Question Distribution & Your Strength

Chapter Questions / Marks Your Score (%) Priority Action
Electrostatics 4 / 10 85 Revise formulas
Magnetism 5 / 12 78 More MCQs
EMI & AC 6 / 14 60 Solve numericals
EM Waves 2 / 4 90 Quick revision
Optics 8 / 16 55 Concept videos
Dual Nature 4 / 8 70 Past papers
Atoms & Nuclei 2 / 5 88 Maintain pace
Semiconductors 2 / 5 65 Clarify basics

Source: CBSE Physics Sample Paper 2024-25 & Your Mock Test Analytics

Electric Charges & Fields – Tough Spot

Class 12 Physics • CBSE

Question

Using Gauss’s law, derive the electric field \(E\) at a distance \(r\) from an infinitely long straight wire carrying uniform line charge density \(\lambda\).

Progressive Hints

Try on your own first. Reveal hints one by one if stuck.

Hint 1: Gaussian surface

Choose a cylindrical Gaussian surface of radius \(r\) and length \(L\) coaxial with the wire.

Hint 2: Symmetry check

By cylindrical symmetry, \(E\) is radial and constant on the curved surface; flux through flat end-caps is zero.

Hint 3: Apply Gauss’s law

\(E(2\pi rL)=\lambda L/\varepsilon_0\) ⇒ \(E=\dfrac{\lambda}{2\pi\varepsilon_0 r}\).

Things to Consider

  • Flux is only through the curved cylindrical surface.
  • Do not cancel \(2\pi\); keep the full \(2\pi rL\) area.
  • Charge enclosed is \(\lambda L\), matching cylinder length.
  • Final field varies as \(1/r\); check your r-dependence.

Concept Notes

Gauss’s Law

Net electric flux through a closed surface equals enclosed charge divided by \(\varepsilon_0\). Powerful for high-symmetry cases.

Cylindrical Symmetry

For an infinite line charge, field magnitude depends only on radial distance; direction is radial outward (or inward for negative \(\lambda\)).

Resulting Field:

\(E=\dfrac{\lambda}{2\pi\varepsilon_0 r}\)

Moving Charges & Magnetism – Challenge

CBSE Class 12 Physics

Question

An electron describes a circle of radius \(r_0\) with speed \(v_0\) and period \(T_0\) in a uniform magnetic field \(B\). Predict qualitatively how the radius \(r\) and period \(T\) change if its speed is doubled to \(2v_0\).

Progressive Hints

Try first, then reveal hints one at a time.

Hint 1: Identify the forces

Magnetic (Lorentz) force supplies centripetal force: \(qvB = \dfrac{mv^{2}}{r}\).

Hint 2: Relate radius to speed

From the above, \(r = \dfrac{mv}{qB}\). For fixed \(B,\,q,\,m\), radius is directly proportional to speed.

Hint 3: Check the period

Angular frequency \( \omega = \dfrac{qB}{m}\) is speed-independent, so \(T = 2\pi/\omega\) remains \(T_0\). Radius doubles.

Things to Consider

  • Compare magnetic force with required centripetal force for uniform circular motion.
  • Decide which variables are fixed by the field \((B,q,m)\) and which vary with speed.
  • Period depends on angular frequency, not on linear speed.
  • Avoid mixing up how radius and period respond to speed change.

Concept Notes

Magnetic Force

A charge moving with velocity \( \mathbf{v}\) in a field \( \mathbf{B}\) experiences \( \mathbf{F}=q\,\mathbf{v}\times\mathbf{B}\) perpendicular to motion.

Uniform Circular Motion

When \( \mathbf{F}\) is always perpendicular to \( \mathbf{v}\), speed stays constant and the particle follows a circle with constant angular frequency.

Key Formulas:

\( qvB = \dfrac{mv^{2}}{r}\)   |   \( T = \dfrac{2\pi m}{qB}\)

Alternating Current – Quick Probe

Series LCR • Impedance & Reactance

Question

In a series LCR circuit, the RMS voltages across R, C and L are each 10 V. The capacitor is suddenly short-circuited. Estimate the new RMS voltage across the inductor.

Progressive Hints

Try to solve first. Reveal hints as needed.

Hint 1: Spot the condition

Equal voltages on R, L and C imply resonance, so \(X_L = X_C\) and circuit current is \(I = V/R\).

Hint 2: New impedance

Short-circuiting \(C\) removes \(X_C\). Net impedance becomes \(Z = \sqrt{R^{2}+X_L^{2}}\).

Hint 3: Re-compute voltage

Compare currents: \(I_{\text{before}} = V/R\), \(I_{\text{after}} = V/Z\). New inductor voltage is \(V_L' = I_{\text{after}}\,X_L\).

Things to Consider

  • Current must be recalculated after C is removed.
  • \(X_L\) stays unchanged; frequency and L are constant.
  • At resonance, \(R = V/I\); keep this value for later steps.
  • Voltage redistribution follows impedance, not supply change.

Concept Notes

Resonant Series LCR

At resonance \(X_L = X_C\); impedance is purely resistive and minimum, equal to \(R\).

Impedance After Change

Removing a reactive element adds its reactance vectorially: \(Z = \sqrt{R^{2}+X^{2}}\). Current and element voltages adjust accordingly.

Important Formulas

\(X_L = \omega L\)   •   \(Z = \sqrt{R^{2} + (X_L - X_C)^{2}}\)

Wave Optics – Interference Twist

CBSE Class 12 • Physics

Question

In a Young’s double-slit experiment, light of wavelengths 400 nm and 600 nm is used together. Find the smallest distance from the central bright fringe where both colours give a dark (minimum-intensity) fringe simultaneously.

Progressive Hints

Reveal hints one at a time only if you need them.

Hint 1: Start with the condition

For a dark fringe in YDS, the path difference must equal an integer multiple of the wavelength: \( \Delta = m\lambda \).

Hint 2: Simultaneous minima

Find integers \(m,n\) such that \(m\lambda_{1}=n\lambda_{2}\) with \(\lambda_{1}=400\text{ nm},\,\lambda_{2}=600\text{ nm}\).

Hint 3: Lowest orders

The least non-zero solution is \(3 \times 400 = 2 \times 600 = 1200\text{ nm}\). Hence \(m=3, n=2\). The required distance = \(m\beta_{1}=n\beta_{2}\), where \(\beta=\lambda D/d\).

Things to Consider

  • Use the destructive-interference condition, not the bright-fringe rule.
  • Ignore the trivial \(m=n=0\) case; it is the central maximum.
  • Find the lowest common multiple of the two wavelengths.
  • Translate fringe order to distance using \(\beta = \lambda D/d\).

Concept Notes

Path Difference & Minima

In a YDS experiment, dark fringes occur when \( \Delta = m\lambda \) with m = 1, 2, 3… .

Simultaneous Minima

Two wavelengths give a common dark fringe when integer orders satisfy \( m\lambda_{1}=n\lambda_{2} \).

Important Formula:

\( \displaystyle \beta = \frac{\lambda D}{d} \)  –  fringe width (distance between successive minima).

Nuclei – Energy Curve Insight

Binding Energy per Nucleon & Nuclear Stability

Question

From the binding-energy curve, nuclei W (190), X (90), Y (60) and Z (30) are marked. Which one favours (i) fission and (ii) fusion? State why.

Binding-energy curve (schematic)

Progressive Hints

Solve first, then reveal hints in order.

Hint 1: Peak Position

The BE/A curve peaks near A ≈ 56 with about 8.8 MeV per nucleon.

Hint 2: Heavy Side

Nuclei with A > 140 gain stability by splitting into fragments nearer the peak.

Hint 3: Light Side

Very light nuclei (A < 10) release energy when they fuse toward A ≈ 56.

Things to Consider

  • Compare each mass number with the 56-region peak.
  • Energy is released when BE/A increases after a reaction.
  • Heavy >140 ➜ fission; light <10 ➜ fusion trends.
  • Avoid picking the already most stable nucleus for fission.

Concept Notes

Binding Energy per Nucleon

Higher BE/A means stronger binding and greater nuclear stability. The maximum lies near iron (A≈56).

Predicting Reaction Type

A nucleus will fission or fusion if products sit higher on the BE/A curve, releasing the difference as energy.

Important Rule:

Energy released ≈ A × Δ(BE/A)

Electrostatic Potential & Capacitance – Composite Dielectric

Class 12 Physics • CBSE

Question

A parallel-plate capacitor has plate area \(A\) and separation \(d\). Derive its capacitance when (i) a dielectric slab of relative permittivity \(\kappa\) and thickness \(t\), and (ii) a metallic slab of thickness \(t\,(<d)\) are fully inserted between the plates. Which insertion yields the larger capacitance? Give reason.

Progressive Hints

Reveal one hint at a time.

Hint 1: Identify the model

Treat the system as two capacitors in series: the slab region and the remaining air gap.

Hint 2: Write individual capacitances

\(C_{\text{slab}}=\kappa \varepsilon_0 A / t\), \(C_{\text{air}}=\varepsilon_0 A /(d-t)\).
Use \(1/C=1/C_{\text{slab}}+1/C_{\text{air}}\).

Hint 3: Compare with metal insert

For a metal, \(\kappa\to\infty\) ⇒ effective gap \(d-t\).
Dielectric: \(C=\varepsilon_0A /(d-t+t/\kappa)\).
Since \(d-t+t/\kappa > d-t\), the metal insert gives higher capacitance.

Things to Consider

  • Series combination: \(1/C=1/C_1+1/C_2\).
  • Dielectric constant \(\kappa\) scales \(\varepsilon_0\) inside the slab.
  • For conductors, \(\kappa\to\infty\) ⇒ zero potential drop inside.
  • Avoid mixing parallel and series models.

Concept Notes

Composite Capacitors in Series

When different media fill the space between plates along the field, treat each region as a separate capacitor connected in series.

Role of Dielectric Constant

The dielectric constant \(\kappa\) multiplies permittivity, reducing electric field and increasing capacitance by a factor \(\kappa\) for the same geometry.

Important Formulae:

\(C_{\text{dielectric}}=\dfrac{\varepsilon_0 A}{d - t + t/\kappa}\)
\(C_{\text{metal}}=\dfrac{\varepsilon_0 A}{d - t}\)

Electromagnetic Waves – Displacement Current

CBSE Class 12 Physics

Question

A parallel-plate capacitor of area 0.001 m² and plate gap 0.0001 m is being charged so that the voltage across it rises at \(10^{8}\,\text{V s}^{-1}\). Estimate the displacement current flowing through the gap.

Progressive Hints

Reveal hints step-by-step only if needed.

Hint 1: Key Formula

\(I_d = \varepsilon_0 A\,\dfrac{dV/dt}{d}\)

Hint 2: Substitute Values

Use \(\varepsilon_0 = 8.85\times10^{-12}\,\text{F m}^{-1}\), \(A = 0.001\,\text{m}^2\), \(d = 1\times10^{-4}\,\text{m}\), \(dV/dt = 10^{8}\,\text{V s}^{-1}\).

Hint 3: Final Evaluation

Calculate: \(I_d \approx 8.85\times10^{-4}\,\text{A}\). Choose the option closest to this value.

Common Pitfalls

  • Applying conduction current \(I = C\,dV/dt\) instead of displacement formula.
  • Forgetting to divide by plate separation \(d\).
  • Mixing electric field and voltage rate units.
  • Dropping powers of ten when multiplying constants.

Concept Notes

Maxwell’s Correction

In free space \( \nabla \times \mathbf{B} = \mu_0\left(\mathbf{J} + \varepsilon_0 \dfrac{\partial \mathbf{E}}{\partial t}\right) \). The extra term represents displacement current, ensuring continuity in time-varying fields.

Charging Capacitor

Between plates, no real charges move. A changing electric field substitutes with \(I_d\), making Ampère’s law valid for the loop linking the capacitor.

Important Formula:

\(I_d = \varepsilon_0\,A\,\dfrac{dE}{dt} = \varepsilon_0\,A\,\dfrac{1}{d}\dfrac{dV}{dt}\)

Dual Nature – Reading Photoelectric Graphs

Photoelectric Effect · Einstein Equation

Question

Photoelectric I–V curves A, B and C show different saturation currents and stopping potentials. Relate these differences to changes in (a) light intensity and (b) light frequency for the same metal surface.

Progressive Hints

Reveal hints one by one only when needed.

Hint 1: Getting Started

Saturation current depends on how many photoelectrons are emitted per second, i.e. on photon flux (intensity).

Hint 2: Key Approach

Einstein’s equation \( eV_0 = h\nu - \phi \) shows stopping potential \(V_0\) varies with frequency, not with intensity, for a given work function \( \phi \).

Hint 3: Almost There

Curves with equal \(V_0\) but larger saturation current were produced by the same frequency at higher intensity. Curves with higher \(V_0\) came from higher-frequency light.

Things to Consider

  • Saturation current ∝ intensity, not frequency.
  • Stopping potential reveals maximum electron energy and depends only on \( \nu \).
  • Work function \( \phi \) and threshold frequency remain fixed for the metal.
  • Vertical shift of I–V curve = intensity change; horizontal shift = frequency change.

Concept Notes

Einstein’s Photoelectric Equation

Energy of one photon \(h\nu\) is partitioned into work function \( \phi \) and kinetic energy \( eV_0 \). Thus \(V_0\) tracks frequency.

Shape of I–V Curve

Current rises quickly, saturates when every emitted electron is collected, then reverses sign past stopping potential.

Important Formula:

\( eV_0 = h\nu - \phi \)

Semiconductors – Band Diagram & Bulb Test

CBSE Class 12 Physics

Question

Sketch the energy-band diagram of a P-type semiconductor at (a) 0 K and (b) room temperature. For the circuit shown, will the bulb glow when the switch is (i) open and (ii) closed? Justify.

Circuit with PN diode, switch and bulb

Progressive Hints

Try on your own first. Reveal hints one by one if needed.

Hint 1 • Getting Started

At 0 K the valence band is full and no holes exist; mark the Fermi level just above it.

Hint 2 • Key Approach

At room temperature acceptor atoms ionise, create holes, and pull the Fermi level nearer the valence band.

Hint 3 • Almost There

The diode conducts only when its P-side (anode) is at higher potential. Check polarity and confirm that current flows—and the bulb glows—only with switch closed.

Things to Consider

  • Locate the Fermi level for a P-type crystal at different temperatures.
  • Remember: holes appear when acceptor levels donate electrons.
  • Forward bias ⇒ P-side positive, depletion region narrows, current flows.
  • Bulb needs a complete circuit; open switch breaks the path.

Concept Notes

Energy Bands in P-Type

Acceptor level lies just above valence band. Ionisation creates holes, shifting Fermi level slightly above the valence band as temperature rises.

PN Diode Biasing

Under forward bias, majority carriers cross the junction and current flows; reverse bias widens depletion region and blocks current.

Important Rule:

Forward conduction when \(V_{\text{anode}}-V_{\text{cathode}} > V_B\), where \(V_B≈0.7\text{ V (Si)}\).

Current Electricity – Heating Element Puzzle

Class 12 • CBSE Physics

Question

A heater draws 10 A from a 100 V battery whose internal resistance is 1 Ω at 20 °C. When the heater reaches 320 °C, calculate the power dissipated inside the battery. Take the temperature coefficient of resistance \( \alpha = 3.7 \times 10^{-4}\,\text{°C}^{-1} \).

Progressive Hints

Solve first, then reveal hints one by one.

Hint 1: Cold resistance

Include the source resistance: \(R_0 = \dfrac{V}{I} - r\).

Hint 2: Hot resistance

Use temperature rise \( \Delta T = 300\,^\circ\text{C} \): \(R_T = R_0[1 + \alpha \Delta T]\).

Hint 3: Current & power

Find \(I = \dfrac{V}{R_T + r}\), then battery loss \(P = I^{2} r\).

Things to Consider

  • Ohm’s law assumes constant resistance; temperature breaks this limit.
  • The battery’s internal resistance \( r \) is always in series.
  • Use \( \alpha \) with \( \Delta T \) in °C for consistency.
  • Required power is \( I^{2} r \), not the heater’s power.

Concept Notes

Temperature Coefficient of Resistance

For small ranges, \(R = R_0(1 + \alpha \Delta T)\); resistance rises with temperature for metals.

Internal Resistance & Joule Heating

A real battery behaves as \(E\) in series with \(r\); heat inside it equals \(I^{2} r\), reducing useful output.

Key Formula:

\(R_0 = \dfrac{V}{I} - r\)

Electromagnetic Induction – Generator Math

CBSE Class 12 Physics

Question

In an AC generator, a coil with N turns and area A spins at angular speed ω in a uniform magnetic field B. Derive the instantaneous emf and name the energy source that keeps the generator running.

[No diagram provided]

Progressive Hints

Reveal hints one at a time as you need them.

Hint 1: Magnetic Flux

Write flux through the rotating coil: \( \Phi(t)=N B A \cos \omega t \).

Hint 2: Apply Faraday’s Law

Use \( \varepsilon(t)= -\dfrac{d\Phi}{dt} \) to differentiate the flux with respect to time.

Hint 3: Final Expression & Source

After differentiating, \( \varepsilon(t)=N B A \omega \sin \omega t\). Peak emf \( \varepsilon_0=N B A \omega\). The coil is driven by mechanical energy from a turbine or engine.

Things to Consider

  • Keep the negative sign in Faraday’s law when differentiating.
  • Emf varies sinusoidally; it is not constant.
  • Axis of rotation must be perpendicular to B for maximum flux change.
  • Mechanical work converts to electrical energy—verify energy conservation.

Concept Notes

Faraday’s Law

Induced emf is proportional to the negative time rate of change of magnetic flux through a circuit.

Rotational emf

A coil rotating in a uniform field produces a sinusoidal emf because the projected area facing the field varies as \(\cos \omega t\).

Important Formula:

\( \varepsilon(t)=\varepsilon_0 \sin \omega t,\; \varepsilon_0=N B A \omega \)

Question with Hints and Concept Notes

Ray Optics – Telescope Calculations

Question

A refracting telescope has an objective of focal length 15 m and an eyepiece of 1 cm.
(a) Find its angular magnification in normal adjustment.
(b) Calculate the diameter of the Moon’s image formed by the objective lens.

No diagram provided

Progressive Hints

Try to solve the problem on your own first. If you need help, reveal the hints one by one.

Hint 1: Getting Started

Convert \(f_e = 1\,\text{cm}\) to metres and remember the final image is at infinity in normal adjustment.

Hint 2: Key Approach

Use \(M = f_o / f_e\). With \(f_o = 15\,\text{m}\) and \(f_e = 0.01\,\text{m}\), find \(M\).

Hint 3: Almost There

The Moon’s angular diameter is \(\theta \approx D/R = 3.48\times10^{6} / 3.8\times10^{8}\,\text{rad}\). Image diameter: \(y = f_o \theta\).

Things to Consider

  • Use small-angle approximation: \( \sin\theta \approx \theta\) (radians).
  • Only the objective decides image size; tube length is not used.
  • Keep focal lengths in the same unit system throughout.
  • Express answers with proper significant figures and units.

Concept Notes

Angular Magnification

For an astronomical telescope in normal adjustment, \(M = -\dfrac{f_o}{f_e}\). The sign indicates an inverted image; magnitude is used.

Image Formation

A distant object subtends angle \( \theta \). The objective forms a real image of height \(y = f_o \theta\) at its focal plane.

Important Formula:

\( \theta_{\text{Moon}} \approx \dfrac{D_{\text{Moon}}}{R_{\text{orbit}}} \)

Atoms – Spectral Lines Riddle

Bohr Model & Atomic Spectra

Question

Why does hydrogen, with only one electron, exhibit many spectral lines in its emission spectrum?

Progressive Hints

Try to solve the problem on your own first. If you need help, reveal the hints one by one.

Hint 1: Getting Started

A discharge tube holds countless hydrogen atoms, each with its own electron.

Hint 2: Key Approach

Different atoms get excited to different Bohr levels \(n = 2,3,4,\dots\).

Hint 3: Almost There

Each electron can drop between two levels, releasing photons of energy \(E = 13.6\,(1/n_1^2-1/n_2^2)\:\text{eV}\), creating many lines.

Things to Consider

  • One photon line = one electron transition.
  • Sample contains billions of atoms, not a single atom.
  • Bohr energies follow \(E_n\propto -1/n^2\).
  • Series (Lyman, Balmer, Paschen) arise from different lower levels.

Concept Notes

Bohr Energy Levels

Allowed orbits have fixed energies; electrons jump between them by absorbing or emitting photons.

Transition Rules

Downward jumps produce emission lines; energy difference sets photon frequency \( \nu = \Delta E/h \).

Important Formula/Rule:

\(E_n = -\dfrac{13.6\ \text{eV}}{n^2}\;\;\;\Delta E = 13.6\Bigl(\dfrac{1}{n_1^2}-\dfrac{1}{n_2^2}\Bigr)\)

Key Take-aways

Constants Cheat-Sheet

Write c, h, e, μ0, ε0 and NA on the answer sheet margin for instant recall.

Formula First, Numbers Next

Always state the equation before substitution to secure partial marks.

Dimensional Cross-Check

Verify unit consistency to spot errors quickly, especially in MCQs.

Smart Time Split

45 min (A+B), 60 min (C), 35 min (D), 40 min (E); keep last 10 min for review.

Attempt Low-Risk First

Start with sure-shot one-markers to build momentum and confidence.

Mental Math Matters

Practice quick powers-of-ten estimates; calculators are not allowed.