Nuclear density (\(\rho\)) is defined as the ratio of nuclear mass to nuclear volume.


Nuclear Mass \(M \propto A\) (Mass Number).


Nuclear Volume \(V = \frac{4}{3}\pi R^3 = \frac{4}{3}\pi (R_0 A^{1/3})^3 \propto A\).


Therefore, \(\rho = \frac{M}{V} \propto \frac{A}{A} = constant\).


It is approximately \(2.3 \times 10^{17} \, kg/m^3\) for all nuclei.
Quick Tip: This incredibly high density implies that the nucleus is extremely compact.

A transistor has three regions: Emitter, Base, and Collector.


Doping levels are: Emitter \(>\) Collector \(>\) Base.


The Emitter is heavily doped to supply carriers.


The Base is lightly doped (minimum impurity) and very thin to reduce recombination.


The Collector is moderately doped.
Quick Tip: Order of Doping: Emitter \(>\) Collector \(>\) Base. Order of Size: Collector \(>\) Emitter \(>\) Base.

Coulomb's law states that the force between two point charges is directly proportional to the product of charges and inversely proportional to the square of the distance between them.


It is strictly applicable only to stationary point charges.


For distributed or dispersed charges, integration methods (calculus) based on Coulomb's law are used, but the law itself is defined for points.
Quick Tip: If spheres are charged, they behave as point charges concentrated at their centers only if they are uniformly charged and seen from outside.

Electric Flux \(\phi_E = \vec{E} \cdot \vec{A}\).


The unit of Electric Field \(E\) is Volt/meter (\(V/m\)) or Newton/Coulomb (\(N/C\)).


The unit of Area \(A\) is square meter (\(m^2\)).


So, unit of \(\phi_E = (V/m) \times m^2 = Vm\) (Volt-meter).


Alternative unit: \((N/C) \times m^2 = N m^2 C^{-1}\).
Quick Tip: Do not confuse with Magnetic Flux, which is measured in Weber (Wb) or Tm\(^2\).

(Note: While maximum current strictly flows when \(R=0\), this question refers to the Maximum Power Transfer Theorem, which is a standard board exam question format often phrased this way).


Current in the circuit is \(I = \frac{E{R+r}\).


Power delivered to the external resistance is \(P = I^2 R = \left(\frac{E}{R+r}\right)^2 R\).


For maximum power transfer, the external resistance must equal the internal resistance (\(R = r\)).


Given the options, \(R=r\) is the intended answer corresponding to the critical condition for maximum output (power).
Quick Tip: Maximum Power Transfer Theorem: Output power is maximum when Load Resistance = Internal Resistance.

The quantity \(L/R\) is known as the Time Constant (\(\tau\)) of an RL circuit.


It represents the time taken for the current to reach approximately 63% of its maximum value.


Therefore, its unit is the unit of time, which is the Second.


Dimensional verification: \(L = [ML^2T^{-2}A^{-2}]\), \(R = [ML^2T^{-3}A^{-2}]\). \(L/R = [T]\).
Quick Tip: Similarly, for an RC circuit, the time constant is \(RC\), which also has the unit of seconds.

In metallic conductors, the atoms form a lattice where the nuclei and inner electrons are fixed.


The valence electrons are loosely bound and become "free electrons" that can move throughout the lattice.


These free electrons are the charge carriers responsible for electric current in metals.


Protons are bound in the nucleus and do not move.
Quick Tip: In electrolytes, both positive and negative ions act as charge carriers.

The energy stored per unit volume in a magnetic field (Magnetic Energy Density) is given by the formula \(u_B = \frac{B^2}{2\mu_0}\).


Therefore, the quantity \(\frac{B_0^2}{\mu_0}\) has the same dimensions as Energy Density.


Energy Density = \(\frac{Energy}{Volume} = \frac{ML^2T^{-2}}{L^3} = [ML^{-1}T^{-2}]\).
Quick Tip: This is analogous to Electric Energy Density: \(u_E = \frac{1}{2}\epsilon_0 E^2\).

Ferromagnetic materials are those which are strongly attracted by magnetic fields.


The most common examples of ferromagnetic materials are Iron (Fe), Cobalt (Co), and Nickel (Ni).


Therefore, Nickel is ferromagnetic.
Quick Tip: Above the Curie temperature, ferromagnetic substances become paramagnetic.

A half-wave rectifier passes only one half-cycle of the AC input.


This produces one pulse of output for every one full cycle of input.


Therefore, the period of the output wave is the same as the input wave.


Consequently, the output frequency is equal to the input frequency (50 Hz).
Quick Tip: For a Full-Wave Rectifier, the output frequency is \textbf{double} the input frequency (100 Hz in this case).

Using Einstein's mass-energy equivalence relation: \(E = mc^2\).


Given Mass \(m = 1 mg = 1 \times 10^{-3} g = 1 \times 10^{-6} kg\).


Speed of light \(c = 3 \times 10^8 m/s\).

\(E = (10^{-6}) \times (3 \times 10^8)^2\).

\(E = 10^{-6} \times 9 \times 10^{16} = 9 \times 10^{10} J\).
Quick Tip: Always convert mass to kg (SI unit) before calculating energy in Joules.

The de Broglie wavelength of an electron accelerated through potential \(V\) is given by \(\lambda = \frac{12.27}{\sqrt{V}} \AA\).


So, \(\lambda \propto \frac{1}{\sqrt{V}}\).

\(\frac{\lambda_1}{\lambda_2} = \sqrt{\frac{V_2}{V_1}}\).


Given \(V_1 = 49\) V and \(V_2 = 64\) V.

\(\frac{\lambda_1}{\lambda_2} = \sqrt{\frac{64}{49}} = \frac{8}{7}\).
Quick Tip: Remember the inverse square root relationship: Higher voltage \(\rightarrow\) Higher momentum \(\rightarrow\) Shorter wavelength.

When a ray falls normally (perpendicularly) on a mirror, it travels along the normal.


The angle of incidence (\(i\)) is the angle between the incident ray and the normal. Here, \(i = 0^\circ\).


According to the law of reflection, Angle of Incidence (\(i\)) = Angle of Reflection (\(r\)).


Therefore, \(r = 0^\circ\). The ray retraces its path.
Quick Tip: Don't confuse angle with the surface (\(90^\circ\)) with angle of incidence (\(0^\circ\)).

The relationship between Phase Difference (\(\phi\)) and Path Difference (\(\Delta x\)) is:

\(\phi = \frac{2\pi}{\lambda} \times \Delta x\)


Rearranging for Path Difference (\(\Delta x\)):

\(\Delta x = \frac{\lambda}{2\pi} \times \phi\)


The question asks for the expression equal to the path difference (denoted confusingly as \(\lambda\) in Hindi text or just asking for relation). Option (C) matches this form.
Quick Tip: A path difference of one full wavelength (\(\lambda\)) corresponds to a phase difference of \(2\pi\) radians.

James Clerk Maxwell unified electricity and magnetism into a single theory.


He formulated a set of equations (Maxwell's Equations) which mathematically predicted the existence of electromagnetic waves.


Heinrich Hertz later proved their existence experimentally.
Quick Tip: Maxwell showed that light is an electromagnetic wave.

The resistance (\(R\)) of a conductor is given by \(R = \rho \frac{L}{A}\).


It depends on:

1. Length (\(L\)).

2. Cross-sectional Area (geometry).

3. Material (Resistivity \(\rho\)).

4. Temperature (affects \(\rho\)).


It is a property of the conductor itself and does not depend on the applied Voltage (\(V\)) or Current (\(I\)), although \(V=IR\) defines the relationship between them.
Quick Tip: Ohm's law states \(V/I\) is constant (\(R\)), implying \(R\) is independent of V.

When a dielectric is placed in an electric field \(E_0\), the molecules get polarized.


This polarization induces an internal electric field \(E_p\) in the opposite direction to the external field.


The net electric field inside is \(E_{net} = E_0 - E_p\).


Thus, the field decreases. The reduced field is \(E = E_0 / K\), where \(K\) is the dielectric constant.
Quick Tip: For conductors, the internal field becomes zero. For dielectrics, it decreases but remains non-zero.

The relationship between charge (\(Q\)), capacitance (\(C\)), and potential difference (\(V\)) is \(Q = CV\).


Given \(C = 1 \, \muF = 1 \times 10^{-6} \, F\).


Given \(V = 1 \, V\).

\(Q = (1 \, \muF) \times (1 \, V) = 1 \, \muC\).
Quick Tip: \(1 \, \muF = 10^{-6} \, F\). Always check prefixes.

The velocity component perpendicular to the magnetic field is \(v_{\perp} = v \sin \theta\).


This perpendicular component is responsible for the circular motion.


The magnetic force provides the centripetal force: \(q v_{\perp} B = \frac{m v_{\perp}^2}{r}\).


Radius \(r = \frac{m v_{\perp}}{qB} = \frac{mv \sin \theta}{qB}\).
Quick Tip: The parallel component \(v \cos \theta\) is responsible for the pitch of the helix.

An ammeter is connected in series in a circuit to measure current.


To ensure it does not alter the current it is measuring, its resistance must be as low as possible.


For an ideal ammeter, the resistance is defined as zero.
Quick Tip: An ideal Voltmeter (connected in parallel) has Infinite resistance.

Television remote controls transmit signals using light emitting diodes (LEDs) that emit Infrared (IR) radiation.


Infrared waves are used because they are invisible to the eye but can be easily detected by sensors on the TV.
Quick Tip: Our eyes can't see IR, but many smartphone cameras can detect the purple/white flash from a remote's IR LED.

The electric field intensity on the surface of a conducting sphere is given by \(E = \frac{\sigma}{\epsilon_0}\).


Here, \(\sigma\) is the surface charge density.


Since \(\sigma\) is given to be equal for both spheres, and \(\epsilon_0\) is a universal constant, the Electric Field \(E\) depends only on \(\sigma\).


Therefore, \(E_1 = E_2\), and the ratio is 1:1.
Quick Tip: \(E\) depends on radius only if Charge \(Q\) is constant (\(E \propto 1/R^2\)). If \(\sigma\) is constant, \(E\) is constant.

Ohm's law states that \(V \propto I\) provided all physical conditions, especially temperature, remain constant.


Resistance \(R\) changes with temperature.


If the temperature changes, the ratio \(V/I\) is not constant, and Ohm's law is not strictly applicable in its linear form.
Quick Tip: Devices that do not follow Ohm's law are called Non-ohmic devices (e.g., diodes, transistors).

Radio and Television are human-made technologies utilizing electromagnetic waves.


Battery charging is an electrochemical process.


Lightning is a massive natural electrical discharge. The rapid change in current during a lightning strike produces strong changing magnetic fields, which induce electromagnetic currents (EMPs) in nearby conductors. This is a natural manifestation of induction principles.
Quick Tip: The Earth itself acts as a giant magnet, and interactions with solar wind (Aurora) also involve natural electromagnetic phenomena.

A transformer is used to change the voltage level of AC.


A step-down transformer decreases the output voltage compared to the input voltage (\(V_{secondary} < V_{primary}\)).


Since Power (\(P=VI\)) is conserved (ideally), if Voltage decreases, Current increases.


Frequency remains unchanged.
Quick Tip: Step-Down: Voltage \(\downarrow\), Current \(\uparrow\). Step-Up: Voltage \(\uparrow\), Current \(\downarrow\).

Refractive index (\(n\)) = Speed of light in vacuum (\(c\)) / Speed of light in medium (\(v\)).

\(n = 1.33 \approx \frac{4}{3}\).

\(v = \frac{c}{n} = \frac{3 \times 10^8}{4/3}\).

\(v = 3 \times \frac{3}{4} \times 10^8 = \frac{9}{4} \times 10^8 = 2.25 \times 10^8\) m/s.
Quick Tip: Light slows down in denser media. The speed cannot exceed \(3 \times 10^8\) m/s.

Einstein's Photoelectric Equation: \(K_{max} = h\nu - \Phi\).

\(K_{max}\) depends linearly on frequency \(\nu\).


It does not depend on the intensity of light. Intensity determines the \textit{number of photoelectrons emitted, not their energy.
Quick Tip: Below the threshold frequency, no emission occurs regardless of intensity.

Nucleons are the particles that reside inside the nucleus of an atom.


The atomic nucleus consists of Protons and Neutrons.


Therefore, protons and neutrons are collectively called nucleons.
Quick Tip: Mass number (A) is the total count of nucleons.

Sir Tim Berners-Lee invented the World Wide Web (WWW) in 1989.


John Logie Baird is associated with TV, and Marconi with Radio.
Quick Tip: WWW uses HTTP protocol to link documents over the internet.

Electrical Power \(P = V \cdot I\).


Electrical Energy \(E = Power \times time = V \cdot I \cdot t\).


Option (C) has a typo (\(I \cdot R^2 \cdot t\) instead of \(I^2 Rt\)) and (D) represents Power. Option (A) is the correct expression for Energy.
Quick Tip: \(E = VIt = I^2 Rt = \frac{V^2}{R}t\). All are valid formulas for energy.

Alpha particles are positively charged (\(+2e\)).


When brought closer, the electrostatic repulsive force acts between them.


Work must be done against this repulsive force to bring them closer.


This work is stored as Electrostatic Potential Energy. \(U = \frac{k q_1 q_2}{r}\). As \(r\) decreases, \(U\) increases.
Quick Tip: Potential energy increases when moving against a conservative force (repulsion).

The dielectric constant (Relative Permittivity, \(K\) or \(\epsilon_r\)) of pure water is approximately 80 (specifically 81 at \(20^\circ\)C).


This high value is due to the polar nature of water molecules.


Air/Vacuum is \(\approx 1\). Metals are \(\infty\).
Quick Tip: High dielectric constant explains why water is such a good solvent for ionic compounds (reduces force between ions).

This is the microscopic or vector form of Ohm's Law.


Current Density \(\vec{J} = \frac{I}{A}\). Electric Field \(\vec{E} = \rho \vec{J}\).


Conductivity \(\sigma = \frac{1}{\rho}\).


Therefore, \(\vec{J} = \sigma \vec{E}\).
Quick Tip: Standard Ohm's law \(V=IR\) is the macroscopic form derived from this relation.

Mnemonic: B B R O Y Great Britain Very Good Wife.

Black - 0

Brown - 1

Red - 2

Orange - 3

Yellow - 4

Green - 5
Quick Tip: Remember the sequence: 0 to 9.

Absolute permeability (\(\mu\)) of a medium is defined as \(\mu = \mu_r \mu_0\).


Where \(\mu_0\) is permeability of free space and \(\mu_r\) is relative permeability.


Therefore, \(\mu_r = \frac{\mu}{\mu_0}\).
Quick Tip: \(\mu_r\) is a dimensionless quantity. For vacuum, \(\mu_r = 1\).

Cutting a magnet transverse to its length (perpendicular to the axis) results in two smaller magnets.


The pole strength (\(m\)) remains the same.


The length (\(2l\)) is halved to (\(l\)).


Original Magnetic Moment \(M = m \times 2l\).


New Magnetic Moment \(M' = m \times l = \frac{M}{2}\).


Thus, it behaves as a new magnet with reduced (half) magnetic moment.
Quick Tip: Cutting along the axis (longitudinally) halves the pole strength, also resulting in half the magnetic moment.

In a pure capacitive AC circuit, the current (\(I\)) leads the voltage (\(V\)) by a phase angle of \(90^\circ\) (\(\pi/2\)).


"Leading" means the current is "forward" relative to the voltage.


(Conversely, in an inductive circuit, current lags or is backward).
Quick Tip: Mnemonic: ICE (Current I leads in Capacitance C, EMF E leads in Inductance L - ELI).

Capacitive Reactance \(X_C = \frac{1}{2\pi f C}\).


For Direct Current (DC), frequency \(f = 0\).


Therefore, \(X_C = \frac{1}{0} = \infty\).


Infinite resistance implies it acts as a perfect insulator, blocking steady DC current (after charging).
Quick Tip: Capacitors "block DC and pass AC".

Displacement current is a quantity appearing in Maxwell's equations defined in terms of the rate of change of electric flux.


Physically, it has the same dimension and unit as conduction current.


Its unit is Ampere (A).
Quick Tip: Displacement current makes the path of current continuous across capacitor plates where conduction current is zero.

The light emitted by the sun's core (photosphere) is a continuous spectrum.


However, as it passes through the cooler outer atmosphere of the sun (chromosphere), certain wavelengths are absorbed by the elements present there.


This results in dark lines appearing on the continuous background, known as Fraunhofer lines.


Therefore, it is an absorption spectrum, best described here as a "spectrum of black lines".
Quick Tip: These lines help identify elements present in the sun's atmosphere (e.g., Helium was discovered this way).

A convex mirror curves outward.


This curvature diverges the light rays, allowing it to capture images from a much wider angle compared to plane or concave mirrors.


This wide field of view is why convex mirrors are used as rear-view mirrors in vehicles.
Quick Tip: Images in a convex mirror are always virtual, erect, and diminished.

According to Cauchy's relation, refractive index \(\mu \propto \frac{1}{\lambda^2}\).


Wavelength of red light (\(\lambda_R\)) is greater than blue light (\(\lambda_B\)).


Therefore, refractive index for red (\(\mu_R\)) is less than that for blue (\(\mu_B\)).


Ratio \(\frac{\mu_R}{\mu_B} < 1\).
Quick Tip: Violet/Blue bends more than Red in a prism because \(\mu_V > \mu_R\).

According to de Broglie's hypothesis and Planck's theory, the momentum \(p\) of a photon or particle is related to its wavelength \(\lambda\).

\(p = \frac{h}{\lambda}\)


Where \(h\) is Planck's constant.
Quick Tip: Energy \(E = h\nu = \frac{hc}{\lambda} = pc\).

The radius of the \(n^{th}\) Bohr orbit is given by \(r_n \propto n^2\).

\(r_n = n^2 r_0\) (where \(r_0\) is the radius of the first orbit, \(n=1\)).


For the second orbit, \(n=2\).

\(r_2 = 2^2 \times r_0 = 4r_0\).
Quick Tip: Radii are in ratio 1 : 4 : 9 : 16...

Magnetic Flux \(\phi = B \cdot A \cos \theta\).


If the Magnetic Field \(B\) is constant and the Area \(A\) increases, the product \(BA\) increases.


Therefore, the flux increases.
Quick Tip: Change in flux induces EMF.

AC voltages and currents specified in domestic and industrial contexts are always Root Mean Square (RMS) values unless stated otherwise.


220 V is \(V_{rms}\).


Peak voltage \(V_0 = \sqrt{2} \times 220 \approx 311\) V.


(Note: Option C and D are identical in the text provided, both correct).
Quick Tip: RMS value is the DC equivalent that produces the same heating effect.

A point source emits waves in all directions uniformly in 3D space.


The locus of points having the same phase at a finite distance \(r\) is a sphere.


Therefore, the wavefront is spherical.
Quick Tip: As the distance becomes infinite (\(r \to \infty\)), a small portion of the spherical wavefront appears as a Plane Wavefront.

Destructive interference (dark fringe) occurs when the two waves meet out of phase.


This happens when the path difference is an odd integral multiple of half wavelength.

\(\Delta x = (2n+1) \frac{\lambda}{2}\) (where \(n = 0, 1, 2...\)).


Constructive interference occurs at \(n\lambda\).
Quick Tip: Odd multiples of \(\lambda/2\) lead to cancellation (Destructive). Integer multiples of \(\lambda\) lead to addition (Constructive).

The relation between Electric Field and Potential is \(E = -\frac{dV}{dr}\).


If \(E = 0\), then \(\frac{dV}{dr} = 0\).


The derivative of a quantity is zero only if the quantity is constant.


Therefore, Potential \(V\) is constant.
Quick Tip: This is why the entire volume of a conductor is at the same potential (Equipotential), as the internal field is zero.

In parallel combination, the reciprocal of equivalent resistance is the sum of reciprocals of individual resistances: \(\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ...\)


The equivalent resistance \(R_{eq}\) is always less than the smallest individual resistance in the combination.


Therefore, equivalent resistance decreases compared to individual resistors.
Quick Tip: Series increases resistance (Add); Parallel decreases resistance.

Inside a long, tightly wound current-carrying solenoid, the magnetic field is uniform.


Uniform magnetic field lines are represented by parallel and equidistant straight lines.


They run along the axis of the solenoid.
Quick Tip: Outside an ideal solenoid, the field is approximately zero.

Two sources are said to be coherent if they emit waves of the same frequency and have a constant phase difference (independent of time).


While same frequency is a prerequisite, the defining characteristic is the constant phase relationship.


Option (A) is the standard definition found in most physics texts for multiple choice questions. (Note: Option D is also technically true as constant phase difference implies same frequency, but A is the direct definition).
Quick Tip: Independent light sources (two bulbs) can never be coherent.

Light emitted by thermal sources like the Sun, flames, or bulbs consists of waves vibrating in all possible planes perpendicular to the direction of propagation.


Such light is called unpolarised light.
Quick Tip: Unpolarised light can be polarised using polaroids or by reflection at Brewster's angle.

Radioactivity was accidentally discovered by the French physicist Henri Becquerel in 1896 while working with uranium salts.


Marie and Pierre Curie later coined the term "radioactivity" and discovered Polonium and Radium.
Quick Tip: Becquerel shared the 1903 Nobel Prize with the Curies.

In broadcast mode, a single transmitter sends signals to a large number of receivers simultaneously.


Radio and Television are classic examples of broadcasting.


Telephone is a point-to-point communication mode (one user to another).
Quick Tip: Broadcast is one-to-many; Point-to-point is one-to-one.

A Zener diode is a special purpose p-n junction diode designed to operate in the reverse breakdown region.


In this region, the voltage across the diode remains nearly constant even for large changes in current.


This property is utilized for voltage regulation.
Quick Tip: It is connected in parallel to the load in reverse bias.

The torque \(\tau\) acting on a dipole is given by \(\tau = pE \sin \theta\).


Here, \(\theta\) is the angle between dipole moment \(\vec{p}\) and electric field \(\vec{E}\).


Torque is maximum when \(\sin \theta\) is maximum, i.e., \(\sin \theta = 1\).


This corresponds to \(\theta = 90^\circ\).
Quick Tip: At \(0^\circ\) and \(180^\circ\), torque is zero (Equilibrium positions).

The magnetic dipole moment (\(M\)) of a current loop is given by \(M = Current \times Area\).

\(M = I \times A\).


For a circular loop of radius \(R\), Area \(A = \pi R^2\).


So, \(M = I (\pi R^2)\).
Quick Tip: Direction of M is given by Right Hand Thumb Rule.

The magnifying power (\(M\)) of a compound microscope is approximately given by \(M = \frac{L}{f_o} \frac{D}{f_e}\).


Here, \(L\) is the tube length (distance between focal points of objective and eyepiece).


Since \(M \propto L\), increasing the tube length increases the magnifying power.
Quick Tip: Usually, tube length is fixed in lab microscopes, but theoretical relation shows proportionality.

J.J. Thomson experiments showed that cathode rays consist of negatively charged particles.


These particles were later named electrons.


So, cathode rays are streams of high-speed electrons.
Quick Tip: Canal rays (Anode rays) are streams of positive ions.

When a soap bubble is given a charge (positive or negative), the like charges on the surface repel each other.


This electrostatic repulsion creates an outward pressure (mechanical stress).


This outward pressure acts against surface tension and external pressure, causing the bubble to expand in radius.
Quick Tip: \(P_{excess} = \frac{4T}{R} - \frac{\sigma^2}{2\epsilon_0}\). Expansion continues until equilibrium.

The capacitance of a parallel plate capacitor is \(C = \frac{\epsilon_0 A}{d}\).


If the area \(A\) is halved (\(A' = A/2\)), keeping \(d\) constant:

\(C' = \frac{\epsilon_0 (A/2)}{d} = \frac{1}{2} \left( \frac{\epsilon_0 A}{d} \right) = \frac{C}{2}\).


Therefore, capacitance becomes half.
Quick Tip: \(C \propto A\) and \(C \propto 1/d\).

A choke coil is an inductor with high inductance and negligible resistance.


It is used to control current in AC circuits.


It works on the principle of Self Induction, generating a back EMF that opposes the flow of alternating current, thereby limiting it without significant power loss.


(Note: The current flowing is often called wattless current, but the working \textit{principle is self-induction).
Quick Tip: Choke coils are used in tube lights to reduce voltage without heat loss (unlike a resistor).

Hypermetropia (Far-sightedness) is a defect where a person can see distant objects clearly but near objects appear blurry because the image is formed behind the retina.


To correct this, a converging lens is required to converge the rays sooner.


A convex lens is a converging lens.
Quick Tip: Myopia (Near-sightedness) uses Concave lens.

For a point source, light spreads out in spherical wavefronts.


The surface area of a sphere is \(4\pi r^2\).


Since energy is conserved, Intensity \(I = \frac{Power}{Area} = \frac{P}{4\pi r^2}\).


Therefore, \(I \propto \frac{1}{r^2}\).
Quick Tip: For a linear source (cylindrical wavefront), \(I \propto 1/r\). For a plane source, \(I\) is constant.

In Bohr's model:

Potential Energy \(U = -\frac{Ze^2}{r}\).

Kinetic Energy \(K = \frac{Ze^2}{2r}\).


Comparing the two: \(U = -2K\) or \(K = -\frac{U}{2}\).


Total Energy \(E = K + U = -K = U/2\).
Quick Tip: Kinetic energy is always positive; Potential and Total energies are negative for bound states.

When a strong current flows through a semiconductor, it causes significant heating.


This thermal energy breaks covalent bonds, generating a massive number of electron-hole pairs (avalanche effect).


The sudden surge in charge carriers ("excess of electrons") leads to a breakdown of the crystal structure and permanent damage.
Quick Tip: This is why appropriate resistors are used to limit current in diode/transistor circuits.

Germanium is a Group 14 element (tetravalent).


To create a p-type (positive type) semiconductor, we need to create holes (electron vacancies).


This is done by doping with a Trivalent impurity (Group 13 elements like Boron, Aluminium, Gallium, Indium).


Trivalent elements have a valency of 3.
Quick Tip: For n-type, Pentavalent (valency 5) impurities are used.

A repeater acts to extend the range of a communication system.


It picks up the signal from the transmitter (acts as a Receiver), amplifies and reconditions it, and then re-transmits it to the receiver (acts as a Transmitter).


Therefore, it is a combination of both.
Quick Tip: Repeaters are essential in optical fiber and mobile networks to overcome signal attenuation.

Let the initial frequency be \(\nu\) and max velocity be \(v\).
\(K_1 = \frac{1}{2}mv^2 = h\nu - \Phi\) \quad ...(1)


In the second case, frequency is \(2\nu\) and max velocity is \(2v\).
\(K_2 = \frac{1}{2}m(2v)^2 = 4 (\frac{1}{2}mv^2) = 4K_1\).
\(4K_1 = h(2\nu) - \Phi\) \quad ...(2)


Substitute (1) into (2):
\(4(h\nu - \Phi) = 2h\nu - \Phi\)
\(4h\nu - 4\Phi = 2h\nu - \Phi\)
\(2h\nu = 3\Phi\)
\(\Phi = \frac{2h\nu}{3}\)
Quick Tip: Work function (\(\Phi\)) is a property of the material and does not change with incident light.