JEE Challenger
More from Units and Measurements
PhyEasyAdv 2018 • Paper 1

Dimensional Relationship Between Electric and Magnetic Fields

Comprehension Passage

In electromagnetic theory, electric and magnetic phenomena are interconnected. Consequently, the dimensions of electric and magnetic quantities are interrelated. In the following problems, [E][E] and [B][B] denote the dimensions of electric and magnetic fields, respectively, whereas [ϵ0][\epsilon_0] and [μ0][\mu_0] denote the dimensions of free space permittivity and permeability, respectively. [L][L] and [T][T] represent the dimensions of length and time, respectively. SI units are used throughout.

Determine the correct relation between the dimensions [E][E] and [B][B].

Options

A

[E]=[B][L][T][E] = [B] [L] [T]

B

[E]=[B][L]−1[T][E] = [B] [L]^{-1} [T]

C

[E]=[B][L][T]−1[E] = [B] [L] [T]^{-1}

Correct
D

[E]=[B][L]−1[T]−1[E] = [B] [L]^{-1} [T]^{-1}

Topics & Concepts

Units and MeasurementsElectromagnetic WavesElectric Field

Step-by-Step Solution

To determine the relationship between the dimensions of the electric field [E][E] and the magnetic field [B][B], we can consider the Lorentz force acting on a particle with electric charge qq moving with velocity v⃗\vec{v}:

F⃗=q(E⃗+v⃗×B⃗)\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})

Since both terms inside the parenthesis represent quantities that can be added together, they must have the same physical dimensions:

[E]=[v×B]=[v][B][E] = [v \times B] = [v] [B]

The dimension of velocity [v][v] in terms of length [L][L] and time [T][T] is:

[v]=[L][T]−1[v] = [L] [T]^{-1}

Substituting this into the dimensional relation gives:

[E]=[B][L][T]−1[E] = [B] [L] [T]^{-1}

Alternatively, for an electromagnetic wave propagating in free space, the magnitudes of the electric and magnetic fields are related by the speed of light cc:

c=EB  ⟹  [E]=[c][B]c = \frac{E}{B} \implies [E] = [c] [B]

Since [c]=[L][T]−1[c] = [L][T]^{-1}, we similarly obtain:

[E]=[B][L][T]−1[E] = [B] [L] [T]^{-1}

Therefore, the correct option is (C).

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