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CONCEPT:

Electrical Meters and Resistance

• Electrical meters are devices used to measure electrical quantities such as current, voltage, and resistance. The resistance of these meters is an important factor in their design and functionality.
• Ammeter, milliameter, galvanometer, and voltmeter are different types of electrical meters, each with varying resistances.

EXPLANATION:

• Let's examine the given options:
• Option 1: Ammeter
• An ammeter is used to measure electric current. It is designed to have very low resistance so that it does not affect the current it is measuring.
• Option 2: Milliameter
• A milliameter is a type of ammeter that measures small currents in milliamperes. It also has very low resistance, similar to an ammeter.
• Option 3: Galvanometer
• A galvanometer is an instrument for detecting and measuring small electric currents. It has higher resistance compared to an ammeter and milliameter.
• Option 4: Voltmeter
• A voltmeter is used to measure electrical potential difference between two points. It is designed to have high resistance to minimize the current draw from the circuit being measured.

Therefore, the correct answer is option 2: Milliameter.

Concept:

Potential Gradient in a Wire:

• The potential gradient ΔV/Δx is given by the formula:
• ΔV/Δx = I × ρ / A, where:
• I = Current through the wire (A)
• ρ = Resistivity of the wire (Ω·m)
• A = Cross-sectional area of the wire (m²)
• Potential gradient represents how much the potential changes per unit length along the wire.

Calculation:

Given:

• Current I = 0.5 A
• Resistivity ρ = 4 × 10⁻⁷ Ω·m
• Area A = 8 × 10⁻⁶ m²

We can calculate the potential gradient using the formula:

ΔV/Δx = (0.5 A × 4 × 10⁻⁷ Ω·m) / (8 × 10⁻⁶ m²)

ΔV/Δx = 25 × 10⁻³ V/m = 25 mV/m

∴ The potential gradient in the wire is 25 mV/meter, which corresponds to Option 1.

CONCEPT:

Vibration Magnetometer

• A vibration magnetometer measures the period of oscillation of a magnet in a magnetic field.
• The period of oscillation (T) is inversely proportional to the square root of the magnetic moment (M) of the magnet.
• Mathematically, T∝1M√T∝1M

.

EXPLANATION:

• Given:
• The period of oscillation of the first magnet, T1=2T1=2

sec.

• The magnetic moment of the second magnet, M2=4M1M2=4M1

.

• Since T∝1M√T∝1M

:

• T2=T14√=22=1T2=T14=22=1

sec.

Therefore, the correct answer is option 1: 1 sec.

CONCEPT:

AC Generator

• An AC generator is a device that converts mechanical energy into electrical energy using the principle of electromagnetic induction.
• The emf (electromotive force) generated in a coil rotating in a magnetic field depends on the number of turns in the coil, the area of the coil, the magnetic field strength, and the angular frequency of rotation.

EXPLANATION:

• Let's examine the given options:
• Option 1: N.A.B.R.ω
• This option includes resistance (R), which is not part of the formula for the maximum emf generated.
• Option 2: N.A.B.
• This option is missing the angular frequency (ω), which is necessary for calculating the maximum emf.
• Option 3: N.A.B.R.
• This option incorrectly includes resistance (R), which does not influence the maximum emf generated.
• Option 4: N.A.B.ω
• This option correctly includes the number of turns (N), the area of the coil (A), the magnetic field strength (B), and the angular frequency of rotation (ω).

Therefore, the correct answer is option 4: N.A.B.ω.

CONCEPT:

Frequency of Oscillations in an LC Circuit

• The frequency of oscillations in a resonant LC circuit is given by the formula:

f = 1 / (2π√(LC))

where:

• f is the frequency of oscillation
• L is the inductance
• C is the capacitance

EXPLANATION:

• Initially, the frequency of the oscillator is given by:

f = 1 / (2π√(LC))

• When the inductance L is doubled and the capacitance C is changed to 4C, the new frequency f' is given by:

f' = 1 / (2π√(2L * 4C)) = 1 / (2π√(8LC))

Now, factor out the 8:

f' = 1 / (2π√(8)√(LC))

Since √8 = 2√2, we have:

f' = 1 / (2π * 2√2 * √(LC)) = 1 / (2√2 * 2π√(LC))

Therefore:

f' = f / (2√2)

Therefore, the correct answer is option 2: f / 2√2.

CONCEPT:

Electromagnetic Spectrum

• The electromagnetic spectrum encompasses all types of electromagnetic radiation, each with varying wavelengths and frequencies.
• Different parts of the spectrum have different applications, from radio waves used in communication to gamma rays used in medical treatments.

EXPLANATION:

• Let's examine the given options:
• Option 1: U.V rays - finger prints detection
• Ultraviolet (U.V) rays are indeed used in forensic science to detect and analyze fingerprints.
• Option 2: I.R. rays - Secret writing on ancient walls
• Infrared (I.R.) rays can be used to reveal hidden layers of writing or paintings on ancient walls, as they penetrate the surface layers and highlight the underlying details.
• Option 3: X-rays - Atomic structure
• X-rays are used in X-ray crystallography to determine the atomic structure of crystals, including biological molecules like proteins and DNA.
• Option 4: Microwaves - forged document detection
• Microwaves are typically not used for detecting forged documents. This application does not align with the usual uses of microwaves.

Therefore, the correct answer is option 4: Microwaves - forged document detection.

CONCEPT:

Angle of Deviation in a Prism

• The angle of deviation is the angle between the incident ray and the emergent ray when light passes through a prism.
• For an equilateral prism, the angle of the prism (A) is 60°.
• If the angle of incidence (i) is equal to the angle of emergence (e) and e = 3/4 A, then we can use these relationships to find the angle of deviation (D).

EXPLANATION:

• Let’s analyze the given conditions:
• The angle of the prism (A) = 60°.
• The angle of incidence (i) is equal to the angle of emergence (e).
• The angle of emergence (e) = (3/4)A = (3/4) × 60° = 45°.
• Using the prism formula:
• For minimum deviation, i = e and A = 60°.
• The formula for angle of deviation (D) in a prism is:
• D = i + e - A
• Since i = e = 45°:
• D = 45° + 45° - 60°
• D = 90° - 60°
• D = 30°

Therefore, the correct answer is option 4: 30°.

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.