of Ohm's Law: \( \vec{J} = \sigma \vec{E} \) (where \(\sigma\) is conductivity and \(\vec{E}\) is electric field).

Electromotive Force (EMF, \(E\)): The maximum potential difference between the electrodes of a cell when no current is drawn from it (open circuit).

Internal Resistance (\(r\)): The resistance offered by the electrolyte and electrodes of a cell to the flow of current within the cell.

Relation: Terminal voltage (\(V\)) is the potential difference when current \(I\) is drawn. It is less than EMF due to the voltage drop across internal resistance: \( V = E - Ir \).

Kirchhoff's Current Law (Junction Rule): The algebraic sum of currents meeting at any junction in a circuit is zero (\(\sum I = 0\)). It is based on the Law of Conservation of Charge.

Kirchhoff's Voltage Law (Loop Rule): The algebraic sum of changes in potential around any closed loop in an electrical circuit is zero (\(\sum \Delta V = 0\)). It is based on the Law of Conservation of Energy.

Wheatstone Bridge: An arrangement of four resistances (\(P, Q, R, S\)) used to measure an unknown resistance accurately.

Balanced Condition: The bridge is balanced when no current flows through the galvanometer. At this state, the ratio of arms is equal: \( \frac{P}{Q} = \frac{R}{S} \).

Electric Power (\(P\)): The rate at which electrical energy is consumed or dissipated in an electrical circuit.
Formula: \( P = \frac{W}{t} = VI = I^2R = \frac{V^2}{R} \) (SI Unit: Watt).

Electrical Energy (\(W\)): The total work done by the source in maintaining the current in the circuit for a given time.
Formula: \( W = P \times t = VIt = I^2Rt \) (SI Unit: Joule).

Magnetic Field (\(B\)): The space around a magnet or a current-carrying conductor in which its magnetic influence can be experienced.

SI Unit: Tesla (T) or Weber/m². | Dimension: \( [M^1 L^0 T^{-2} A^{-1}] \).

Biot-Savart Law: It states that the magnetic field (\(dB\)) due to a small current element (\(Idl\)) at a distance \(r\) is directly proportional to the current, the length of the element, the sine of the angle between the element and the position vector, and inversely proportional to the square of the distance.

Formula: \( dB = \frac{\mu_0}{4\pi} \frac{I dl \sin\theta}{r^2} \)

Ampere's Circuital Law: It states that the line integral of the magnetic field (\(\vec{B}\)) around any closed path (loop) in free space is equal to absolute permeability (\(\mu_0\)) times the total net current (\(I\)) threading through the area bounded by the closed path.

Mathematical Expression: \( \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enclosed} \)

Lorentz Force: The total force experienced by a charged particle moving in a region where both electric and magnetic fields are present simultaneously.

Formula: \( \vec{F}_{total} = \vec{F}_e + \vec{F}_m = q\vec{E} + q(\vec{v} \times \vec{B}) = q[\vec{E} + (\vec{v} \times \vec{B})] \)

Magnetic Dipole Moment (\(\vec{M}\)): For a current-carrying loop, it is defined as the product of the current flowing through it and the area of the loop. It acts as a magnetic dipole.

Formula: \( \vec{M} = N I \vec{A} \) (where \(N\) is the number of turns).

SI Unit: Ampere-meter² (\(A \cdot m^2\)). | Dimension: \( [M^0 L^2 T^0 A^1] \).

Current Sensitivity (\(I_s\)): The deflection produced in the galvanometer per unit current flowing through it.
Formula: \( I_s = \frac{\theta}{I} = \frac{NAB}{k} \) (where \(k\) is torsional constant).

Voltage Sensitivity (\(V_s\)): The deflection produced in the galvanometer per unit voltage applied across its ends.
Formula: \( V_s = \frac{\theta}{V} = \frac{NAB}{kR} \) (where \(R\) is the resistance of the galvanometer).

Magnetic Field Lines: Imaginary continuous curves in a magnetic field such that the tangent at any point gives the direction of the magnetic field at that point.

Properties: 1) They form closed continuous loops (unlike electric field lines). 2) Two magnetic field lines never intersect each other.

Magnetic Flux (\(\Phi_B\)): The total number of magnetic field lines crossing a given area normally.
Formula: \( \Phi_B = \vec{B} \cdot \vec{A} = BA \cos\theta \). | SI Unit: Weber (Wb).

• Magnetic Declination (\(\theta\)): The angle between the geographic meridian and the magnetic meridian at a given place on Earth.
• Magnetic Dip or Inclination (\(\delta\)): The angle that the total magnetic field of the Earth (\(B_e\)) makes with the surface of the Earth (horizontal direction) at that place.
• Horizontal Component (\(B_H\)): The component of Earth's total magnetic field acting along the horizontal direction. \( B_H = B_e \cos\delta \).

• Magnetization (\(M\)): The net magnetic dipole moment developed per unit volume of a material. \( M = \frac{m_{net}}{V} \).
• Magnetic Intensity (\(H\)): The ability of an external magnetic field to magnetize a material. \( H = \frac{B_0}{\mu_0} \).
• Magnetic Susceptibility (\(\chi_m\)): A measure of how easily a magnetic material can be magnetized. It is the ratio of Magnetization to Magnetic Intensity. \( \chi_m = \frac{M}{H} \).
• Magnetic Permeability (\(\mu\)): The degree or extent to which magnetic lines of force can penetrate or pass through a material. \( \mu = \frac{B}{H} \).

• Diamagnetic: Weakly repelled by magnets. Susceptibility (\(\chi_m\)) is small and negative (\(-1 \le \chi_m < 0\)). Permeability (\(\mu_r\)) is slightly less than 1.
• Paramagnetic: Weakly attracted by magnets. Susceptibility (\(\chi_m\)) is small and positive (\(0 < \chi_m < \epsilon\)). Permeability (\(\mu_r\)) is slightly greater than 1.
• Ferromagnetic: Strongly attracted by magnets. Susceptibility (\(\chi_m\)) is very large and positive (\(\chi_m \gg 1\)). Permeability (\(\mu_r\)) is much greater than 1 (\(\mu_r \gg 1\)).

Electromagnetic Induction: The phenomenon of producing an induced EMF (and hence an induced current) in a closed circuit due to a change in the magnetic flux linked with it over time.

Faraday's Laws:

• First Law: Whenever the magnetic flux linked with a circuit changes, an induced EMF is generated in it.
• Second Law: The magnitude of the induced EMF is directly proportional to the rate of change of magnetic flux linked with the circuit.

Formula: \( e = -N \frac{d\Phi_B}{dt} \) (where \(N\) is the number of turns).

Lenz's Law: The polarity of the induced EMF is such that it produces a current which opposes the change in magnetic flux that produced it (this is why there is a negative sign in Faraday's formula).

Conservation of Energy: To change the magnetic flux (e.g., pushing a magnet into a coil), mechanical work must be done against the opposing force created by the induced current. This mechanical work is exactly what gets converted into electrical energy (induced current). Thus, it obeys the law of conservation of energy.

Self-Induction: The phenomenon of production of induced EMF in a coil when a varying current flows through the same coil, opposing the change in current.

Self-Inductance (\(L\)): The magnetic flux linked with a coil when a unit current flows through it (\( \Phi = LI \)), or the induced EMF when the rate of change of current is unity (\( e = -L \frac{dI}{dt} \)).

SI Unit: Henry (H). | Dimension: \( [M^1 L^2 T^{-2} A^{-2}] \).

Mutual Induction: The phenomenon of production of induced EMF in one coil (secondary) due to a varying current in a neighboring coil (primary).

Mutual Inductance (\(M\)): The magnetic flux linked with the secondary coil when a unit current flows in the primary coil (\( \Phi_s = M I_p \)), or the induced EMF in the secondary when the rate of change of current in the primary is unity (\( e_s = -M \frac{dI_p}{dt} \)).

Coefficient of Coupling (\(k\)): A measure of how well two coils are magnetically linked. \( k = \frac{M}{\sqrt{L_1 L_2}} \) (where \(L_1, L_2\) are self-inductances of the two coils).

Alternating Current (AC): An electric current whose magnitude changes continuously with time and whose direction reverses periodically.

RMS Value of AC (\(I_{rms}\)): That steady (DC) current which produces the same amount of heat in a given resistor in a given time as is produced by the AC passing through the same resistor for the same time.

Relation: \( I_{rms} = \frac{I_0}{\sqrt{2}} \approx 0.707 I_0 \) (where \(I_0\) is the Peak Value).

• Inductive Reactance (\(X_L\)): The opposition offered by an inductor to the flow of AC. \(X_L = \omega L = 2\pi\nu L\). SI Unit: Ohm (\(\Omega\)).
• Capacitive Reactance (\(X_C\)): The opposition offered by a capacitor to the flow of AC. \(X_C = \frac{1}{\omega C} = \frac{1}{2\pi\nu C}\). SI Unit: Ohm (\(\Omega\)).
• Impedance (\(Z\)): The total effective opposition offered by an LCR circuit to the flow of AC. \(Z = \sqrt{R^2 + (X_L - X_C)^2}\). SI Unit: Ohm (\(\Omega\)).

Resonance: It is the condition in a series LCR circuit when the inductive reactance becomes equal to the capacitive reactance (\(X_L = X_C\)). At this state, the impedance is minimum (\(Z = R\)) and the current in the circuit is maximum.

Resonant Frequency Formula: \( \nu_r = \frac{1}{2\pi \sqrt{LC}} \)

Power Factor (\(\cos \phi\)): It is the ratio of true power to the apparent power in an AC circuit. It is also equal to the ratio of resistance to impedance: \( \cos\phi = \frac{R}{Z} \).

Wattless Current: When the current in an AC circuit flows such that the average power consumed over a complete cycle is zero (which happens in a purely inductive or purely capacitive circuit where \(\phi = 90^\circ\)), the current is called wattless current. It is the \(I_{rms} \sin\phi\) component of the current.

Transformer: A device used to increase or decrease alternating voltage based on the principle of mutual induction.

Transformation Ratio (\(k\)): The ratio of the number of turns in the secondary coil to the number of turns in the primary coil. \( k = \frac{N_s}{N_p} = \frac{V_s}{V_p} \).

• Step-Up Transformer: Increases voltage (\(V_s > V_p\)), decreases current. \(N_s > N_p\) (so \(k > 1\)).
• Step-Down Transformer: Decreases voltage (\(V_s < V_p\)), increases current. \(N_s < N_p\) (so \(k < 1\)).

Displacement Current (\(I_d\)): The current which comes into play in the region where the electric field and the electric flux are changing with time (e.g., between the plates of a charging capacitor).
Formula: \( I_d = \epsilon_0 \frac{d\Phi_E}{dt} \).

Maxwell's Modified Ampere's Law: \( \oint \vec{B} \cdot d\vec{l} = \mu_0 (I_c + I_d) = \mu_0 \left(I_c + \epsilon_0 \frac{d\Phi_E}{dt}\right) \) (where \(I_c\) is conduction current).

Electromagnetic Waves: Waves composed of mutually perpendicular, time-varying electric and magnetic fields that oscillate perpendicular to the direction of propagation.

Characteristics: 1) They do not require a material medium to propagate. 2) They are transverse in nature. 3) They carry energy and momentum.

Speed in Vacuum: \( c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \approx 3 \times 10^8 \) m/s.

Order (Increasing Frequency): Radio waves \(\rightarrow\) Microwaves \(\rightarrow\) Infrared \(\rightarrow\) Visible Light \(\rightarrow\) Ultraviolet \(\rightarrow\) X-rays \(\rightarrow\) Gamma rays.

• Microwaves: Used in Radar systems for aircraft navigation and microwave ovens.
• Infrared: Used in remote controls for TVs/VCRs and physical therapy (heat therapy).
• X-rays: Used as a diagnostic tool in medicine (imaging bones) and for treating certain forms of cancer.

Refraction: The phenomenon of bending of light from its straight-line path when it passes obliquely from one transparent medium to another.

Absolute Refractive Index (\(n\)): The ratio of the speed of light in vacuum (\(c\)) to the speed of light in the medium (\(v\)). \( n = \frac{c}{v} \).

Snell's Law: The ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant for a given pair of media and a given color of light. \( \frac{\sin i}{\sin r} = ^1n_2 = \frac{n_2}{n_1} \).

Total Internal Reflection (TIR): When a light ray traveling from a denser to a rarer medium hits the interface at an angle of incidence greater than the critical angle, it is completely reflected back into the denser medium.

Critical Angle (\(i_c\)): The angle of incidence in the denser medium for which the angle of refraction in the rarer medium is \(90^\circ\).

Relation: \( n = \frac{1}{\sin i_c} \) (where \(n\) is the refractive index of the denser medium with respect to the rarer medium).

Power of a Lens: The measure of a lens's ability to converge or diverge the rays of light falling on it. It is the reciprocal of its focal length in meters.

Formula: \( P = \frac{1}{f (\text{in meters})} \) | SI Unit: Diopter (D).

Power of Combination: When thin lenses are placed in contact, the total power is the algebraic sum of their individual powers. \( P = P_1 + P_2 + P_3 + ... \)

Dispersion: The phenomenon of splitting of white light into its constituent seven colors when passing through a dispersive medium (like a prism).

Angle of Deviation (\(\delta\)): The angle between the incident ray extended forward and the emergent ray extended backward.

Angle of Minimum Deviation (\(\delta_m\)): The smallest angle of deviation produced by the prism. At this angle, the refracted ray inside the prism becomes parallel to its base, and angle of incidence = angle of emergence.

Magnifying Power (\(M\)): The ratio of the angle subtended by the image at the eye to the angle subtended by the object at the unaided eye.

• Simple Microscope: \( M = 1 + \frac{D}{f} \) (image at near point) or \( M = \frac{D}{f} \) (image at infinity).
• Compound Microscope: \( M = M_o \times M_e = -\frac{v_o}{u_o} \left(1 + \frac{D}{f_e}\right) \) (image at near point).
• Astronomical Telescope: \( M = -\frac{f_o}{f_e} \) (normal adjustment, image at infinity).

Wavefront: The continuous locus of all points in a medium vibrating in the same phase.

• Spherical: Produced by a point source of light at a finite distance.
• Cylindrical: Produced by a linear source of light (like a slit) at a finite distance.
• Plane: A small part of a spherical or cylindrical wavefront at an infinite distance from the source.

Huygens' Principle: Every point on a given primary wavefront acts as a fresh source of secondary wavelets, spreading out in all directions with the speed of light. The forward envelope of these wavelets forms the new wavefront.

Interference: The phenomenon of redistribution of light energy due to the superposition of light waves from two coherent sources.

• Constructive Interference: When the crest of one wave falls on the crest of another, intensity is maximum. (Path difference \(\Delta x = n\lambda\)).
• Destructive Interference: When the crest of one wave falls on the trough of another, intensity is minimum. (Path difference \(\Delta x = (2n-1)\frac{\lambda}{2}\)).

Fringe Width (\(\beta\)): The linear distance between the centers of two consecutive bright or dark fringes. \( \beta = \frac{\lambda D}{d} \).

Diffraction: The phenomenon of bending of light around the corners of tiny obstacles or apertures and spreading into their geometrical shadow regions.

Dependence: The width of the central maximum (\( \frac{2\lambda D}{a} \)) is inversely proportional to the slit width (\(a\)). As the slit is made narrower, the central maximum becomes wider.

Interference vs Diffraction: Interference is due to superposition of waves from two separate coherent sources. Diffraction is due to superposition of secondary wavelets originating from different parts of the same wavefront.

Polarization: The phenomenon of restricting the vibrations of electric field vectors of a light wave to only one single plane perpendicular to the direction of wave propagation.

Polaroid: A commercially produced optical filter or thin film that transmits only plane-polarized light and blocks light vibrating in other planes. Used in sunglasses and 3D glasses.

Work Function (\(\phi_0\)): The minimum amount of energy required by an electron to just escape from the metal surface. It depends on the nature of the metal and its surface conditions.

Unit: Commonly measured in Electron Volts (eV).

Electron Volt (eV): The kinetic energy gained by an electron when it is accelerated through a potential difference of 1 Volt.
\( 1 \text{ eV} = 1.6 \times 10^{-19} \text{ Joules} \).

Photoelectric Effect: The phenomenon of emission of electrons from a metal surface when light of a suitable frequency falls on it. The emitted electrons are called photoelectrons.

Threshold Frequency (\(\nu_0\)): The minimum frequency of incident light below which no photoelectric emission takes place, regardless of the intensity of the light.

Stopping Potential (\(V_0\)): The minimum negative (retarding) potential given to the anode with respect to the cathode at which the photoelectric current becomes zero. It depends on the frequency of incident light and the nature of the material.

Einstein's Photoelectric Equation: The maximum kinetic energy (\(K_{max}\)) of an emitted photoelectron is the difference between the energy of the incident photon (\(h\nu\)) and the work function (\(\phi_0\)) of the metal.
\( K_{max} = h\nu - \phi_0 \) or \( eV_0 = h\nu - h\nu_0 \).

Particle Nature of Light (Photon): Light radiation consists of tiny packets of energy called photons. Each photon travels at the speed of light and carries a definite energy given by \( E = h\nu \), where \(h\) is Planck's constant and \(\nu\) is the frequency.

Matter Waves: The waves associated with moving material particles (like electrons, protons, or atoms). This establishes the dual nature of matter.

de Broglie Wavelength (\(\lambda\)):
\( \lambda = \frac{h}{p} = \frac{h}{mv} \) (where \(h\) is Planck's constant, \(p\) is momentum, \(m\) is mass, and \(v\) is velocity).

Distance of Closest Approach (\(r_0\)): The minimum distance from the nucleus up to which an \(\alpha\)-particle heading directly towards the nucleus can approach before momentarily stopping and retracing its path. Here, Kinetic Energy totally converts to Electrostatic Potential Energy.

Impact Parameter (\(b\)): The perpendicular distance of the initial velocity vector of the \(\alpha\)-particle from the center of the nucleus. It determines the angle of scattering (smaller \(b\) leads to larger scattering angle).

Bohr's Postulates:

• Electrons revolve around the nucleus in specific, stable, circular orbits without radiating energy (stationary states).
• The angular momentum of an electron in a stable orbit is quantized. \( mvr = \frac{nh}{2\pi} \) (where \(n = 1, 2, 3...\)).
• Energy is radiated or absorbed only when an electron jumps from one orbit to another. \( E_2 - E_1 = h\nu \).

Radius of \(n^{th}\) Orbit (\(r_n\)): \( r_n = \frac{n^2 h^2 \epsilon_0}{\pi m e^2 Z} \) (For Hydrogen, \(r_n \propto n^2\)).

Ionization Energy: The minimum energy required to completely remove an electron from an atom (moving it from its ground state \(n=1\) to infinity). For a hydrogen atom, it is \( +13.6 \text{ eV} \).

Excitation Energy: The energy required by an electron to jump from the ground state (or a lower energy state) to any higher energy state. (e.g., Energy to jump from \(n=1\) to \(n=2\) is \(10.2 \text{ eV}\)).

Atomic Mass Unit (amu or u): Defined as exactly \(1/12^{th}\) the mass of an unbonded carbon-12 atom. \( 1 \text{ amu} \approx 1.66 \times 10^{-27} \text{ kg} \approx 931.5 \text{ MeV} \).

Mass Defect (\(\Delta m\)): The difference between the sum of the masses of individual nucleons (protons and neutrons) forming a nucleus and the actual mass of the nucleus.
\( \Delta m = [Z m_p + (A - Z)m_n] - M \).

Nuclear Binding Energy (\(E_b\)): The minimum energy required to separate the nucleons of a nucleus to an infinite distance apart. It is the energy equivalent of the mass defect.

Relation (Einstein's Mass-Energy Equivalence): \( E_b = \Delta m \cdot c^2 \).

Nuclear Force: The strong attractive force that holds protons and neutrons tightly together inside a tiny nucleus, overcoming the electrostatic repulsion between protons.

Properties: 1) It is the strongest fundamental force in nature. 2) It is a very short-range force (effective only up to \(2-3 \text{ fm}\)). 3) It is charge-independent (acts equally between p-p, n-n, and p-n).

Binding Energy per Nucleon (\(\frac{E_b}{A}\)): It is the average energy required to remove a single nucleon from the nucleus. A higher value indicates a more stable nucleus (e.g., Iron-56 has the highest value).

Radioactivity: The spontaneous emission of high-energy radiation (such as \(\alpha\), \(\beta\) particles, or \(\gamma\) rays) by an unstable, heavy nucleus in order to achieve greater stability.

Nuclear Fission: The process in which a heavy, unstable nucleus (like Uranium-235) is bombarded with slow neutrons, splitting it into two lighter, more stable nuclei of roughly equal mass, accompanied by the release of massive energy and more neutrons.

Nuclear Fusion: The process in which two very light nuclei (like Hydrogen isotopes) combine at extremely high temperatures to form a single, heavier, and more stable nucleus, releasing an enormous amount of energy (the principle behind the Sun's energy and hydrogen bombs).

• Conductors: The conduction band and valence band overlap. There is no forbidden energy gap (\(E_g = 0\)). Electrons can easily flow.
• Semiconductors: There is a small forbidden energy gap (\(E_g < 3 \text{ eV}\)) between the valence band and the empty conduction band. At room temperature, some electrons cross the gap, allowing limited conduction.
• Insulators: There is a very large forbidden energy gap (\(E_g > 3 \text{ eV}\)). Electrons cannot jump to the conduction band under normal conditions, so no current flows.

Intrinsic Semiconductors: A semiconductor in its purest form (e.g., pure Silicon or Germanium) where the number of electrons equals the number of holes (\(n_e = n_h\)).

Extrinsic Semiconductors: An impure semiconductor created by adding desirable impurity atoms to increase its electrical conductivity.

Doping: The deliberate process of adding impurities to a pure semiconductor to alter its electrical properties.

• n-type: Formed by doping with a pentavalent impurity (e.g., Phosphorus, Arsenic). Electrons are the majority charge carriers, and holes are minority carriers.
• p-type: Formed by doping with a trivalent impurity (e.g., Boron, Indium). Holes are the majority charge carriers, and electrons are minority carriers.

p-n Junction: It is a single crystal of semiconductor in which one side is doped with acceptor (p-type) impurity and the other side with donor (n-type) impurity.

• Depletion Region: A thin region created near the junction which is completely devoid of free mobile charge carriers. It contains only immobile positive and negative ions.
• Barrier Potential (\(V_B\)): The potential difference developed across the depletion region due to the accumulated immobile ions. It acts as a barrier and opposes any further diffusion of electrons and holes across the junction.

Forward Bias: When the p-side of the diode is connected to the positive terminal and the n-side is connected to the negative terminal of the battery. The depletion region narrows, the barrier potential decreases, and significant current flows.

Reverse Bias: When the p-side is connected to the negative terminal and the n-side is connected to the positive terminal. The depletion region widens, the barrier potential increases, and only a negligible leakage current flows.

Rectifier: An electrical device (using p-n junction diodes) that converts alternating current (AC) into direct current (DC).

• Half-Wave Rectifier: It uses a single diode and allows current to flow only during the positive half-cycle of the input AC. The negative half-cycle is blocked. The output frequency equals the input frequency.
• Full-Wave Rectifier: It uses two (or more) diodes and allows current to flow in the same direction during both the positive and negative half-cycles of the input AC. The output frequency is double the input frequency.