Question

Electromagnetic waves travelling in a medium which has relative permeability \(1.3\) and relative permittivity \(2.14\) speed of electromagnetic waves in this medium will be (A) \(3.6 \times 10^{8} \mathrm{~m} / \mathrm{s}\) (B) \(1.8 \times 10^{8} \mathrm{~m} / \mathrm{s}\) (C) \(1.8 \times 10^{6} \mathrm{~m} / \mathrm{s}\) (D) \(13.6 \times 10^{6} \mathrm{~m} / \mathrm{s}\)

Short answer

The short answer is: The speed of electromagnetic waves in the medium will be (B) \(1.8 \times 10^8 ~\mathrm{m} / \mathrm{s}\).

Step-by-step solution

Identify the formula for the speed of light in a medium

The formula for the speed of light in a medium is given by: \(v = \frac{c}{\sqrt{\mu_r \epsilon_r}}\), where \(v\) is the speed of light in the medium, \(c\) is the speed of light in vacuum (\(3 \times 10^8 \mathrm{~m} / \mathrm{s}\)), \(\mu_r\) is the relative permeability, and \(\epsilon_r\) is the relative permittivity.

Plug in the given values

Now, plug in the values of relative permeability and relative permittivity given in the problem: \(\mu_r = 1.3\) \(\epsilon_r = 2.14\)

Calculate the speed of light in the medium

Substitute the values of \(\mu_r\) and \(\epsilon_r\) into the formula: \(v = \frac{3 \times 10^8}{\sqrt{1.3 \times 2.14}}\) Now, compute the value of \(v\): \(v \approx 1.8 \times 10^8 ~\mathrm{m} / \mathrm{s}\)

Compare the result with the given options

The calculated value of the speed of electromagnetic waves in the medium is approximately \(1.8 \times 10^8 ~\mathrm{m} / \mathrm{s}\). This matches option (B). Hence, the correct answer is (B) \(1.8 \times 10^8 ~\mathrm{m} / \mathrm{s}\).

Key concepts

Relative Permeability

Relative permeability (\( \mu_r \)) is a measure of how a material can support the formation of a magnetic field in its body. It's a dimensionless quantity that compares the magnetic permeability of a medium to the magnetic permeability of vacuum. The permeability of vacuum is considered as the baseline and has a value of one.

• If the relative permeability is greater than one, the medium can store more magnetic energy compared to a vacuum.
• If it's less than one, the medium is less effective at storing magnetic energy than a vacuum.

In the context of electromagnetic waves, relative permeability plays a vital role in determining the speed at which these waves travel through different materials. The value given in the exercise, which is 1.3, indicates that the medium is slightly more effective at propagating magnetic fields compared to a vacuum.
The speed of electromagnetic waves in such a medium is affected by this property, as it directly gets incorporated into the formula mentioned earlier. Understanding relative permeability helps in designing materials with desired magnetic properties for various applications, such as in transformers, inductors, and microwave devices.

Relative Permittivity

Relative permittivity (\( \epsilon_r \)), also known as the dielectric constant, refers to how much a material can store electrical energy in an electric field compared to a vacuum. Like relative permeability, relative permittivity is a unitless ratio that provides insight into how electric fields interact with different substances.

• A higher relative permittivity means that the material can store more electric energy compared to a vacuum.
• A lower value indicates lesser capability.

In the exercise, the given relative permittivity of 2.14 suggests that the medium is more capable of storing electric energy than a vacuum. The effect of this property is crucial in the computation of the speed of light in a medium, as seen in the equation provided in the step-by-step solution.
By enhancing our understanding of relative permittivity, we improve how we use materials in electronics, such as in capacitors and insulators, which rely on a material's ability to store and manage electric fields efficiently. It also plays an essential role in the science of optics, affecting the behavior of lenses and filters.

Speed of Light in Vacuum

The speed of light in a vacuum (\( c \)) is one of the fundamental constants in physics, with an approximate value of \(3 \times 10^8\) meters per second. This constant speed is due to the absence of obstacles or interactions in a vacuum that might slow down light.
Light, which is an electromagnetic wave, travels fastest in a vacuum because there is no matter to obstruct its propagation.
When light moves from a vacuum into another medium, its speed decreases. This change in speed depends on the optical density of the new medium, which is determined by its relative permeability and permittivity.

• The tighter the atomic and molecular structure of the medium, the slower the speed.
• Light's speed in a medium can be efficiently calculated using the formula \(v = \frac{c}{\sqrt{\mu_r \epsilon_r}}\).

Understanding the speed of light in a vacuum is crucial because it serves as a basis for comparison when studying light propagation in other media. This parameter is instrumental across various fields, from engineering to astronomy, enabling precise calculations and helping us understand phenomena ranging from straightforward optics to the relativistic effects in physics.