Due to presence of an em-wave whose electric component is given by E = 100 (ω t - kx)NC⁻¹$\mathrm{E} = 100\sin (\omega t - kx)\mathrm{NC}^{-1}$ , a cylinder of length 200~cm$200~\mathrm{cm}$ holds certain amount of em-energy inside it. If another cylinder of same length but half diameter than previous one holds the same amount of em-energy, the magnitude of the electric field of the corresponding em-wave should be modified as

Core Logic

Let's review the fundamental Maxwell's equations:

• Ampere - Maxwell Law relates the magnetic path integral to conduction current and displacement current:

• Faraday's Law of Induction states that changing magnetic flux induces an electromotive force (EMF):

• Gauss's Law for Electricity relates net electric flux to enclosed charge:

• Gauss's Law for Magnetism states that magnetic monopoles do not exist:

Step 1: Match Evaluation

The match configurations are:

• A arrow$\rightarrow$ IV
• B arrow$\rightarrow$ III
• C arrow$\rightarrow$ I
• D arrow$\rightarrow$ II

This perfectly corresponds to Option (3).

Pattern Recognition

Understand the integral geometries: Path integrals (line integrals ∮ · d l$\oint \cdot d\vec{l}$) correspond to circulating fields (induction laws like Ampere/Faraday). Surface integrals (flux integrals ∮ · d A$\oint \cdot d\vec{A}$) correspond to bounded charge states (Gauss laws).

Chapter Mix

Class 12 Physics: Electromagnetic Waves Class 12 Physics: Electrostatics Class 12 Physics: Magnetism and Matter

Related Formula

Core Logic

For an electromagnetic wave incident on a surface that is completely absorbed, the total momentum transferred is equal to the total energy transferred divided by the speed of light in vacuum (c$c$).

Step 1: Calculate Momentum

Given total energy E = 6.48 × 10⁵ ~J$E = 6.48 \times 10^5 \mathrm{~J}$. Speed of light c = 3 × 10⁸ ~m/s$c = 3 \times 10^8 \mathrm{~m/s}$.

Pattern Recognition

Always check for "complete absorption" versus "perfect reflection". For complete absorption, p = E/c$p = E/c$. For perfect reflection, p = 2E/c$p = 2E/c$.

Chapter Mix

Class 12 Physics: Electromagnetic Waves

Core Logic

Match each law with its corresponding mathematical expression (Maxwell's equations). (A) Gauss's law of magnetostatics: The net magnetic flux through any closed surface is zero. ∮ B · d a = 0$\oint \vec{B} \cdot \mathrm{d}\vec{a} = 0$ (Matches II). (B) Faraday's law of electromagnetic induction: The induced electromotive force in any closed circuit is equal to the negative of the time rate of change of the magnetic flux. ∮ E · d l = - ddt ∫ B · d a$\oint \vec{E} \cdot \mathrm{d}\vec{l} = -\frac{\mathrm{d}}{\mathrm{d}t} \int \vec{B} \cdot \mathrm{d}\vec{a}$ (Matches III). (C) Ampere's law: The line integral of the magnetic field around a closed loop is proportional to the electric current passing through the loop. ∮ B · d l = μ₀ I$\oint \vec{B} \cdot \mathrm{d}\vec{l} = \mu_0 I$ (Matches IV). (D) Gauss's law of electrostatics: The electric flux through any closed surface is proportional to the enclosed electric charge. ∮ E · d a = (1)/(ε₀) ∫ ρ dV$\oint \vec{E} \cdot \mathrm{d}\vec{a} = \frac{1}{\varepsilon_0} \int \rho \mathrm{d}V$ (Matches I).

Step 1: Final Match

A arrow$\rightarrow$ II B arrow$\rightarrow$ III C arrow$\rightarrow$ IV D arrow$\rightarrow$ I This matches option (4).

Pattern Recognition

These are the fundamental Maxwell equations in integral form. Memorizing their direct mappings guarantees quick marks.

Chapter Mix

Class 12 Physics: Electromagnetic Waves Class 12 Physics: Electromagnetic Induction

Related Formula

Core Logic

In an electromagnetic wave in free space, the magnitudes of the electric and magnetic fields are related by E₀ = c B₀$E_0 = c B_0$. Thus, B₀ = E₀ / C$B_0 = E_0 / C$.

The direction of wave propagation is given by the cross product of the electric field and magnetic field vectors: C = E × B$\hat{C} = \hat{E} \times \hat{B}$.

Step 1: Determine Wave Direction and Magnetic Field Direction

Given the phase term (ω t - kz)$(\omega t - kz)$, the wave propagates in the +z$+z$ direction, so C = k$\hat{C} = \hat{k}$. The electric field oscillates in the +x$+x$ direction, so E = i$\hat{E} = \hat{i}$. We know:

Since i × j = k$\hat{i} \times \hat{j} = \hat{k}$, the magnetic field must oscillate in the +y$+y$ direction (j$\hat{j}$).

Step 2: Construct Final Vector

The full magnetic field vector shares the same phase and applies the above amplitude and direction:

Pattern Recognition

Phase remains identical. Amplitude scales by 1/c$1/c$. Direction satisfies the right-hand triad ( E, B, v)$(\vec{E}, \vec{B}, \vec{v})$ where v = E × B$\vec{v} = \vec{E} \times \vec{B}$.

Chapter Mix

Class 12 Physics: Electromagnetic Waves

Related Formula

pᵣ = (I)/(c)$$p_r = \frac{I}{c}$$ (radiation pressure for completely absorbing surface)

Core Logic

Statement I: In an EM wave, energy is stored in both the electric and magnetic fields. The energy densities are equal: UE = UB$U_E = U_B$ because E = cB$E = cB$ and c² = (1)/(μ₀ ε₀)$c^2 = \frac{1}{\mu_0 \epsilon_0}$. This is a correct fact.

Statement II: Electromagnetic waves carry momentum (p = U/c$p = U/c$). When they strike a surface, they transfer this momentum, creating radiation pressure. This is also a correct fact.

Step 1: Final Evaluation

Since both facts are universally true theoretical properties of EM waves, both statements are correct.

Pattern Recognition

Standard NCERT theory. Energy density is strictly symmetric 50-50 between E and B. Momentum transfer →$\to$ pressure is the definitive signature of the particle-like nature (photons) of EM waves.

Chapter Mix

Class 12 Physics: Electromagnetic Waves