Question

Fernando detects the electric field from an isotropic source that is \(22 \mathrm{km}\) away by tuning in an electric field with an rms amplitude of \(55 \mathrm{mV} / \mathrm{m} .\) What is the average power of the source?

Short answer

The average power of the source is approximately 24.45 kW.

Step-by-step solution

Understand the Relationship

For an isotropic source, the power spreads out uniformly in all directions. The power in the electric field at a distance is related to the intensity, which is given by the Poynting vector. The average power is given by integrating the intensity over a sphere of radius equal to the distance.

Identify the Variables

We have the electric field rms amplitude, which is \(E_{rms} = 55\; \text{mV/m} = 0.055\; \text{V/m}\) and the distance to the source is \(r = 22\; \text{km} = 22,000\; \text{m}\).

Use the Poynting Vector Formula

The Poynting vector \(S\) represents the intensity and is given by \(S = \frac{E_{rms}^2}{2Z_0}\), where \(Z_0 = 377\; \Omega\) (impedance of free space). Substitute \(E_{rms} = 0.055\; \text{V/m}\) to find \(S\).\[ S = \frac{(0.055)^2}{2 \times 377} \approx 4.02 \times 10^{-6}\; \text{W/m}^2 \]

Calculate the Total Power

The total power \(P\) radiated by the source is the intensity \(S\) times the area of the sphere. The area \(A\) of a sphere is \(4\pi r^2\).\[ A = 4\pi (22000)^2 \approx 6.08 \times 10^9\; \text{m}^2 \]Thus, power \(P\) is:\[ P = S \cdot A \approx 4.02 \times 10^{-6}\; \text{W/m}^2 \times 6.08 \times 10^9\; \text{m}^2 \approx 24.45\; \text{kW} \]

Conclusion

The average power produced by the isotropic source is 24.45 kW.

Key concepts

Poynting Vector

The Poynting vector is a fundamental concept in electromagnetism that describes the energy flow per unit area in an electromagnetic field. Think of it as a way to measure how much power is radiating through a given area from an electromagnetic source. For physicists, this is crucial because it connects the electric and magnetic fields to the power they transport.

For a plane electromagnetic wave, the Poynting vector \( S \) is expressed by the formula:

• \( S = \frac{E_{rms}^2}{2Z_0} \)

Here, \( E_{rms} \) is the root mean square value of the electric field, and \( Z_0 \) is the impedance of free space, which is approximately \( 377 \; \Omega \). This implies that the intensity, or energy flow, is partly a consequence of the electric field strength.

By using the Poynting vector, you can compute how much power passes through a particular surface in the field, which helps in understanding the behavior and dynamics of electromagnetic waves.

Isotropic Source

An isotropic source is a theoretical ideal where power is radiated uniformly in all directions. Imagine a perfect light bulb that emits light equally in every direction. This model makes it easier to analyze and calculate the distribution of power over a distance, as it simplifies the geometry by assuming symmetry in the emission.

In physics, analyzing an isotropic source involves considering how energy spreads out as you move away from the source. Because power is distributed uniformly, the intensity decreases with the square of the distance due to the enlargement of the area over which it's spread. This effect is often used in calculations involving electromagnetic waves and light, applying the inverse square law, which states:

• As distance \( r \) increases, intensity \( I \) decreases at a rate proportional to \( \frac{1}{r^2} \).

This principle is crucial to finding the total power output of an isotropic source by calculating the surface area of a sphere (\( 4\pi r^2 \)) at a given radius.

Physics Problem Solving

Solving physics problems can be a rewarding yet challenging endeavor. It requires a methodical approach to break down the concepts and apply the appropriate formulas accurately.

When tackling a problem like determining the average power from an isotropic source based on detected electric fields, analyze the following steps:

• Understand the relationships in the problem. Identify the fundamental physics concepts like the role of the Poynting vector and isotropic source assumptions.
• List all known variables, such as electric field amplitude and distance to the source. Ensure you are working with consistent units for calculations.
• Apply relevant mathematical formulas, like the Poynting vector formula and surface area of a sphere, to translate physical concepts into solvable equations.

Physics problem solving isn't just about getting an answer. It's about understanding how to use principles and equations accurately and developing insights into the natural phenomena they describe. With practice, anyone can improve their problem-solving skills by learning to approach problems systematically and logically.