Question

The intensity of the sunlight that reaches Earth's upper atmosphere is approximately \(1400 \mathrm{W} / \mathrm{m}^{2} .\) (a) What is the average energy density? (b) Find the rms values of the electric and magnetic fields.

Short answer

(a) Energy density is \(4.67 \times 10^{-6} \text{ J/m}^3\). (b) RMS values: \(E_{rms} = 725.8 \text{ V/m}\), \(B_{rms} = 2.42 \times 10^{-6} \text{ T}\).

Step-by-step solution

Understanding Intensity and Energy Density Relation

The energy density (u) of electromagnetic radiation can be found from the intensity (I) using the relation:\[I = c \cdot u\]where \(c\) is the speed of light in vacuum, approximately \(3 \times 10^8 \text{ m/s}\). Given that the intensity \(I\) is \(1400 \text{ W/m}^2\), we can solve for \(u\).

Solving for Average Energy Density

Rearrange the equation from Step 1 to solve for the energy density \(u\):\[u = \frac{I}{c} = \frac{1400}{3 \times 10^8} \text{ J/m}^3\]Calculate this to find the energy density.

Calculation of Energy Density

Substitute the given values and compute\[u = \frac{1400}{3 \times 10^8} = 4.67 \times 10^{-6} \text{ J/m}^3\].This is the average energy density.

Electric Field and Intensity Relation

The intensity of an electromagnetic wave is related to its electric field (E) by the equation:\[I = \frac{1}{2} \varepsilon_0 c E^2\]where \(\varepsilon_0\) is the permittivity of free space, approximately \(8.85 \times 10^{-12} \text{ C}^2/\text{N m}^2\). We can rearrange to solve for the RMS value of the electric field \(E\).

Solving for RMS Electric Field

Rearrange the formula to solve for \(E\):\[E_{rms} = \sqrt{\frac{2I}{\varepsilon_0 c}}\]Substitute the values to compute.

Calculation of RMS Electric Field

Substitute the known values:\[\varepsilon_0 = 8.85 \times 10^{-12}, \; c = 3 \times 10^8, \; I = 1400 \text{ W/m}^2\]\[E_{rms} = \sqrt{\frac{2 \times 1400}{8.85 \times 10^{-12} \times 3 \times 10^8}} = 725.8 \text{ V/m}\].This is the RMS value of the electric field.

Magnetic Field and Electric Field Relation

The RMS values of the electric (E) and magnetic (B) fields are related by the speed of light:\[c = \frac{E}{B}\]Solve for \(B_{rms}\):\[B_{rms} = \frac{E_{rms}}{c}\].

Calculation of RMS Magnetic Field

Using the value of the electric field from Step 6:\[B_{rms} = \frac{725.8}{3 \times 10^8} = 2.42 \times 10^{-6} \text{ T}\].This is the RMS value of the magnetic field.

Key concepts

Energy Density

Energy density is a crucial concept when dealing with electromagnetic waves, as it tells us how much energy is stored in a wave per unit volume. It is a measure of the energy associated with electromagnetic fields in a given space. Whenever sunlight, or any other electromagnetic radiation, travels through space, it carries energy with it.
The relationship between the intensity of electromagnetic waves and energy density is given by the formula:

• \( I = c \cdot u \)

Where:

• \( I \) is the intensity of the wave.
• \( c \) is the speed of light (approximately \( 3 \times 10^8 \text{ m/s} \)).
• \( u \) is the energy density.

To find the average energy density, you can rearrange this formula to get:

• \( u = \frac{I}{c} \)

So, if the intensity of sunlight reaching Earth is \( 1400 \text{ W/m}^2 \), by plugging in the numbers, the energy density becomes \( 4.67 \times 10^{-6} \text{ J/m}^3 \). The smaller magnitude of energy density shows that the energy per unit volume is quite small, as expected in electromagnetic waves spread over large distances like sunlight.

Intensity of Sunlight

The intensity of sunlight is vital in understanding how much solar power reaches the Earth. When talking about intensity in electrostatics or wave physics, we refer to the power per unit area carried by a wave. This intensity is usually measured in watts per square meter (W/m^2). For sunlight, an approximate intensity value at Earth's atmosphere is \( 1400 \text{ W/m}^2 \).
The sun's intensity directly impacts the energy we receive, influencing various phenomena such as climate and photovoltaic electricity generation. It also plays a key role in calculating the energy density, as discussed earlier. Since intensity quantifies how strong a wave is over a certain area, it is a crucial parameter when dealing with electromagnetic waves, enabling us to compute or understand their effects on any surface, especially Earth.

Electric Field

An electric field is a fundamental concept when dealing with waves, especially electromagnetic ones. It is represented by a vector field that way we can express the force felt by a test charge in space. When we talk about electromagnetic waves, the electric field often forms oscillating waves along with the magnetic field. The strength of this electric field contributes to the intensity of the wave.
The relationship between the electric field and intensity can be described by the formula:

• \( I = \frac{1}{2} \varepsilon_0 c E^2 \)

Where:

• \( \varepsilon_0 \) is the permittivity of free space (approximately \( 8.85 \times 10^{-12} \text{ C}^2/\text{N m}^2 \)).
• \( E \) is the electric field strength.

We can calculate the root mean square (RMS) value of the electric field using:

• \( E_{rms} = \sqrt{\frac{2I}{\varepsilon_0 c}} \)

For the given sunlight intensity \( 1400 \text{ W/m}^2 \), the RMS electric field is around \( 725.8 \text{ V/m} \). Understanding the electric field's role in wave propagation helps us better grasp how electromagnetic radiation operates and interacts with matter.

Magnetic Field

In electromagnetic waves, alongside the electric field, there exists a magnetic field. These fields oscillate perpendicularly to each other and the direction of wave propagation. Analysis of the magnetic field is just as essential as that of the electric field when studying electromagnetic waves.
The speed of light ties the electric and magnetic fields together, with the equation:

• \( c = \frac{E}{B} \)

Where:

• \( B \) is the magnetic field strength.

To find the RMS value of the magnetic field, it is often calculated using:

• \( B_{rms} = \frac{E_{rms}}{c} \)

Given an electric field strength from previous calculations as \( 725.8 \text{ V/m} \), the RMS magnetic field is approximately \( 2.42 \times 10^{-6} \text{ T} \) (Teslas). Understanding both the electric and magnetic fields enables scientists and engineers to design and predict the behavior of systems influenced by electromagnetic waves, such as in communications and optics.