Correct Relationship Between Velocity of Light, Permittivity and Magnetic Permeability

Correct Answer: v = 1/√(με)

Understanding the Speed of Light in a Medium

Light is an electromagnetic wave consisting of mutually perpendicular oscillating electric and magnetic fields. The speed at which an electromagnetic wave propagates through a medium is determined by the electrical and magnetic properties of that medium.

The two important properties that control the propagation speed are the permittivity ε and the magnetic permeability μ of the medium. The correct relationship among these quantities is:

v = 1/√(με)

Here, v represents the velocity of light or any electromagnetic wave in the medium, ε represents the permittivity of the medium, and μ represents its magnetic permeability.

Correct Formula for the Velocity of Light in a Medium

The velocity of an electromagnetic wave in a material medium is given by:

v = 1/√(με)

This equation shows that the speed of light is inversely proportional to the square root of the product of permittivity and magnetic permeability.

In proportional form:

v ∝ 1/√(με)

Therefore, if the product με increases, the velocity of light decreases. If the product με decreases, the velocity of light increases.

Physical Meaning of Permittivity

Permittivity, represented by ε, describes the electrical response of a medium to an applied electric field. It indicates how the medium interacts with and responds to the electric component of an electromagnetic wave.

A medium with greater permittivity generally produces a lower propagation speed for an electromagnetic wave when other factors remain unchanged. This dependence appears directly in the denominator of the expression v = 1/√(με).

The SI unit of permittivity is farad per metre, written as F m⁻¹.

Physical Meaning of Magnetic Permeability

Magnetic permeability, represented by μ, describes how a medium responds to a magnetic field. Since an electromagnetic wave contains an oscillating magnetic field as well as an oscillating electric field, the magnetic properties of the medium also influence the wave velocity.

An increase in magnetic permeability, with other quantities remaining unchanged, decreases the speed of the electromagnetic wave because μ also appears in the denominator under the square root.

The SI unit of magnetic permeability is henry per metre, written as H m⁻¹.

How the Formula Follows from Electromagnetic Wave Theory

According to Maxwell’s electromagnetic theory, changing electric fields produce magnetic fields, while changing magnetic fields produce electric fields. These continuously varying fields sustain each other and propagate through space as an electromagnetic wave.

When Maxwell’s equations are combined to obtain the electromagnetic wave equation, the wave speed is found to depend on the permittivity and permeability of the medium. The resulting relationship is:

v = 1/√(με)

This result was historically important because, for free space, the calculated electromagnetic wave speed was found to be equal to the measured speed of light. This established that light itself is an electromagnetic wave.

Velocity of Light in Vacuum

For vacuum, the permittivity and permeability are represented by ε₀ and μ₀, respectively. Therefore, the speed of light in vacuum is:

c = 1/√(μ₀ε₀)

The numerical value of the speed of light in vacuum is approximately:

c = 3.0 × 10⁸ m s⁻¹

This is the maximum speed at which light propagates in vacuum.

Relationship with Relative Permittivity and Relative Permeability

For a material medium, the permittivity and permeability can be expressed as:

ε = ε_rε₀

and:

μ = μ_rμ₀

Here, ε_r is the relative permittivity of the medium and μ_r is its relative permeability.

Substituting these expressions into the wave velocity formula gives:

v = 1/√(μ_rμ₀ε_rε₀)

Using c = 1/√(μ₀ε₀), we obtain:

v = c/√(μ_rε_r)

This equation shows how the velocity of light in a medium is related to the speed of light in vacuum and the electromagnetic properties of the material.

Why Are μ and ε Under the Square Root?

The velocity of an electromagnetic wave does not vary directly as the reciprocal of μ or ε individually. Instead, it depends on the reciprocal square root of their product.

Therefore, doubling either μ or ε alone does not reduce the wave speed to one-half. For example, if the product με becomes four times its original value, the velocity becomes:

v_new = v_old/√4 = v_old/2

This inverse square-root dependence is essential when comparing different possible mathematical relationships in multiple-choice questions.

Dimensional Verification of the Relationship

The correctness of the equation can also be checked using dimensions. The product of magnetic permeability and permittivity has dimensions such that its square root has the dimensions of inverse velocity.

Therefore:

1/√(με)

has the dimensions of velocity, confirming that the equation is dimensionally consistent.

Final Answer

The correct relationship among the velocity of light in a medium (v), permittivity (ε), and magnetic permeability (μ) is:

v = 1/√(με)

The velocity of light in a medium is inversely proportional to the square root of the product of the medium’s permittivity and magnetic permeability. Therefore, a larger value of με results in a lower velocity of electromagnetic waves in that medium.