Problem 2146 - JEE Physics

Question

Energy density of an electromagnetic wave of intensity \(0.02 \mathrm{Wm}^{-2}\) is (A) \(6.67 \times 10^{-11} \mathrm{Jm}^{-3}\) (B) \(6 \times 10^{6} \mathrm{Jm}^{-3}\) (C) \(1.5 \times 10^{10} \mathrm{Jm}^{-3}\) (D) none of the above

Step-by-Step Solution

Answer

The energy density (u) of an electromagnetic wave with an intensity of \(0.02 \,\mathrm{Wm}^{-2}\) can be found using the formula \(I = u \times c\), where I is the intensity and c is the speed of light. Plugging in the given values, we get \(u = \frac{0.02}{3 \times 10^8}\), resulting in an energy density of \(u = 6.67 \times 10^{-11} \, \mathrm{Jm}^{-3}\). Therefore, the correct answer is (A) \(6.67 \times 10^{-11} \, \mathrm{Jm}^{-3}\).

Step 1: Write the formula relating intensity, energy density, and the speed of light

The formula relating intensity (I), energy density (u), and the speed of light (c) is: \[ I = u \times c \]

Step 2: Plug in given values and solve for energy density

We are given the intensity \(I = 0.02 \,\mathrm{Wm}^{-2}\) and the speed of light \(c = 3 \times 10^8 \, \mathrm{m/s}\). Plugging in the values into the formula, we have: \[ 0.02 = u \times (3 \times 10^8) \] To solve for energy density (u), we divide both sides by \(3 \times 10^8\): \[ u = \frac{0.02}{3 \times 10^8} \]

Step 3: Calculate the value of energy density

Performing the division, we get the value of energy density: \[ u = 6.67 \times 10^{-11} \, \mathrm{Jm}^{-3} \]

Step 4: Choose the correct option

Comparing the calculated value of energy density with the provided options, we find that the correct answer is: (A) \(6.67 \times 10^{-11} \, \mathrm{Jm}^{-3}\)

Key Concepts

Intensity of Electromagnetic Waves

The intensity of electromagnetic waves is an essential concept in understanding how energy propagates through space. It represents the power per unit area carried by the wave, measured in watts per square meter (W/m²).
In simple terms, intensity tells us how much electromagnetic energy is passing through a specific area over time. This is crucial when studying phenomena like light, microwaves, or X-rays found in electromagnetic waves.

• Power Density: Intensity quantifies the concentration of wave power, helping us assess potential impacts on materials it encounters.
• Relevance: Knowledge of intensity is vital, particularly in fields like telecommunications, where signal strength and safety standards are based on intensity measurements.

Recognizing intensity enables us to calculate other important properties like energy density, using the relationship with the speed of light.

Speed of Light

The speed of light is one of the most important constants in physics, defining the speed at which electromagnetic waves travel through a vacuum. It has the exact value of approximately \(3 \times 10^8 \) meters per second (m/s).
This constant allows us to interrelate various properties of electromagnetic waves.

• Universal Speed Limit: The speed of light is the maximum speed at which all energy, matter, and information in the universe can travel.
• Affects Time and Space: This speed influences theories, such as Einstein's relativity, which states that time and space are interwoven, affected by the speed of light.
• Relationship with Intensity and Energy Density: In electromagnetic waves, the speed of light has a direct connection to intensity and energy density, as captured in formulas like \( I = u \times c \).

Understanding this speed is fundamental for exploring how electromagnetic waves behave in different mediums and the universe.

Electromagnetic Wave Formula

Electromagnetic waves can be analyzed using essential formulas that bridge different wave properties. One key formula is \( I = u \times c \), where:

• \( I \) represents the intensity of the wave.
• \( u \) denotes the energy density.
• \( c \) stands for the speed of light.

This formula allows us to link intensity, energy density, and the speed of electromagnetic waves. Here is how it works:
- Calculating Energy Density: Given intensity and known speed of light, we rearrange the formula to find energy density, \( u = \frac{I}{c} \). This aids in discovering how much energy is contained per unit volume as waves travel.
- Practical Implications: Accurate energy density calculations from intensity using this formula help in determining energy transfer rates in systems like solar panels or broadcast signals.
The electromagnetic wave formula simplifies complex wave interactions, enabling efficient calculations and deeper insights into wave dynamics.