Sign in
12thPass logo
New chatPYQ LibraryDoubtsRank report
Sign in to see Recents

Your guest activity stays on this device

Sign in to save progress →
Sign in

Electromagnetic Waves question

2018 · Shift 1 · Q53
Guest · filters and generic practice availableBrowsing as a guest · PYQ filters and generic practice are available. Sign in only for personalised features and saved progress.
  1. PYQ Library
  2. /JEE Advanced
  3. /Physics
  4. /Electromagnetic Waves
  5. /2018 · Shift 1 · Q53

Electromagnetic Waves question

2018 · Shift 1 · Q53

JEE AdvancedPhysicsElectromagnetic WavesMCQ+3 / −1
In electromagnetic theory, the electric and magnetic phenomena are related to each other. Therefore, the dimensions of electric and magnetic quantities must also be related to each other. In the questions below, [E][E][E] and [B][B][B] stand for dimensions of electric and magnetic fields respectively, while [ε0]\left[ {{\varepsilon _0}} \right][ε0​] and [μ0]\left[ {{\mu _0}} \right][μ0​] stand for dimensions of the permittivity and permeability of free space respectively. [L]\left[ L \right][L] and [T]\left[ T \right][T] are dimensions of length and time respectively. All the quantities are given in SISISI units. The relation between [E][E][E] and [B][B][B] is
  1. A
    [E]=[B][L][T]\left[ E \right] = \left[ B \right]\left[ L \right]\left[ T \right][E]=[B][L][T]
  2. B
    [E]=[B][L]−1[T]\left[ E \right] = \left[ B \right]{\left[ L \right]^{ - 1}}\left[ T \right][E]=[B][L]−1[T]
  3. C
    [E]=[B][L][T]−1\left[ E \right] = \left[ B \right]\left[ L \right]{\left[ T \right]^{ - 1}}[E]=[B][L][T]−1
  4. D
    [E]=[B][L]−1[T]−1\left[ E \right] = \left[ B \right]{\left[ L \right]^{ - 1}}{\left[ T \right]^{ - 1}}[E]=[B][L]−1[T]−1
View written solutionFree

Correct answer: C

Step-by-step Derivation

  1. Identify the governing physical principle. In the theory of electromagnetic waves, the magnitudes of the electric field (EEE) and the magnetic field (BBB) are related through the speed of light in a vacuum, ccc. The relationship is given by: E=cBE = cBE=cB where ccc is the speed of light.

  2. Express the relationship in terms of dimensions. To find the relation between the dimensions of EEE and BBB, we write the dimensional equation corresponding to the formula above: [E]=[c][B]\left[ E \right] = \left[ c \right]\left[ B \right][E]=[c][B]

  3. Determine the dimensions of the speed of light, ccc. The speed of light, ccc, is a speed. The dimension of any speed is length divided by time. In the given notation: [c]=[L][T]=[L][T]−1\left[ c \right] = \frac{{\left[ L \right]}}{{\left[ T \right]}} = \left[ L \right]{\left[ T \right]^{ - 1}}[c]=[T][L]​=[L][T]−1

  4. Substitute the dimensions of ccc into the dimensional equation. Now, we substitute the dimensions of ccc from Step 3 into the equation from Step 2: [E]=([L][T]−1)[B]\left[ E \right] = \left( {\left[ L \right]{{\left[ T \right]}^{ - 1}}} \right)\left[ B \right][E]=([L][T]−1)[B] Rearranging the terms, we get: [E]=[B][L][T]−1\left[ E \right] = \left[ B \right]\left[ L \right]{\left[ T \right]^{ - 1}}[E]=[B][L][T]−1

Alternative Method: Using the Lorentz Force

  1. Recall the Lorentz force equation. The total force (Lorentz force) on a charge qqq moving with velocity vvv in the presence of an electric field EEE and a magnetic field BBB is given by: F⃗=qE⃗+q(v⃗×B⃗)\vec F = q\vec E + q(\vec v \times \vec B)F=qE+q(v×B)

  2. Apply the principle of dimensional homogeneity. For this equation to be dimensionally consistent, the dimensions of the electric force term (qE⃗q\vec EqE) must be the same as the dimensions of the magnetic force term (q(v⃗×B⃗)q(\vec v \times \vec B)q(v×B)). [qE]=[qvB]\left[ {qE} \right] = \left[ {qvB} \right][qE]=[qvB] Note that the cross product itself doesn't change the dimensions, only the magnitude (via a factor of sin⁡θ\sin\thetasinθ, which is dimensionless).

  3. Solve for the relationship between [E] and [B]. We can cancel the dimension of charge, [q][q][q], from both sides: [E]=[v][B]\left[ E \right] = \left[ v \right]\left[ B \right][E]=[v][B] The dimension of velocity, [v][v][v], is length per time, which is [L][T]−1\left[ L \right]{\left[ T \right]^{ - 1}}[L][T]−1.

  4. Substitute the dimensions of velocity. [E]=([L][T]−1)[B]\left[ E \right] = \left( {\left[ L \right]{{\left[ T \right]}^{ - 1}}} \right)\left[ B \right][E]=([L][T]−1)[B] [E]=[B][L][T]−1\left[ E \right] = \left[ B \right]\left[ L \right]{\left[ T \right]^{ - 1}}[E]=[B][L][T]−1

Conclusion

Both methods lead to the same dimensional relationship: [E]=[B][L][T]−1\left[ E \right] = \left[ B \right]\left[ L \right]{\left[ T \right]^{ - 1}}[E]=[B][L][T]−1. Comparing this result with the given options:

A: [E]=[B][L][T]\left[ E \right] = \left[ B \right]\left[ L \right]\left[ T \right][E]=[B][L][T] - Incorrect

B: [E]=[B][L]−1[T]\left[ E \right] = \left[ B \right]{\left[ L \right]^{ - 1}}\left[ T \right][E]=[B][L]−1[T] - Incorrect

C: [E]=[B][L][T]−1\left[ E \right] = \left[ B \right]\left[ L \right]{\left[ T \right]^{ - 1}}[E]=[B][L][T]−1- Correct

D: [E]=[B][L]−1[T]−1\left[ E \right] = \left[ B \right]{\left[ L \right]^{ - 1}}{\left[ T \right]^{ - 1}}[E]=[B][L]−1[T]−1 - Incorrect

Thus, the correct option is C.

PreviousNext

More from Electromagnetic Waves

  • In terms of potential difference V, electric current I, permittivity ε0​, permeability μ0​ and speed of light c, the dimensionally correct equation(s) is(are) :2015 · Multiple correct
  • A pulse of light of duration 100 ns is absorbed completely by a small object initially at rest. Power of the pulse is 30 mW and the speed of light is 3 × 108 ms − 1. The final momentum of the object is2013 · MCQ
  • A cube of unit volume contains 35×107 photons of frequency 1015 Hz. If the energy of all the photons is viewed as the average energy being contained in the electromagnetic waves within the same volume, then the…2025 · Numerical
  • The electric field associated with an electromagnetic wave propagating in a dielectric medium is given by E=30(2x^+y^​)sin[2π(5×1014t−3107​z)]Vm−1. Which of…2023 · Multiple correct
  • In electromagnetic theory, the electric and magnetic phenomena are related to each other. Therefore, the dimensions of electric and magnetic quantities must also be related to each other. In the questions below, [E] and [B] stand for…2018 · MCQ