UP Board Class 12 Physics Question Paper 2024 (Code 347 FV) Available- Download Solution PDF with Answer Key

The electric field intensity is defined as the force experienced by a unit positive charge in a field, and its units are newton per coulomb (\(N/C\)) or volt per meter (\(V/m\)). Quick Tip: Understand unit relationships to identify corresponding physical quantities effectively.

In a pure resistive circuit, the voltage and current are in phase, meaning the phase difference between them is \(0\). Quick Tip: In resistive circuits, voltage and current always align perfectly in time.

Applying a high negative voltage at the \(n\)-terminal of a \(p\)-\(n\) diode reverse biases it. If the applied voltage exceeds the breakdown voltage of the diode (typically around -2.2 V for a standard diode), it results in breakdown, allowing current to flow. Quick Tip: Remember that exceeding the breakdown voltage in reverse bias causes the diode to conduct via avalanche or Zener breakdown.

In a fusion reaction, mass is converted into energy, resulting in the final mass (\(m\)) being less than the initial total mass (\(m_1 + m_2\)). This mass defect is the source of the released energy. Quick Tip: Mass defect occurs in nuclear reactions due to energy release as per Einstein’s equation \(E = \Delta m c^2\).

The dynamic mass of a photon can be derived from the relation \(E = mc^2\). Since \(E = h\nu\), we get \(m = \frac{h\nu}{c^2}\). Using the speed of light \(c\), dynamic mass simplifies to \(\frac{h\nu}{c}\). Quick Tip:For photons, remember \(E = h\nu\) and \(E = mc^2\) to derive \(m = \frac{h\nu}{c^2}\).

In a hollow metallic sphere, the electric field inside the conductor is zero due to electrostatic shielding. The charge resides only on the outer surface, ensuring no electric field within. Quick Tip: Inside any conductor, the electric field is always zero in electrostatic equilibrium.

Convex mirrors are used in vehicles for rearview because they provide a wider field of view, which helps the driver see more traffic behind.

Quick Tip: Convex mirrors always form virtual, erect, and diminished images, making them ideal for rearview applications.

Work function is the minimum energy required to eject an electron from the surface of a photosensitive material. It is represented as $W = h\nu_0$, where $\nu_0$ is the threshold frequency. Only photons with energy greater than or equal to the work function can cause photoemission.

Quick Tip: For photoelectric effect: $E_{\text{photon}} \geq W$; otherwise, no electron ejection occurs.

The size of the nucleus is proportional to the cube root of its mass number. This can be expressed as $R = R_0 A^{1/3}$, where $R_0$ is a constant.

Quick Tip: This formula arises from the assumption that the nucleus is roughly spherical and its density is constant.

^{Kilowatt-hour is a unit of energy. $1$ kilowatt-hour corresponds to $1000$ watts of power used over $1$ hour, which equals $3600 \times 1000 = 3.6 \times 10^6$ joules.}

^{Quick Tip: To convert kilowatt-hours to joules, use $1 \text{ kWh} = 3.6 \times 10^6 \text{ J}$.}

Since the uncharged rod does not contribute any charge to the charged rod, the potential of the charged rod remains unaffected.

Quick Tip: The potential depends on the charge and distance; since the uncharged rod has no charge, there is no influence.

The direction of the magnetic field is given by curling the fingers of your right hand around the wire while your thumb points in the direction of the current. The field is stronger near the wire and weaker as you move away.

Quick Tip: Use the right-hand thumb rule for determining the direction of magnetic field lines around a straight current-carrying conductor.

Current sensitivity of a galvanometer is defined as the deflection produced in the galvanometer per unit current flowing through it. It is given by: \[ S_I = \frac{\theta}{I}, \] where $\theta$ is the deflection and $I$ is the current.

Increasing current sensitivity:

• Increasing the number of turns in the galvanometer coil.
• Using a stronger magnetic field.
• Reducing the torsional constant of the suspension wire.

Quick Tip: Increasing current sensitivity alone may not necessarily improve the overall sensitivity because reducing the damping can cause instability.

1. Power Factor:
The power factor of an AC circuit is defined as the cosine of the phase angle $\phi$ between the voltage and the current. It is given by: \[ \text{Power Factor} = \cos\phi = \frac{\text{True Power}}{\text{Apparent Power}}. \] 2. Wattless Current:
The component of the alternating current that contributes no power to the circuit is called the wattless current. It is the current component perpendicular to the voltage.

Quick Tip: A high power factor close to 1 indicates efficient power usage, whereas a low power factor suggests significant reactive power.

1.Wavelength ($\lambda$):

From the equation, the wave number $k = 0.5 \times 10^3 \, \text{m}^{-1}$. The wavelength is given by: \[ \lambda = \frac{2\pi}{k} = \frac{2\pi}{0.5 \times 10^3} = 12.57 \, \text{m}. \] 2. Magnetic Field ($B_z$):
The magnetic field is perpendicular to both the electric field and the direction of wave propagation. The amplitude of the magnetic field is: \[ B_0 = \frac{E_0}{c} = \frac{3 \times 10^{-7}}{3 \times 10^8} = 1 \times 10^{-15} \, \text{T}. \] The equation for the magnetic field is: \[ B_z = 1 \times 10^{-15} \sin(0.5 \times 10^3 x + 1.5 \times 10^{11} t) \, \text{T}. \]

Quick Tip: The relationship between $E_0$ and $B_0$ in electromagnetic waves is $B_0 = \frac{E_0}{c}$, where $c$ is the speed of light.

1.Diamagnetic Substances:

- Weakly repelled by a magnetic field.
- Magnetic susceptibility ($\chi$) is negative.
- Examples: Bismuth, Copper. 2. Paramagnetic Substances:
- Weakly attracted by a magnetic field.
- Magnetic susceptibility ($\chi$) is positive.
- Examples: Aluminum, Oxygen.

Quick Tip: Diamagnetic substances exhibit induced magnetism in the opposite direction of the external field, while paramagnetic substances align slightly with the field.

1.Need of Displacement Current:

Displacement current is needed to account for a time-varying electric field in cases where conduction current does not exist. It ensures that Maxwell's equations remain consistent, particularly in explaining electromagnetic waves' propagation. 2. Same Physical Quantity:
The speed of electromagnetic waves in a vacuum, $c$, is the same for X-rays, radio waves, and light waves, regardless of their wavelength. It is approximately $3 \times 10^8 \, \text{m/s}$.Quick Tip: Displacement current arises due to a time-varying electric flux and is given by $\epsilon_0 \frac{d\Phi_E}{dt}$.

1.Isotopes:

Nuclei of the same element with the same atomic number but different mass numbers.
Example: ${}^{12}_{6}\text{C}$ and ${}^{14}_{6}\text{C}$. 2. Isobars:
Nuclei of different elements with the same mass number but different atomic numbers.
Example: ${}^{14}_{6}\text{C}$ and ${}^{14}_{7}\text{N}$. 3. Isotones:
Nuclei with the same number of neutrons but different atomic and mass numbers.
Example: ${}^{14}_{6}\text{C}$ and ${}^{15}_{7}\text{N}$.

Quick Tip: Remember: - Isotopes → Same protons. - Isobars → Same total nucleons. - Isotones → Same neutrons.

1.Electric Dipole:

An electric dipole consists of two equal and opposite charges separated by a small distance. 2. Electric Dipole Moment ($p$):
The product of the magnitude of one of the charges and the separation between them: \[ p = q \cdot d. \] For the given dipole: \[ p = (20 \times 10^{-6}) \cdot (1 \times 10^{-2}) = 2 \times 10^{-7} \, \text{C.m}. \] 3. Electric Field Intensity:
At an axial point, the field is given by: \[ E = \frac{2kp}{r^3}, \] where $k = \frac{1}{4\pi \epsilon_0} = 9 \times 10^9 \, \text{N·m}^2/\text{C}^2$, $p = 2 \times 10^{-7} \, \text{C·m}$, and $r = 1 \, \text{m}$. \[ E = \frac{2 \cdot (9 \times 10^9) \cdot (2 \times 10^{-7})}{1^3} = 3600 \, \text{N/C}. \]

Quick Tip: The axial field is stronger than the equatorial field by a factor of 2 in the case of dipoles.

1.E.m.f. ($E$):

The e.m.f. is the maximum potential difference of a cell when no current flows through it. 2. Terminal Potential Difference ($V$):
The potential difference across the terminals of the cell when current flows through the cell. Calculation:
Using $E = V + Ir$, where $r$ is the internal resistance: \[ V = I \cdot R = 0.5 \cdot 3.9 = 1.95 \, \text{V}. \] \[ 2.0 = 1.95 + (0.5 \cdot r) \implies r = \frac{2.0 - 1.95}{0.5} = 0.1 \, \Omega. \]

Quick Tip: Internal resistance causes the terminal voltage to drop as current increases in the circuit.

1.Focal Length of the Combination:

If $f_1$ and $f_2$ are the focal lengths of two convex lenses in contact, the equivalent focal length is given by: \[ \frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2}. \] 2. Replacing with a Concave Lens:
If one lens is replaced by a concave lens with $f_1 = -f_2$, the net focal length becomes infinite: \[ \frac{1}{F} = \frac{1}{f_1} + \frac{1}{-f_1} = 0 \implies F \to \infty. \] Thus, the combination behaves as a plane glass plate.

Quick Tip: A combination of lenses in contact is more compact but changes behavior significantly with lens types (convex vs concave).

1. Conditions for Interference:
- The two sources of light must be coherent.
- The sources must have the same wavelength and frequency.
- The two waves must have a constant phase difference.
- The two light waves should overlap in space to form a region of constructive or destructive interference.

2. Coherent Source:
A coherent source emits waves with a constant phase difference and the same frequency. Examples of coherent sources include lasers and two slits illuminated by the same monochromatic light source.

1. Potential Energy Formula:
The potential energy $U$ of a magnetic dipole in an external magnetic field is given by: \[ U = -\vec{M} \cdot \vec{B} = -MB \cos \theta, \] where $M$ is the magnetic dipole moment, $B$ is the external magnetic field strength, and $\theta$ is the angle between $\vec{M}$ and $\vec{B}$.

2. Maximum Value:
When $\cos \theta = -1 \; (\theta = 180^\circ)$, the potential energy is: \[ U_{\text{max}} = MB. \]

3. Minimum Value:
When $\cos \theta = 1 \; (\theta = 0^\circ)$, the potential energy is: \[ U_{\text{min}} = -MB.

1. Meaning of Polarization:
Polarization refers to the phenomenon in which the oscillations of the electric field vector in a light wave are restricted to one particular plane perpendicular to the direction of propagation.

2. Representation:
- Unpolarized Light: Oscillations occur in all directions perpendicular to the direction of propagation. Represented as random electric field lines.
- Plane Polarized Light: Oscillations occur only in a single plane perpendicular to the propagation direction.

3. Nature Proved by Polarization:
Polarization of light proves the transverse nature of light waves.

The difference between conductor, semiconductor, and insulator based on energy band diagrams is as follows:

1. Conductor:
- Valence and conduction bands overlap, allowing free movement of electrons.
- There is no energy gap.

2. Semiconductor:
- A small energy gap ($<3$ eV) exists between the valence and conduction bands.
- Electrons can jump from the valence band to the conduction band when external energy is supplied (e.g., thermal energy or light).

3. Insulator:
- A large energy gap ($>5$ eV) exists between the valence and conduction bands, preventing electron flow under normal conditions.

Circuit parameters:

Resistance (\(R\)) = \(200 \, \Omega\)
Capacitance (\(C\)) = \(15 \, \mu\mathrm{F}\)
Voltage (\(V\)) = \(220 \, \mathrm{V}\)
Frequency (\(f\)) = \(50 \, \mathrm{Hz}\)

• (A) Calculate the capacitive reactance (\(X_C\)):} \[ X_C = \frac{1}{2 \pi f C} = \frac{1}{2 \pi (50)(15 \times 10^{-6})} \approx 212 \, \Omega \]
• (B) Calculate the impedance (\(Z\)):} \[ Z = \sqrt{R^2 + X_C^2} = \sqrt{(200)^2 + (212)^2} \approx 291.6 \, \Omega \]
• (C) Calculate the current (\(I\)):} \[ I = \frac{V}{Z} = \frac{220}{291.6} \approx 0.754 \, \mathrm{A} \]
• (D) Voltage across resistance (\(V_R\)):} \[ V_R = I R = (0.754)(200) \approx 150.8 \, \mathrm{V} \]
• (E) Voltage across capacitor (\(V_C\)):} \[ V_C = I X_C = (0.754)(212) \approx 159.8 \, \mathrm{V} \]

Quick Tip: For AC circuits, always consider phase differences between components while calculating the resultant voltage or current.

1.Dielectric Materials:

Dielectric materials are insulating materials that can be polarized when placed in an electric field. They do not conduct electricity but help in storing electrical energy by reducing the effective electric field.

2.Electric Susceptibility:

Electric susceptibility ($\chi_e$) of a medium is a dimensionless measure of the ease with which the medium becomes polarized in response to an applied electric field. It is defined as: \[ \chi_e = \frac{P}{\epsilon_0 E} \] where $P$ is the polarization, $\epsilon_0$ is the permittivity of free space, and $E$ is the electric field intensity.

3.Relative Permittivity:

Relative permittivity ($\epsilon_r$), also called the dielectric constant, is the ratio of the permittivity of the medium ($\epsilon$) to the permittivity of free space ($\epsilon_0$). It is expressed as: \[ \epsilon_r = \frac{\epsilon}{\epsilon_0} \]

4.Relation between $\chi_e$ and $\epsilon_r$:
The relative permittivity and electric susceptibility are related as: \[ \epsilon_r = 1 + \chi_e \]

Quick Tip: Dielectrics are key components in capacitors, where they enhance the storage capacity by reducing the effective electric field between the plates.

1.Drift Velocity Relation:

Drift velocity ($v_d$) is the velocity of electrons due to an applied electric field. The current ($I$) flowing in a conductor is related to the drift velocity by the formula: \[ I = n e A v_d \] where: - $n$ = number of free electrons per unit volume, - $e$ = charge of an electron ($1.6 \times 10^{-19} \, \mathrm{C}$), - $A$ = cross-sectional area of the wire, - $v_d$ = drift velocity.

From the equation: \[ v_d = \frac{I}{n e A} \]

2. Calculation of Drift Velocity:
Substituting the given values: \[ v_d = \frac{3.0}{(8.5 \times 10^{28})(1.6 \times 10^{-19})(2.0 \times 10^{-6})} \] \[ v_d = 1.1 \times 10^{-4} \, \mathrm{m/s} \]

3.Time Taken for Drift ($t$):The time taken for electrons to drift the length of the wire is given by: \[ t = \frac{\text{Length of wire}}{\text{Drift velocity}} \] \[ t = \frac{3.0}{1.1 \times 10^{-4}} \] \[ t \approx 27,273 \, \mathrm{seconds} \approx 7.6 \, \mathrm{hours} \]

Quick Tip: Drift velocity is extremely small compared to the speed of the electric signal propagation, which occurs almost instantaneously throughout the conductor.