Problem 2138 - JEE Physics

Question

A new system of unit is evolved in which the values of \(\mu_{0}\) and \(\varepsilon_{0}\) are 2 and 8 respectively. Then the speed of light in this system will be (A) \(0.25\) (B) \(0.5\) (C) \(0.75\) (D) 1

Step-by-Step Solution

Answer

The speed of light in the new system of units can be found using the formula \(c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}\). Substituting the given values of \(\mu_0 = 2\) and \(\varepsilon_0 = 8\), we get \(c = \frac{1}{\sqrt{(2)(8)}} = \frac{1}{\sqrt{16}} = \frac{1}{4}\). The correct answer is (A) \(0.25\).

Step 1: Write down the formula for the speed of light in terms of permeability and permittivity constants

In this step, we will write down the formula relating the speed of light to the permeability and permittivity constants: \(c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}\).

Step 2: Substitute the given values

Now we will substitute the given values of \(\mu_0\) and \(\varepsilon_0\) into the formula: \(c = \frac{1}{\sqrt{(2)(8)}}\).

Step 3: Calculate the speed of light

Let's simplify the expression to find the value of the speed of light: \(c = \frac{1}{\sqrt{16}} = \frac{1}{4}\).

Step 4: Choose the correct answer from the options

Since the speed of light is \(\frac{1}{4}\), the correct answer is (A) \(0.25\).

Key Concepts

Understanding the Permeability Constant

The permeability constant, also known as the magnetic constant, measures how a material can support the formation of a magnetic field. It's essential for understanding how magnetic fields interact with materials.

• Symbol: Often denoted by \( \mu_0 \).
• Role in Physics: In a vacuum, \( \mu_0 \) is a fundamental constant used to calculate magnetic forces in the equations of electromagnetism.
• Use in Speed of Light Formula: The permeability constant is crucial for the speed of light equation: \( c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} \). It interacts with permittivity to determine how fast electromagnetic waves travel through a medium.

In the given exercise, changing the value of \( \mu_0 \) affects the speed of light in the new unit system. Remember that this doesn't change the nature of light, just how we measure it.

Exploring the Permittivity Constant

The permittivity constant, or electric constant, indicates how an electric field affects, and is affected by, a dielectric medium. This concept is vital for understanding electrical interactions.

• Symbol: Represented as \( \varepsilon_0 \).
• Functionality: It defines the ability of a material to permit electric field lines. In free space, it serves as a reference point for measuring permittivity.
• Connection to Speed of Light: In the speed of light formula \( c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} \), \( \varepsilon_0 \) is paired with \( \mu_0 \) to determine light's velocity in a medium.

For the problem here, assigning a new value to \( \varepsilon_0 \) means adjusting how light propagates in this new system. It's a pivotal part of how we calculate and understand wave speed in different environments.

Unit System Conversion and Its Importance

Unit system conversion isn't just about changing numbers. It involves understanding how these conversions impact the relationships between physical constants and measurements.

• Why It's Necessary: Different systems may use different base units (like meters or feet), affecting calculations. Conversion ensures consistency and accuracy.
• In the Exercise: The problem gives new values for \( \mu_0 \) and \( \varepsilon_0 \), essentially creating a new unit system. Without conversion, this would radically alter our calculations.
• Impact on Formulas: Adjusting unit values impacts directly on formulas such as the speed of light equation, ensuring that the derived values remain useful and understandable.

By understanding unit conversion, we're able to better grasp how changes in measurement can influence our comprehension of fundamental physics principles. In this exercise, it's how new insights emerge from adjusted constants.