When a plano-convex lens is placed on a glass slab, an air film is formed which is wedge-shaped. The film thickness gradually changes from the point of contact (negligible thickness) outward.

Newton’s rings result from the interference of light in the wedge-shaped air gap between the lens and the glass plate.

A. Linear momentum const. → Net external force = 0 (IV),
B. Angular momentum const. → External torque = 0 (III),
C. Inertial frame → Unaccelerated body appears unaccelerated (I),
D. Non-inertial frame → Unaccelerated body appears accelerated (II).

An inertial frame is one with no net acceleration; a non-inertial frame is accelerating, leading to fictitious forces.

Maxwell's First Equation → Gauss’s Law in Electrostatics (III),
Maxwell's Second Equation → Gauss’s Law in Magnetostatics (IV),
Maxwell's Third Equation → Faraday's Laws of Induction (II),
Maxwell's Fourth Equation → Modified Ampere’s Law (I).

Different textbooks may list Maxwell's equations in alternate orders, but here the question’s official matching yields (4) A–III, B–IV, C–II, D–I.

Steam point = 100°C = 373 K.
Ice point = 0°C = 273 K.
Carnot Efficiency, η = (T_H - T_L) / T_H = (373 - 273) / 373 = 100 / 373 ≈ 0.268 ≈ 26.8%.

Always convert Celsius to Kelvin in thermodynamics. Carnot efficiency = (T_H - T_L) / T_H.

Use Maxwell's relations in thermodynamics:

A → (III), B → (IV), C → (II), D → (I).

Maxwell's relations come from the equality of mixed second partial derivatives of thermodynamic potentials.

For a thermodynamic process to be reversible, the system must be in near-equilibrium at each stage, requiring infinitesimally small differences in pressure/temperature, and hence the process proceeds very slowly without dissipative losses.

Reversible processes require no net energy losses due to friction, conduction, or radiation.

Using Wien’s displacement law: λ_max * T ≈ 2.9 × 10^-3 m·K.

Given λ_max = 446 nm = 446 × 10^-9 m, the temperature is approximately 6500 K.

Wien’s constant is 2.9 × 10^-3 m·K. Make sure to convert nm to m before calculation.

Energy of incident photon, E_γ = hc / λ, and using the work function for potassium, the maximum kinetic energy is approximately 1.3 eV.

The work function (φ) for potassium is about 2.25 eV. The kinetic energy of the photoelectron is found by subtracting the work function from the energy of the incident photon.

Compton effect shows an increase (shift) in the wavelength of scattered X-rays, which depends on the scattering angle and not on the intensity of the incident X-rays. So, statements (A) and (C) are true.

Compton shift: Δλ = λ' - λ = (h / m_e c)(1 - cos θ), which depends on the angle θ, not the intensity of incident radiation.

Clausius-Clapeyron relation is used to describe the change in pressure with temperature for phase transitions.

The equation gives the slope of phase boundaries in a P-T diagram for phase changes at equilibrium.

The Compton shift formula is:
Δλ = (h / m_e c)(1 - cos θ).
For θ = 45°, and using h / m_e c ≈ 2.43 pm,
Δλ = 2.43 pm × (1 - cos 45°) ≈ 2.43 pm × 0.2929 ≈ 0.71 pm.
Hence, λ_scattered = λ_initial + Δλ = 10.0 pm + 0.71 pm = 10.71 pm ≈ 10.7 pm.

Keep in mind that h / m_e c ≈ 2.43 × 10^-12 m. For θ < 90°, the Compton shift is relatively small.

Statements (A) and (C) summarize two well-known impossibility statements of the second law: (1) No heat engine can continuously produce work by just cooling a single reservoir below its surroundings, and (2) no spontaneous (self-acting) device can transfer heat from cold to hot without external energy.

The Second Law has several equivalent formulations (Kelvin–Planck and Clausius statements). They all forbid "something for nothing" from a single reservoir or spontaneous cold-to-hot flow.

Heat absorbed, Q = m × L = 10 g × 540 cal/g = 5400 cal.
At 100°C, the temperature in Kelvin is T = 373 K.
Change in entropy: ΔS = Q / T = 5400 cal / 373 K ≈ 14.48 cal/K.

Often, ΔS = m × L / T. Watch out for potential factor-of-10 mismatches in the given options or question statements.

Compton shift formula: Δλ = (h / m_e c)(1 - cos θ). For backscatter, θ = 180° → cos 180° = -1.
Δλ = 2 × (h / m_e c) = 2 × 2.43 × 10^-12 m = 4.86 × 10^-12 m.

Backscattering (θ = 180°) doubles the maximum Compton shift compared to θ = 0°.

In forward bias: P → +, N → - (statement A is correct; B is not).
In reverse bias: P → -, N → + (statement C is correct; D is not).

Forward bias lowers the junction barrier, letting current flow; reverse bias heightens it, blocking conduction (except tiny leakage).

For a series LCR circuit, the total impedance is:
Z = √[R² + (X_L - X_C)²], and the phase angle Φ (between voltage and current) satisfies:
tan Φ = (X_L - X_C)/R.

X_L = ωL, X_C = 1/(ωC). If X_L > X_C, circuit is inductive and voltage leads current.

The quality factor Q is defined as:
Q = f₀ / (f₂ - f₁),
where f₀ = 300 Hz, f₁ = 150 Hz, f₂ = 450 Hz.
Thus, Q = 300 / (450 - 150) = 300 / 300 = 1.0.

Half-power frequencies occur where the power drops to half its peak value, so the bandwidth is the difference between these frequencies. Q = f₀ / bandwidth.

The radius of the first dark ring (r₁) is given by:
r₁ = 1.22 × (λ × f) / D,
where D = 2a = 0.1 cm. Substituting the values:
r₁ = 1.22 × (5 × 10^-5 × 20) / 0.1 = 12.20 × 10^-3 cm.

Ensure to double the radius to find the diameter if only the radius is given. This calculation assumes ideal conditions.

Brewster’s law states that the polarizing angle i_p satisfies:
tan i_p = μ,
where μ is the refractive index of the medium.

At Brewster's angle, the reflected beam is fully plane-polarized, and θ_Brewster + θ_refracted = 90°.

The grating equation is: d sin θ = nλ, where (sin θ ≤ 1). Given d = 1 / 5000 cm = 2 × 10^-4 cm and λ = 5 × 10^-5 cm, we find:
n ≤ d / λ = 2 × 10^-4 / 5 × 10^-5 = 4.
Thus, the highest observable order is n = 4.

For a diffraction grating, sin θ ≤ 1 implies nλ ≤ d. The maximum observable integer n is ⌊ d / λ ⌋.

For a function to be piecewise continuous according to Dirichlet's conditions, it must have only a finite number of extremities (maxima and minima) and discontinuities within any finite interval. This specifically excludes infinite discontinuities which would make the function not piecewise continuous.

Piecewise continuity is a crucial concept in mathematical analysis, particularly in Fourier analysis, where it ensures that a function can be represented as a Fourier series.

Fraunhofer diffraction (A) occurs at large distances where the light waves are parallel (II). Fresnel diffraction (B) happens when the source and screen are relatively close to the obstacle or aperture, requiring consideration of wave curvature (IV). Interference of light (C) involves interactions between waves from different sources (I). Polarization (D) relates to the orientation of light waves in a plane (III).

Understanding the different types of optical phenomena and their configurations helps in correctly categorizing and predicting their effects in practical applications, such as lenses and other optical devices.

To calculate the moment of inertia I for a diatomic molecule such as hydrogen (H₂), use the formula:
I = μ r², where μ is the reduced mass and r is the inter-molecular distance. Substituting the given values:

The moment of inertia for molecular systems is crucial in determining their rotational spectra, which is foundational in molecular physics and chemistry for identifying molecular structures.

Fresnel half period zones are used to describe the contribution of different portions of a wavefront to the intensity at a point in the wave's path. Each successive zone contributes an equal phase shift but alternates in sign, effectively contributing to the constructive and destructive interference pattern.

Understanding the contributions of different zones helps in designing lenses and understanding diffraction effects in various media.

The thickness d needed to introduce a phase difference Δφ = π/2 in a quarter-wave plate is given by:
d = λ / (4 × |n_e - n_o|), where n_e and n_o are the refractive indices for the extraordinary and ordinary rays, respectively. Substituting the given values:

Quarter-wave plates are used to create circular polarization from linear polarization by introducing a phase shift of π/2 between orthogonal polarization components.

For a rolling solid sphere, the acceleration a down an inclined plane can be calculated using the formula that accounts for both the translation and rotation of the sphere:

a = (5/7) * g * sin(θ) where θ is the angle of inclination and g is the acceleration due to gravity. Here, θ = 30° and g = 9.8 m/s². Plugging in the values:

a = (5/7) * 9.8 * sin(30°) = (5/7) * 9.8 * 0.5 = 3.5 m/s²

Rolling motion on an incline demonstrates the conservation of energy and the interplay between rotational and translational kinetic energy. This principle shows how rotation affects acceleration on an incline.

The quality factor Q of an LCR circuit is given by:

Q = (ω₀ L) / R where ω₀ = 1/√(LC) is the resonant frequency. Calculating ω₀:

ω₀ = 1 / √(2 × 10⁻³ × 2 × 10⁻⁶) ≈ 500 rad/s

Substituting in Q's formula:

Q = (500 × 2 × 10⁻³) / 0.2 = 5 × 10 = 50

The quality factor indicates how underdamped an oscillator is and the sharpness of the resonance peak.

The orbital period T of a satellite is inversely proportional to the square root of the mean radius r (which includes Earth's radius R plus altitude h). The relationship is given by Kepler's third law:

T ∝ 1 / √r, where r = R + h

The larger the altitude h, the larger the distance r from the center of Earth, and the longer the orbital period T will be.

According to Kepler's third law, the orbital period increases as the distance from the Earth increases.

To find the position of a point half a second earlier in a frame moving at 10 cm/s, subtract the displacement due to the frame's motion from the current position. Here, the displacement in half a second is:

Δx = 10 cm/s × 0.5 s = 5 cm

Thus, the position half a second earlier, moving backward along the x-axis, is:

xₑₐʳʹ = 11 cm + 5 cm = 16 cm

When calculating position changes due to uniform motion, always consider the direction of motion and coordinate axes involved.

The duration of one day is determined by the rotational period of the Earth. If the radius is halved while the mass remains constant, the Earth's moment of inertia I will change according to:

I = kMR² where k is a constant, M is mass, and R is radius.

Reducing R by half means:

Iₙ = kM(R/2)² = ¼Iₒ

With a smaller moment of inertia and the same angular momentum (since angular momentum is conserved), the Earth would spin faster. The new rotational period Tₙ can be estimated by considering the conservation of angular momentum:

Tₙ = Tₒ / 2

Thus, the new period would be 12h, and further adjustments lead to 6h.

This problem illustrates the principle of conservation of angular momentum and how changes in a body's structure affect its rotational dynamics.

For Newton's rings, the interference condition for constructive and destructive interference is based on the thickness t of the air film and the wavelength of light. The pattern of colored rings can only be observed if the light is polychromatic, as monochromatic light will produce a simple fringe pattern without any color.

Polychromatic light leads to the formation of colored rings due to the different wavelengths that produce constructive interference at different points.

The Michelson interferometer produces interference fringes based on the optical path difference. The change in the interference pattern is related to the change in the optical path length, which corresponds to a difference in the wavelength of the two lines. The formula for the change in wavelength is:

Δλ = 2 * Δx / m where Δx = 0.289 mm is the distance traveled by the mirror and m is the fringe order. Based on the given values, we find Δλ = 12 Å.

In interferometry, the fringe shift is proportional to the change in the optical path length, which can be used to measure minute differences in wavelength.

The width of the space charge region W in a pn junction is given by the formula:

W = √(2εV_bi / q * (1/N_A + 1/N_D)) where ε is the permittivity, V_bi is the built-in voltage, and N_A and N_D are the doping concentrations. Substituting the values:

W ≈ 9.51 μm

The width of the space charge region depends on doping concentrations and the built-in potential of the pn junction.

In a bipolar junction transistor (BJT):

• Cut-off mode: Both junctions are reverse biased.
  Saturation mode: Both junctions are forward biased.
  Inverse active mode: The base-emitter junction is forward biased and the base-collector junction is reverse biased.
  Forward active mode: Both junctions are forward biased.








 

The operating mode of a BJT is determined by the biasing of the base-emitter and base-collector junctions.

• A: ∇⋅E = ρ/ε₀ is Gauss's Law in electrostatics (III)
  B: ∇⋅B = 0 is Gauss's Law in magnetostatics (I)
  C: ∇×E = -∂B/∂t is Faraday's Law (II)
  D: ∇×B = μ₀J + μ₀ε₀∂E/∂t is the Modified Ampere's Law (IV)








 

Maxwell’s equations describe the behavior of electric and magnetic fields in both static and dynamic situations. The correct match of the equations is important for understanding electromagnetism.

The fraction of electrons in the donor states can be determined using the formula for the concentration of electrons in the conduction band, given the donor concentration and temperature. For N_d = 10¹⁶ cm^-3, the fraction of electrons in donor states is approximately 41% at T = 300 K.

Doping introduces donor or acceptor states in a semiconductor, altering the electron distribution and the fraction of electrons in these states.

For an ideal PN junction, the depletion layer approximation assumes a sharp boundary, while the Maxwell-Boltzmann statistics applies to the carrier distribution. The low injection assumption simplifies the behavior of the junction under forward bias.

Ideal PN junctions follow certain simplifications like the abrupt depletion model and low-injection conditions, which are valid for small forward bias and low current densities.

The displacement current is related to the rate of change of electric flux density. It accounts for the current associated with the changing electric field in regions where there is no actual charge movement, like in a capacitor.

Displacement current is a term introduced by Maxwell to ensure consistency in Ampère's Law when there is a changing electric field.

• Intrinsic semiconductors are pure and do not have any doping (A - II).
  N-type semiconductors are created by doping with pentavalent impurities (B - III).
  P-type semiconductors are created by doping with trivalent impurities (C - IV).
  A P-N junction diode is created by combining P-type and N-type semiconductors (D - I).








 

Understanding the doping process in semiconductors is crucial for designing electronic devices like diodes and transistors.

The relationship between the intensity I and the amplitude E₀ of the electric field for an electromagnetic wave is given by:

I = (1/2) ε₀ c E₀² where ε₀ = 8.854 × 10^-12 C²/N·m² is the permittivity of free space and c = 3 × 10⁸ m/s is the speed of light. Solving for E₀:

E₀ = √(2I / ε₀ c) = √(2 × 2.0 / (8.854 × 10^-12 × 3 × 10⁸)) ≈ 388.8 NC^-1

The electric field amplitude is related to the intensity of the electromagnetic wave, with higher intensity corresponding to higher field amplitudes.

For a compound pendulum, the equation of motion is given by:

I θ'' + mgl sin(θ) = 0

For small angles, sin(θ) ≈ θ, simplifying the equation to a linear form for simple harmonic motion.

Hamilton's equations describe the evolution of generalized coordinates q_j and momenta p_j. They are fundamental in classical mechanics for systems in phase space.

Hamiltonian mechanics provides a reformulation of classical mechanics, focusing on energy rather than force.

Statement (I) is incorrect because linear momentum is conserved when no external force is applied, regardless of the force component. Statement (II) is correct; if no external torque acts on a system, angular momentum is conserved.

Angular momentum conservation is a consequence of the absence of external torque. Linear momentum conservation follows from Newton's first law.

The Poisson bracket properties are well-defined in Hamiltonian mechanics and satisfy all the given relations: (A) is true because the Poisson bracket is antisymmetric. (B) is true because any quantity commutes with itself. (D) follows from the Leibniz rule.

Poisson brackets are essential for describing the time evolution of dynamical variables in classical mechanics.

To find the shortest distance between two skew lines, use the formula: \( d = \frac{| \mathbf{a_1} - \mathbf{a_2} \cdot (\mathbf{b_1} \times \mathbf{b_2}) |}{|\mathbf{b_1} \times \mathbf{b_2}|} \) where \( \mathbf{a_1} \) and \( \mathbf{a_2} \) are points on the lines and \( \mathbf{b_1} \) and \( \mathbf{b_2} \) are direction vectors of the lines.

For skew lines, the shortest distance formula requires both the position vectors and direction vectors to be known.

The reflection of a point across a plane can be found by determining the foot of the perpendicular from the point to the plane and then doubling that distance to find the reflected point. The correct image point is (-3, 5, 2).

To find the image of a point, use the formula for the reflection of a point across a plane and calculate the corresponding coordinates.

The general equation of a sphere can be written in the form: (x - h)² + (y - k)² + (z - l)² = r² where h, k, and l are the coordinates of the center, and r is the radius. The radius is obtained by completing the square for each term.

The radius of the sphere is determined after converting the general equation into the standard form by completing the square.

The standard equations for the different surfaces are: Ellipsoid: x²/a² + y²/b² + z²/c² = 1, Hyperboloid of one sheet: x²/a² - y²/b² - z²/c² = 1, Hyperboloid of two sheets: -x²/a² + y²/b² + z²/c² = 1, Central conicoid: x²/a² + y²/b² + z = 1.

Each surface equation has a distinct form depending on the signs and coefficients of x², y², and z².

A homogeneous equation of the second degree in x, y, and z represents a cone with the vertex at the origin. The second statement is correct, as a cone can be generated by making a first-degree equation homogeneous with a guiding curve equation.

Homogeneous second-degree equations define conic surfaces, and a cone can be obtained by combining linear and second-degree equations.

Using the Laplace transform method, we take the transforms of the given differential equation and solve for Y(s). After applying the initial conditions, we find: y(t) = e^t - 3e^-t + 2e^-2t

For higher-order differential equations, Laplace transforms simplify solving by converting the equation into algebraic form.

The concept of orthogonal trajectories is correct in Statement (I), as two families of curves that intersect at right angles are indeed called orthogonal. For Statement (II), the orthogonal trajectory of the hyperbola xy = c is y = 1/x, which is correct.

Orthogonal trajectories can be found by solving the differential equation of one family of curves, and then solving for the other family that satisfies the condition of orthogonality.

Using the Mean Value Theorem, we can write: f(8) - f(3) = f'(3) × (8 - 3). Given that f'(x) ≤ 32, we get: f(8) - 21 = 32 × 5, hence f(8) = 21 + 160 = 181.

The Mean Value Theorem helps in finding the maximum or minimum values of a function given certain conditions on its derivative.

We solve the given non-homogeneous differential equation by finding the complementary function and the particular integral. The complementary function comes from the homogeneous equation, and the particular solution is found using the method of undetermined coefficients.

Use the method of undetermined coefficients to find the particular solution for equations with exponential and trigonometric forcing functions.

The complementary function for each differential equation is identified based on the form of the equation, and the solutions are matched accordingly.

For second-order linear equations with constant coefficients, the complementary function is found by solving the characteristic equation.

We simplify the given expression by first evaluating the complex logarithm and applying the properties of trigonometric functions. After simplification, we find the value of the expression to be 4√3.

When dealing with complex logarithms, the argument of the logarithm must be simplified carefully, especially when dealing with imaginary parts.

The given integral is a line integral around a closed path. We apply Green’s Theorem to convert the line integral into a double integral over the area enclosed by the curve. After calculating the double integral, we find that the result is zero.

Using Green's Theorem simplifies evaluating line integrals by converting them to double integrals over the enclosed region.

We calculate the surface integral using the given vector field and the surface of the cylinder. The flux through the surface is found by integrating the dot product of the vector field and the normal vector over the surface area.

Surface integrals can be simplified by using symmetry and parameterizing the surface in cylindrical coordinates.

The curl of the gradient of any scalar field is always zero, i.e., ∇ × ∇ f = 0. Therefore, the value of curl (grad f) is 0.

The curl of a gradient is always zero, a fundamental identity in vector calculus.

Statement (I) is incorrect because irrotational fields have zero curl, which does not imply the angular velocity is always greater than zero. Statement (II) is correct because for a solenoidal vector field (a field with no divergence), the divergence is always zero.

In vector calculus, irrotational fields have zero curl, and solenoidal fields have zero divergence.

We first compute the gradient of φ(x, y) and then find the directional derivative along the given direction. The gradient is:

∇φ = (∂φ/∂x, ∂φ/∂y)

After calculating the gradient, we compute the magnitude of the directional derivative and then find 1/a².

The directional derivative gives the rate of change of a function in the direction of a vector, and it is computed using the dot product of the gradient and the unit vector in the direction of interest.

The volume can be calculated using triple integration, first setting up the bounds based on the curves and surface, and then integrating to find the volume enclosed by the solid.

Use triple integrals for calculating volumes in the presence of complex boundaries like curves and surfaces.

To maximize the speed, we take the derivative of the function S(x) with respect to x and set it to zero. This results in the optimal ratio of x = 1/√e.

When optimizing physical systems, use calculus to find critical points where derivatives equal zero.

The spiral (r = α/θ) asymptotically approaches a straight line, and its asymptote is given by the relation (r sin θ = α).

In polar coordinates, the asymptote of a spiral can be found by analyzing its behavior at large values of θ.

The radius of curvature is found using the formula for curvature at a point on the curve. After computing the radius, we apply the given formula to find the final value of (b^2 + 2a + 1).

Use the formula for the radius of curvature to determine the geometric properties of curves.

The nth derivative of (cos x cos 2x cos 3x) is correctly given in Statement (II), while Statement (I) contains an error in the derivative form of the given function.

Check for patterns in derivatives of exponential and trigonometric functions, as they often follow established identities.

The value of (n) is determined by applying the given partial derivatives. After performing the calculations, we find that (n = -3/2) satisfies the given equation.

Solve partial differential equations by computing the necessary derivatives and comparing both sides of the equation.

The principal value of (i^i) can be computed by using the logarithmic representation of complex numbers. Using Euler's formula, we find that (i^i = e^{-π/2}).

The principal value of powers of complex numbers involves using logarithms and Euler’s formula to simplify expressions.

We use the logarithmic identity: (log(1) + log(2) = log(ab)). Thus, the expression (log(1 + i) + log(1 - i)) becomes: (log[(1 + i)(1 - i)] = log[1^2 - i^2] = log[1 + 1] = log 2). So, the general value is (log 2), with the imaginary component arising from the argument of the product of (1 + i) and (1 - i), which adds an imaginary factor of (4ηπ i).

This question involves the use of logarithmic properties and complex numbers. Always simplify the argument inside the logarithm before applying the logarithmic properties.

Statement (I) is true because the determinant of a matrix and its transpose are always equal. Statement (II) is also true by the property of determinants, which states that (det(AB) = det(A)det(B)).

Both properties of determinants are standard facts and hold for any square matrices.

By testing the values of n, we find that when n = 2, the equation holds true. This can be verified by directly substituting into the equation and calculating both sides.

Substitute different values of n into the equation to check for consistency.

If A is a singular matrix and B is a null matrix, it implies that the system of equations has infinitely many solutions. This is because a singular matrix means that the system does not have a unique solution, but since B is a null matrix, there are multiple solutions that satisfy the equation.

Singular matrices lead to either no solutions or infinitely many solutions, depending on the consistency of the system.

We can solve this matrix equation using Gaussian elimination or substitution. The system of equations is:
3x - y = 4 and 2x + 5y = -3. Solving these equations, we get x = 1 and y = -1.

For a 2x2 matrix, solving the linear equations directly gives us the values of the variables.

For a skew-symmetric matrix of odd order, the determinant is always zero. This is a property of skew-symmetric matrices.

A skew-symmetric matrix of odd order always has a determinant of 0.

Since P is non-singular and PQ = 0, the matrix Q must be singular. This is because a non-singular matrix cannot multiply with another matrix to produce the zero matrix unless the other matrix is singular.

If a non-singular matrix multiplied by another matrix results in the zero matrix, the second matrix must be singular.

Symmetric matrices have a_ij = a_ji.
Hermitian matrices have the same property as symmetric matrices, but they also have complex entries.
Skew-Hermitian matrices have a_ij = -a_ji.
Skew-Symmetric matrices also have a_ij = -a_ji, but are specifically real matrices.

Symmetric matrices satisfy a_ij = a_ji, while skew-symmetric matrices satisfy a_ij = -a_ji.