Q.24
Which one of the following shows the CORRECT relationship among velocity of light in a medium (\( v \)), permittivity of medium (\( \varepsilon \)) and magnetic permeability of medium (\( \mu \))?
v = 1 / √(με), so the correct option is (D).
Introduction
Understanding the velocity of light in a material medium is essential for mastering electromagnetism, optics and competitive exam physics.
The speed of an electromagnetic wave depends on the electrical permittivity (ε) and magnetic permeability (μ) of the medium, and Maxwell’s equations give a precise mathematical relation between these quantities.
Correct relationship from Maxwell’s theory
From the electromagnetic wave equation derived from Maxwell’s equations, the speed of propagation of an EM wave in any homogeneous medium is
v = 1 / √(με).
- μ is the magnetic permeability of the medium, which measures how the medium responds to magnetic fields.
- ε is the electric permittivity of the medium, which measures how the medium permits electric field lines.
Thus, if either μ or ε increases, the denominator √(με) increases, so the velocity v decreases, which matches the fact that light generally travels slower in optically denser media.
Therefore, option (D), v = 1 / √(με), is the only correct expression for the velocity of light in a medium in terms of permittivity and permeability.
Why the other options are incorrect
Option (A): v = 1 / (με)
In this expression the product με is not under a square root, so the dimensions become T²L⁻² instead of LT⁻¹, which is required for velocity.
Dimensionally this represents the square of a time per length squared, not a speed, so option (A) is incorrect.
Option (B): v = 1 / (με)²
Here με is squared in the denominator, giving dimensions that are even farther from those of velocity, effectively proportional to T⁴L⁻⁴.
There is no theoretical basis in Maxwell’s equations for a fourth power of με, so option (B) is physically meaningless in this context.
Option (C): v = 1 / (με)⁻²
Using algebra, (με)⁻² = 1 / (με)², so v = 1 / (με)⁻² = (με)², which again has completely wrong dimensions for speed.
This option effectively states that velocity increases with the square of με, opposite to the physical trend that higher μ or ε slows down light, so option (C) is also incorrect.
Key exam takeaway
Remember the compact formula v = 1 / √(με), and in vacuum, μ = μ0, ε = ε0, giving c = 1 / √(μ0ε0).
Any expression for the speed of light must be inversely proportional to √(με); if you see με without a square root or with higher powers, it is dimensionally and physically incorrect.
