Question

A point source of electromagnetic radiation has an average output power of \(800 \mathrm{~W}\). The maximum value of electric field at a distance of \(4.0 \mathrm{~m}\) from the source is (A) \(64.7 \mathrm{Vm}^{-1}\) (B) \(57.8 \mathrm{Vm}^{-1}\) (C) \(56.72 \mathrm{Vm}^{-1}\) (D) \(54.77 \mathrm{Vm}^{-1}\)

Short answer

The maximum value of the electric field at a distance of 4.0 m from the point source of electromagnetic radiation is approximately 54.77 Vm^{-1} (Option D).

Step-by-step solution

Write down the given variables

Average output power (P) = 800 W Distance from the source (r) = 4.0 m

Find the intensity of radiation at the given distance

The intensity (I) of radiation is the power emitted per unit area (A). Since the radiation is emitted radially in all directions: Area (A) = 4πr^2 Now we can calculate the intensity (I) as: I = P / A

Calculate the intensity

Substitute the given values of power and distance into the formula and find the intensity: I = 800 / (4π × (4.0)^2) I = 200 / (π × 16)

Write down the formula for the Poynting vector

The Poynting vector (S) is given by: S = I = (1/2)ε₀cE_max^2 Where S is the Poynting vector, I is the intensity, ε₀ is the vacuum permittivity (\(8.854 × 10^{-12} \mathrm{F m}^{-1}\)), c is the speed of light \(3 × 10^8 \mathrm{ms}^{-1}\), and E_max is the maximum electric field.

Calculate the maximum electric field

Rearrange the Poynting vector formula to find the maximum value of the electric field: E_max^2 = (2I) / (ε₀c) Substitute the intensity, vacuum permittivity, and speed of light into the equation: E_max^2 = \(2 × (200 / (π × 16))\) / ( \(8.854 × 10^{-12} × 3 × 10^8\) ) E_max = \(\sqrt{25 / (π × 4)} × 3 × 10^5 Vm^{-1}\) E_max ≈ 54.77 Vm^{-1}

Choose the correct answer

Comparing the value of E_max with the options given in the problem, the maximum value of the electric field at a distance of 4.0 m from the source is: (D) 54.77 Vm^{-1}

Key concepts

Electric Field

In the context of electromagnetic radiation, the electric field is a vector field that describes the electrical force experienced by a charged particle at any point in space. It is an essential component of electromagnetic waves, such as light and radio waves. These waves are made up of oscillating electric and magnetic fields that propagate through space. The strength of the electric field at a point can change depending on the distance from the source of the radiation. For instance, in the original exercise, the electric field strength at a distance of 4.0 meters is given as an option, calculated to identify the correct value. The maximum value of the electric field, often denoted as \( E_{max} \), represents the peak field strength that occurs as the wave oscillates. This peak value is crucial for determining the intensity of the radiation and the power it can deliver.

Intensity of Radiation

The intensity of radiation refers to the power per unit area that the radiation delivers to a surface perpendicular to the direction of the wave. It is a measure of how much energy is passing through a certain area per unit time, typically expressed in watts per square meter (\( ext{W/m}^2 \)). In the original exercise, the intensity at a distance from a point source was calculated by dividing the total power emitted by the source by the surface area of a sphere centered around that source.

• The formula used is \( I = \frac{P}{A} \), where \( P \) is the power and \( A \) is the area.
• For a point source radiating uniformly in all directions, \( A \) is the surface area of a sphere, calculated as \( 4 \pi r^2 \).

This shows how the intensity diminishes as the distance from the source increases, since the energy is spread over a larger area.

Poynting Vector

The Poynting vector, named after the physicist John Henry Poynting, is a key concept in understanding electromagnetic waves. It represents the directional energy flux or the rate of energy transfer per unit area within an electromagnetic field. The Poynting vector is important for quantifying how much electromagnetic energy is passing through a particular area at a given time. It is denoted by \( \mathbf{S} \) and is calculated using the formula: \[ S = \frac{1}{2} \varepsilon_0 c E_{max}^2 \]

• \( \varepsilon_0 \) is the vacuum permittivity.
• \( c \) is the speed of light.
• \( E_{max} \) is the maximum electric field.

It provides an understanding of how energy flows in radio, microwave, and other forms of radiation, helping predict how these waves impact the environment or antennas they interact with.

Vacuum Permittivity

Vacuum permittivity, denoted by \( \varepsilon_0 \), is a fundamental constant that characterizes the ability of a vacuum to permit electric field lines. It is sometimes known as the electric constant and represents the baseline level of permittivity encountered in empty space. Its value is approximately \( 8.854 \times 10^{-12} \text{ F/m} \) (farads per meter) and it plays a critical role in defining the relationships within electromagnetic theory, particularly through Maxwell's equations that describe how electric and magnetic fields interact and propagate.Not only does vacuum permittivity link the electric field intensity to the charge density, but it also appears in the formula for calculating the Poynting vector, as illustrated in the original exercise. This showcases its importance in determining the strength and directionality of electromagnetic waves in a vacuum.

Speed of Light

The speed of light, denoted by \( c \), is one of the most well-known constants in physics, valued at approximately \( 3 \times 10^8 \text{ m/s} \). It indicates how fast light and other forms of electromagnetic radiation travel through the vacuum of space. The speed of light is not just a pivotal factor in optics and electromagnetism but also plays a fundamental role in the theories of relativity developed by Albert Einstein.

• It determines the maximum speed at which information or matter can travel.
• Within the Poynting vector formula, \( c \) is crucial for linking the electric field's maximum value to the intensity of the wave.
• This speed underpins many calculations in electromagnetic theory and is central to understanding how electromagnetic waves propagate over distances.

The universal nature of this constant makes it vital for calculations related to light and other waves across the electromagnetic spectrum.