Problem 2192 - JEE Physics

Question

At room temperature, if the relative permittivity of water is 80 and the relative permeability be \(0.0222\) then the velocity of light in water is \(\mathrm{m} / \mathrm{s}\) (A) \(2.5 \times 10^{8}\) (B) \(2.26 \times 10^{8}\) (C) \(3.5 \times 10^{8}\) (D) \(3 \times 10^{8}\)

Step-by-Step Solution

Answer

The velocity of light in water at room temperature, with a relative permittivity of 80 and relative permeability of 0.0222, can be calculated using the formula \(v = \frac{c}{\sqrt{\epsilon_r \mu_r}}\). Plugging in the given values, we find that the velocity of light in water is approximately \(2.26 \times 10^8\ \mathrm{m/s}\) (Option B).

Step 1: Identify the given values

We are given: Relative permittivity of water, \(\epsilon_r = 80\) Relative permeability of water, \(\mu_r = 0.0222\)

Step 2: Write the formula for velocity of light in a medium

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

Step 3: Plug in the given values and calculate the velocity

Now we plug in the given values and the constant speed of light in vacuum: \(v = \frac{3 \times 10^8}{\sqrt{80 \times 0.0222}}\) Compute the value under the square root: \(v = \frac{3 \times 10^8}{\sqrt{1.776}}\) Now, calculate the square root and divide: \(v \approx \frac{3 \times 10^8}{1.3329} \approx 2.25 \times 10^8\ \mathrm{m/s}\)

Step 4: Compare the calculated value with given options

Comparing our calculated value with the given options, we find that it is closest to: (B) \(2.26 \times 10^8\ \mathrm{m/s}\) So, the correct answer is (B) \(2.26 \times 10^8\ \mathrm{m/s}\).

Key Concepts

Understanding Relative Permittivity

Relative permittivity, also known as the dielectric constant, is a measurement of how much an electric field is weakened inside a medium. It's a key concept in understanding how different materials impact electromagnetic fields. When you place a material in an electric field, it becomes polarized. This means that the electric charges within the material align themselves in response to the external field. This effect reduces the overall field strength within the material compared to the strength in a vacuum.Relative permittivity is expressed with the symbol \(\epsilon_r\) and is a unitless number. The value of \(\epsilon_r\) typically starts from 1, which represents a vacuum, and increases with the material's ability to reduce the electric field. A higher relative permittivity indicates a better ability of the material to polarize, thus more effectively reducing the electric field strength.

• In this context, for water, \(\epsilon_r = 80\), meaning water significantly reduces the electric field compared to a vacuum.
• This high value explains why water is a good insulator and impacts the velocity of light as it passes through it.

Exploring Relative Permeability

Relative permeability is a measure of how much a material can support the formation of a magnetic field within itself. Similar to permittivity, it provides an understanding of how materials respond to magnetic fields. Every material responds differently to magnetic fields based on its properties.The symbol used for relative permeability is \(\mu_r\), and it is also a unitless number. A relative permeability of 1 indicates the material behaves like a vacuum in terms of magnetic field interaction. Materials with \(\mu_r > 1\) are attracted by magnetic fields, known as paramagnetic, while those with \(\mu_r < 1\) are repelled, known as diamagnetic.

• For water, the approximate value of \(\mu_r = 0.0222\) shows that water is a diamagnetic material.
• This value suggests that water slightly repels an external magnetic field, which is typical for non-magnetic substances.

Importance of Speed of Light in a Vacuum

The speed of light in a vacuum is one of the fundamental constants of nature. It is symbolized by \(c\) and has a value of approximately \(3 \times 10^8\) m/s. This speed, often referred to in scientific contexts, represents the maximum limit at which information or matter can travel. Light travels at its fastest when it is in a vacuum with no obstructing media. When light passes through any medium, such as water or glass, it slows down. The speed reduction is governed by the medium's properties, specifically its relative permittivity and permeability.Understanding the speed of light in a vacuum is essential for calculating how light interacts with different materials and for understanding various physical principles like refraction and optical illusions.

• The formula \(v = \frac{c}{\sqrt{\epsilon_r \mu_r}}\) demonstrates that the velocity of light depends directly on the speed of light in a vacuum.
• When light enters a medium with higher permittivity or permeability than a vacuum, it reduces its speed accordingly.