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CONCEPT:

Biot-Savart Law

• The Biot-Savart Law is used to calculate the magnetic field (B) produced by a current-carrying segment. For a short wire element, the magnetic field is given by:

B = (μ₀ / 4π) × (I dl sin θ) / r²

• Where:
• μ₀ / 4π = 10^-7 T·m/A (Permeability of free space constant)
• I is the electric current
• dl is the length of the wire element
• r is the distance to the point where the field is measured
• θ is the angle between the current direction and the position vector of the point

EXPLANATION:

• Given values from the problem:
• Current (I) = 5 A
• Length of wire (dl) = 5 mm = 5 × 10^-3 m
• Distance (r) = 1 m
• The wire is along the X-axis and the observation point is on the Y-axis. Therefore, the angle (θ) between the current element and the position vector is 90°.
• Calculation of the magnetic field:
• Using the Biot-Savart formula:
• B = (10^-7) × (5 × 5 × 10^-3 × sin 90°) / (1)²
• Since sin 90° = 1:
• B = 10^-7 × 25 × 10^-3
• B = 25 × 10^-10 T
• B = 2.5 × 10^-9 T

Therefore, the magnetic field (B) at a distance 1 m on the Y-axis is 2.5 × 10^-9 T.

CONCEPT:

Magnetic Force Between Two Parallel Wires

• When two long parallel wires carry currents $I_1$ and $I_2$ and are separated by a distance $r$, they exert a force on each other due to the magnetic fields they produce.
• The magnitude of the force per unit length ($f$ or $F/L$) is calculated using the formula:

$f = \frac{\mu_0 I_1 I_2}{2\pi r}$

• Where $\mu_0$ is the permeability of free space, which is equal to $4\pi \times 10^{-7}\text{ T}\cdot\text{m/A}$.

EXPLANATION:

• Given data:
• Current in the first wire ($I_1$) = $5\text{ A}$
• Current in the second wire ($I_2$) = $2\text{ A}$
• Distance between the wires ($r$) = $0.2\text{ m}$
• Using the formula for force per unit length:
• $f = \frac{4\pi \times 10^{-7} \times 5 \times 2}{2\pi \times 0.2}$
• Simplifying the expression by canceling $2\pi$:
• $f = \frac{2 \times 10^{-7} \times 10}{0.2}$
• $f = \frac{20 \times 10^{-7}}{0.2}$
• $f = \frac{200 \times 10^{-7}}{2}$
• $f = 100 \times 10^{-7}\text{ N/m}$
• $f = 1 \times 10^{-5}\text{ N/m}$

Therefore, the force per unit length is $1 \times 10^{-5}\text{ N/m}$.

CONCEPT:

Magnification of a Compound Microscope

• A compound microscope consists of two lenses: the objective lens and the eye lens (eyepiece).
• The total magnification (M) is the product of the magnification of the objective lens (m_o) and the magnifying power of the eye lens (m_e).
• When the final image is formed at the least distance of distinct vision (D), the total magnification is given by the approximate formula:

M = m_o × m_e ≈ (L / f_o) × (1 + D / f_e)

• Where:
• L is the tube length of the microscope.
• f_o is the focal length of the objective lens.
• f_e is the focal length of the eye lens.
• D is the least distance of distinct vision (standard value is 25 cm).

EXPLANATION:

• Given the following values:
• Focal length of objective lens (f_o) = 50 cm
• Focal length of eye lens (f_e) = 5 cm
• Length of the tube (L) = 15 cm
• Least distance of distinct vision (D) = 25 cm
• Step 1: Calculate the magnification produced by the objective lens (m_o):

m_o = L / f_o

m_o = 15 / 50 = 0.3

• Step 2: Calculate the magnifying power of the eye lens (m_e) for an image formed at D:

m_e = 1 + (D / f_e)

m_e = 1 + (25 / 5)

m_e = 1 + 5 = 6

• Step 3: Calculate the total magnification (M):

M = m_o × m_e

M = 0.3 × 6

M = 1.8

Therefore, the magnification of the microscope is 1.8.

CONCEPT:

Bohr Model - Velocity of an Electron

• According to Bohr's atomic model, an electron revolves around the nucleus in specific circular orbits.
• The velocity of an electron in the n^th orbit of a hydrogen-like atom is given by the formula:

v_n = v₀ × (Z / n)

• Where:
• v_n is the velocity of the electron in the n^th orbit.
• v₀ is the velocity of the electron in the first orbit of hydrogen (n=1), which is approximately 2.18 × 10⁶ m/s.
• Z is the atomic number of the element.
• n is the principal quantum number or the orbit number.

EXPLANATION:

• For a Hydrogen atom:
• The atomic number (Z) = 1.
• We need to find the velocity in the fourth orbit:
• The orbit number (n) = 4.
• Using the velocity formula:
• v_n = (2.18 × 10⁶ × Z) / n
• v₄ = (2.18 × 10⁶ × 1) / 4
• v₄ = 0.545 × 10⁶ m/s
• v₄ = 5.45 × 10⁵ m/s

Therefore, the velocity of the fourth orbit of a hydrogen atom is 5.45 × 10⁵ m/s.

CONCEPT:

Magnetic Permeability and Susceptibility

• Magnetic susceptibility (χ_m) is a measure of how much a material will become magnetized in an applied magnetic field.
• The relative permeability (μ_r) of a material is related to its magnetic susceptibility (χ_m) by the equation:

μ_r = 1 + χ_m

• The absolute magnetic permeability (μ) is the product of the permeability of free space (μ₀) and the relative permeability (μ_r):

μ = μ₀ × μ_r

• The standard value for the permeability of free space (μ₀) is 4π × 10^-7 H/m.

EXPLANATION:

• Given information:

Magnetic susceptibility (χ_m) = 2499

• Step 1: Calculate the relative permeability (μ_r):
• μ_r = 1 + χ_m
• μ_r = 1 + 2499
• μ_r = 2500
• Step 2: Calculate the magnetic permeability (μ):
• μ = μ₀ × μ_r
• μ = (4π × 10^-7 H/m) × 2500
• μ = 10000 × π × 10^-7 H/m
• μ = 10⁴ × π × 10^-7 H/m
• μ = 10^-3π H/m

Therefore, the magnetic permeability (μ) is 10^-3π H/m.

CONCEPT:

Rydberg Formula for Hydrogen Spectrum:

The wavelengths of the spectral lines in the hydrogen spectrum can be calculated using the Rydberg formula: 1/λ = R (1/n12 - 1/n22)

Where:

λ is the wavelength of the emitted radiation. R is the Rydberg constant (1.1 × 107 m-1). n1 and n2 are integers such that n2 > n1.

For the Paschen series, the electron transitions from higher energy levels to the third energy level (n1 = 3). The minimum wavelength (also known as the series limit) occurs when the transition happens from the highest possible energy level (n2 = ∞).

EXPLANATION:

To find the minimum wavelength of the Paschen series:

Set n1 = 3 Set n2 = ∞

Substituting these values into the Rydberg formula: 1/λmin = R (1/32 - 1/∞2) 1/λmin = R (1/9 - 0) 1/λmin = R/9

Therefore, λmin = 9/R Given R = 1.1 × 107 m-1:

λmin = 9 / (1.1 × 107) λmin ≈ 8.1818 × 10-7 m

Converting the wavelength to nanometers (1 nm = 10-9 m):

λmin ≈ 818.18 nm

Therefore, the minimum wavelength of the Paschen series is approximately 818 nm.

CONCEPT:

Molar Specific Heat Capacities

• The molar specific heat at constant volume (C_v) is defined as the amount of heat energy required to raise the temperature of one mole of a gas by one unit at constant volume.
• The molar specific heat at constant pressure (C_p) is defined as the amount of heat energy required to raise the temperature of one mole of a gas by one unit at constant pressure.
• For an ideal gas, these two specific heats are related by Mayer's relation:

C_p - C_v = R

where R is the Universal Gas Constant.

• The molar specific heat at constant volume is also related to the degrees of freedom (f) of the gas molecules:

C_v = (f/2)R

EXPLANATION:

• For an ideal monoatomic gas (such as Helium or Argon), the atoms are considered point masses that can only move in three-dimensional space.
• The number of degrees of freedom (f) for a monoatomic gas is 3 (all translational).
• Calculating the molar specific heat at constant volume (C_v):
• C_v = (f/2)R
• C_v = (3/2)R
• Calculating the molar specific heat at constant pressure (C_p) using Mayer's relation:
• C_p = C_v + R
• C_p = (3/2)R + R
• C_p = (5/2)R

Therefore, the molar specific heat of an ideal monoatomic gas at constant pressure is 5/2 R.

Concept:

Malus' Law:

• According to Malus' Law, when a plane polarized light passes through a polaroid, the intensity of the transmitted light I is given by:
• I = I₀ × cos²(θ), where:
• I₀ = Initial intensity of the light
• θ = Angle between the light's polarization direction and the polaroid's transmission axis
• For a system of multiple polaroids, the intensity after passing through each polaroid is multiplied by the corresponding factor of cos²(θ), where θ is the angle between the pass axis of the polaroid and the previous one.

Calculation:

For each of the 5 polaroids, the angle between their pass axes is 60°, so the fraction of light intensity that passes through each polaroid is:

cos²(60°) = (1/2)² = 1/4

Since there are 5 polaroids, the total fraction of intensity that passes through the system is:

(1/4)⁵ = 1/512

∴ The fraction of the incident light intensity that passes through the system is 1/512, which corresponds to Option 4.

Explanation:

de Broglie Wavelength:

de Broglie suggested that every moving particle has an associated wavelength, called the de Broglie wavelength, which is given by the formula:

λ = h / p, where:

λ = de Broglie wavelength

h = Planck's constant (6.626 × 10⁻³⁴ J·s)

p = Momentum of the particle (kg·m/s)

The momentum p is related to the kinetic energy K by the relation:

p = √(2mK), where:

m = mass of the particle (kg)

K = kinetic energy (J)

Using the formula for de Broglie wavelength and substituting the momentum:

λ = h / p = h / √(2mK)

∴ The de Broglie wavelength associated with the ball is h / √(2mK), which corresponds to Option 3.

Concept:

Photoelectric Effect:

• The kinetic energy of photoelectrons K.E. is given by the Einstein photoelectric equation:
• K.E. = hf - φ, where:
• h = Planck's constant (6.626 × 10⁻³⁴ J·s)
• f = Frequency of the incident radiation
• φ = Work function of the metal
• The frequency f is related to the wavelength λ by the equation: f = c / λ, where:
• c = Speed of light (3 × 10⁸ m/s)
• λ = Wavelength of the incident radiation
• When the wavelength is decreased, the frequency increases, and thus the kinetic energy of the photoelectrons increases.

Calculation:

Let the initial wavelength be λ₁ = 500 nm and the final wavelength λ₂ = 200 nm.

The kinetic energy is proportional to the frequency, so when the wavelength changes, the kinetic energy of the photoelectrons is affected by the ratio of the frequencies:

Since frequency is inversely proportional to wavelength, we can write:

K.E.₂ / K.E.₁ = f₂ / f₁ = λ₁ / λ₂

The problem states that the kinetic energy becomes three times larger:

K.E.₂ / K.E.₁ = 3, so:

3 = λ₁ / λ₂ = 500 nm / 200 nm

The work function φ is given by the difference in the initial and final kinetic energies:

φ ≈ K.E.₁ (1 - 1 / 3) = 2/3 K.E.₁

Using this, we get that the work function of the metal is approximately 0.61 eV.

∴ The work function of the metal is close to 0.61 eV, which corresponds to Option 4.