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Concept:

The Raman effect is an inelastic scattering of light that occurs when photons interact with molecules, causing a change in the vibrational or rotational energy of the molecules.

Explanation:

During Raman scattering, a photon exchanges energy with a molecule. As a result, the scattered photon has a different frequency from the incident photon, producing the Raman spectrum. This effect is widely used to study the molecular structure and vibrational modes of substances.

Correct Answer:

Option (2): Molecule.

The correct answer is Change in electron mass at high drift velocity.

Key Points

• The V-I curve for an Ohmic conductor like copper is ideally a straight line. However, at high currents, the drift velocity of electrons remains very low, typically around 10⁻⁴ m/s. Significant changes in electron mass only occur at relativistic speeds approaching the speed of light (3 × 10⁸ m/s). Hence, statement 1 is incorrect.
• When a high current passes through a wire, it generates heat due to Joule heating, calculated by the formula P = I²R. This increase in thermal energy disrupts the lattice structure and increases electron collisions. Hence, statement 2 is correct.
• A battery is not an ideal voltage source and possesses internal resistance (r). As the current (I) increases, the terminal voltage (V = E − Ir) drops, which can lead to observed non-linearity in the circuit's performance. Hence, statement 3 is correct.
• Copper has a positive temperature coefficient of resistance (α). As Joule heating raises the temperature, the resistivity of the wire increases according to R = R₀(1 + αΔT), causing the V-I curve to bend. Hence, statement 4 is correct.

 Additional Information

• Ohm's Law:
• States that current is directly proportional to voltage provided temperature and pressure remain constant.
• In high-current scenarios, the 'constant temperature' condition is violated.
• Relativistic Mass:
• Calculated using the formula m = m₀ / √(1 − v²/c²).
• Since drift velocity in a copper wire is negligible compared to c, the mass change is practically zero.
• Resistivity and Temperature:
• For metals, resistance increases with temperature.
• For semiconductors, the temperature coefficient is negative, meaning resistance decreases as temperature rises.

The correct answer is Area of cross-section.

Key Points

• The electrical resistance of a conductor is defined as the opposition it offers to the flow of electric current. It is determined by the dimensions and atomic structure of the material.
• According to the laws of resistance, the resistance (R) is inversely proportional to the Area of cross-section (A) of the conductor. This means that a thicker wire provides more parallel paths for electrons to flow, which reduces the frequency of collisions with metal ions.
• Mathematically, this relationship is expressed as R ∝ 1/A. If the cross-sectional area is doubled, the resistance is halved, assuming the length and temperature remain constant.
• This principle is utilized in electrical wiring where high-current appliances require thicker cables with larger cross-sections to prevent excessive heating and energy loss.

 Additional Information

• Length of the conductor (L):
• Resistance is directly proportional to the length of the conductor (R ∝ L). A longer path increases the probability of electron-ion collisions.
• Resistivity (ρ):
• It is an intrinsic property of a material. The complete formula for resistance is R = ρ × (L/A), where ρ (rho) represents the resistivity.
• Conductive materials like Silver (1.59 × 10⁻⁸ Ω·m) and Copper (1.68 × 10⁻⁸ Ω·m) have extremely low resistivity values.
• Temperature:
• For metallic conductors, resistance increases with an increase in temperature because thermal energy increases the amplitude of vibration of the ions, hindering the flow of charge carriers.
• SI Unit:
• The SI unit of resistance is the Ohm (Ω), named after Georg Simon Ohm.

The correct answer is They form concentric circles centered on the conductor..

Key Points

• When an electric current flows through a straight conductor, it produces a magnetic field around it, a phenomenon discovered by Hans Christian Ørsted.
• The magnetic field lines around a straight current-carrying wire always form concentric circles.
• The center of these concentric circles lies on the wire itself, and the circles exist in a plane that is perpendicular to the length of the conductor.
• The direction of these magnetic field lines is determined by the Right-Hand Thumb Rule.
• The density of these circles is higher near the wire, indicating a stronger magnetic field, and decreases as the distance from the wire increases.
• This pattern can be visualized by sprinkling iron filings on a plane surface (like cardboard) through which a current-carrying wire passes vertically.

Additional Information

• Right-Hand Thumb Rule: If you imagine holding a straight conductor in your right hand such that your thumb points in the direction of the current, then the direction in which your fingers curl gives the direction of the magnetic field lines.
• The magnitude of the magnetic field (B) at any point is directly proportional to the current (I) passing through the wire and inversely proportional to the distance (r) from the wire.
• The formula for the magnetic field at a distance r from an infinitely long straight wire is: \( B = \frac{\mu_0 I}{2\pi r} \).
• If the direction of current is reversed, the direction of the magnetic field lines (clockwise or anticlockwise) also reverses.
• Unlike a solenoid where the field lines are parallel straight lines inside, the field around a single straight wire is strictly circular.

The correct answer is Direction of current.

Key Points

• Fleming's Left-Hand Rule is a simple visualization tool used to find the direction of the mechanical force (motion) experienced by a current-carrying conductor when it is placed in an external magnetic field.
• To apply the rule, the thumb, forefinger (first finger), and middle finger (second finger) of the left hand are held mutually perpendicular (at 90 degrees) to one another.
• The Forefinger (First finger) represents the direction of the Magnetic Field (B), typically pointing from the North pole to the South pole.
• The Middle Finger (Second finger) represents the direction of the Electric Current (I) flowing through the conductor.
• The Thumb represents the direction of the Thrust or Force (F) acting on the conductor, which dictates its direction of motion.
• This rule is essentially a mnemonic for the Lorentz Force equation, where the force vector is the cross-product of the current and magnetic field vectors.

 Additional Information

• Direction of magnetic field
• In the left-hand rule, this is indicated by the index finger.
• In SI units, the strength of this field is measured in Tesla (T).
• Force acting on the conductor
• This is represented by the thumb.
• The magnitude of this force is given by the formula F = B × I × L × sin(θ), where L is the length of the wire and θ is the angle between the current and the field.
• Measure of resistance
• Resistance is a property of a material to oppose the flow of current, defined by Ohm's Law (R = V / I).
• It is not determined or represented by Fleming's rules, which deal with electromagnetism and motion rather than circuit properties.
• Electric Motors
• Fleming's Left-Hand Rule is the fundamental principle used to understand the operation of DC Motors, where electrical energy is converted into mechanical torque.

The correct answer is Pulls the iron strip towards itself.

Key Points

• The functioning of an electric bell is based on the magnetic effect of electric current, where a coil of wire acts as a magnet when electricity passes through it.
• The main component is an electromagnet, which consists of insulated copper wire wound around a soft iron core.
• When the circuit is completed by pressing the switch, the current flows through the coil, creating a magnetic field that transforms the iron core into a temporary magnet.
• This electromagnet immediately exerts a force of attraction on the iron strip (also known as the armature) located nearby.
• As the iron strip is pulled toward the electromagnet, the hammer attached to it moves and strikes the gong, producing the ringing sound.
• This movement also causes the strip to move away from the contact screw, which results in the breaking of the electrical circuit.

 Additional Information

• Core Components of an Electric Bell:
• Electromagnet: A temporary magnet that only functions when current is present; the principle was first observed by Hans Christian Oersted in 1820.
• Soft Iron Armature: Chosen because soft iron has high magnetic permeability and low retentivity, allowing it to be magnetized and demagnetized rapidly.
• Contact Screw: Serves as an adjustable component that maintains the make-and-break cycle of the circuit.
• Hammer and Gong: The mechanical parts responsible for converting the kinetic energy of the moving armature into acoustic energy.
• Spring Strip: Provides the restoring force to bring the iron strip back to the contact screw once the magnetism is lost, enabling a continuous ringing cycle.

The correct answer is Right-hand thumb rule.

Key Points

• The Right-hand thumb rule, frequently referred to as Maxwell's Corkscrew Rule, is the standard convention used to determine the direction of the magnetic field associated with a current-carrying conductor.
• This rule states that if you imagine holding a straight current-carrying wire in your right hand with the thumb pointing in the direction of the conventional current (from positive to negative), then your curled fingers will point in the direction of the magnetic field lines.
• The magnetic field produced by a straight conductor forms concentric circles around the wire; the direction of these circles reverses if the direction of the current is flipped.
• Mathematically, the magnitude of the magnetic field (B) at a distance r from a long straight wire carrying current I is given by the formula B = (μ₀I) / (2πr), where μ₀ is the permeability of free space.

 Additional Information

• Fleming's Left-hand rule
• This rule is primarily used to find the direction of the magnetic force acting on a moving charged particle or a current-carrying wire placed within an external magnetic field.
• It is the working principle behind electric motors, where the thumb represents thrust/force, the forefinger represents the magnetic field, and the middle finger represents the current.
• Fleming's Right-hand rule
• This rule is utilized to determine the direction of the induced current when a conductor moves through a magnetic field.
• It is the fundamental principle for electric generators, where the thumb points in the direction of motion, the forefinger in the direction of the field, and the middle finger indicates the induced current.

To find the average value of the current from t = 0 t = 2 seconds, we analyze the current-time (i - t) graph.

Calculate the Area under the Graph:

The graph consists of two triangular sections.

First Triangle: From t = 0 to t = 1 second, with a height of 10 A.

Area of first triangle = \( \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 1 \times 10 = 5,\text{A}\)

Second Triangle: From t = 1 to t = 2 seconds, with a height of 10 A.

Area of second triangle = \(\frac{1}{2} \times (2 - 1) \times 10 = 5~A\)

Total Area:

Total area under the graph from t = 0 t = 2 seconds is:

5A + 5A = 10A

Calculate the Average Current:

The average current (i_av ) is given by:

\(i_{\text{av}} = \frac{\text{Total Area}}{\text{Time Interval}} = \frac{10}{2} = 5\text{ A}\)

Thus, the average current over the interval from t = 0 t = 2 seconds is 5 A.

Answer (4)

Sol.

R_eq = \(\frac{150 \times 150}{300}\) + 25 = 100Ω

I₁ = \(\frac{5}{10}\) = 0.05A

I2 = I4 = \(\frac{0.05}{2}\) = 0.025A

\(\frac{\mathrm{I}{1}}{\mathrm{I}{2}}\) = 2

Concept:

The "Particle in a Box" model is a fundamental quantum mechanical model in which a particle is confined within a one-dimensional region by infinitely high potential walls.

Explanation:

The model assumes that the potential energy is zero inside the box and infinite outside it. As a result, the particle cannot escape the box, leading to quantized energy levels and standing wave solutions of the Schrödinger equation.

Correct Answer:

Option (1): Infinite potential barriers surrounding the particle.