Problem 67 - College Physics

Question

\(\bullet$$\bullet\) A plane sinusoidal electromagnetic wave in air has a wave- length of 3.84 \(\mathrm{cm}\) and an \(\vec{\boldsymbol{E}}\) field amplitude of 1.35 \(\mathrm{V} / \mathrm{m}\) . (a) What is the frequency of the wave? (b) What is the \(\vec{\boldsymbol{B}}\) field amplitude? (c) What is the intensity? (d) What average force does this radiation exert perpendicular to its direction of propagation on a totally absorbing surface with area 0.240 \(\mathrm{m}^{2}\) ?

Step-by-Step Solution

Answer

(a) 7.81 GHz, (b) 4.50 nT, (c) 2.42 W/m², (d) 0.581 N.

Step 1: Calculate Frequency

The frequency \( f \) of a wave is related to the speed of light \( c \) and its wavelength \( \lambda \) by the formula: \( f = \frac{c}{\lambda} \). The speed of light \( c \) is approximately \( 3 \times 10^8 \) m/s, and the given wavelength \( \lambda = 3.84 \times 10^{-2} \) m. Substituting these values, we find:\[ f = \frac{3 \times 10^8}{3.84 \times 10^{-2}} = 7.81 \times 10^9 \text{ Hz} \].

Step 2: Calculate Magnetic Field Amplitude

The relationship between the electric field amplitude \( E \) and the magnetic field amplitude \( B \) is given by: \( c = \frac{E}{B} \). Rearrange it to find \( B \):\[ B = \frac{E}{c} = \frac{1.35}{3 \times 10^8} = 4.50 \times 10^{-9} \text{ T} \].

Step 3: Calculate Intensity

The intensity \( I \) of an electromagnetic wave can be found using the formula: \( I = \frac{1}{2} \varepsilon_0 c E^2 \), where \( \varepsilon_0 = 8.85 \times 10^{-12} \text{ C}^2/\text{N}\cdot\text{m}^2 \) is the permittivity of free space. Substitute \( E = 1.35 \text{ V/m} \):\[ I = \frac{1}{2} (8.85 \times 10^{-12}) (3 \times 10^8) (1.35)^2 = 2.42 \text{ W/m}^2 \].

Step 4: Calculate Average Force on Surface

The average force \( F \) exerted by radiation on a totally absorbing surface is related to intensity \( I \) and area \( A \) by the equation: \( F = I \times A = 2.42 \times 0.240 \). Calculating this gives:\[ F = 0.581 \text{ N} \].

Key Concepts

Frequency Calculation

Frequency is an essential characteristic of any wave, including electromagnetic waves such as light or radio waves. To calculate the frequency of a wave, you can use its wavelength and the speed of light. The frequency \( f \) is calculated using the formula:

\[ f = \frac{c}{\lambda}\]where:

• \( c \) is the speed of light, approximately \( 3 \times 10^8 \, \text{m/s} \)
• \( \lambda \) is the wavelength of the wave in meters.

For an electromagnetic wave with a wavelength of 3.84 cm, first convert the wavelength to meters: \( 3.84 \, \text{cm} = 3.84 \times 10^{-2} \, \text{m} \). Taking these values and substituting them into the formula:

\[ f = \frac{3 \times 10^8}{3.84 \times 10^{-2}} = 7.81 \times 10^9 \, \text{Hz}\]This calculation shows that the frequency is about 7.81 GHz. Frequency tells us how many wave cycles pass a point per second.

Magnetic Field Amplitude

The amplitude of the magnetic field \( \vec{B} \) in an electromagnetic wave is directly related to the electric field amplitude \( \vec{E} \). The speed of light also plays a crucial role in this relationship, dictated by the equation:

\[ c = \frac{E}{B}\]Rearranging it gives:

\[ B = \frac{E}{c}\]Given that \( E = 1.35 \, \text{V/m} \) and \( c = 3 \times 10^8 \, \text{m/s} \), you substitute these into the equation to find \( B \):

\[ B = \frac{1.35}{3 \times 10^8} = 4.50 \times 10^{-9} \, \text{T}\]This tells us that the magnetic field amplitude is approximately \( 4.50 \times 10^{-9} \, \text{tesla} \). The amplitude of the magnetic field is significantly weaker than the electric field but is just as important in defining the wave's properties.

Wave Intensity

Wave intensity describes how much energy an electromagnetic wave carries per unit area per unit time. It is a measure of the strength or power of the wave as it travels. The intensity \( I \) can be calculated using:

\[ I = \frac{1}{2} \varepsilon_0 c E^2\]where:

• \( \varepsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2/\text{N}\cdot\text{m}^2 \) is the permittivity of free space
• \( c \) is the speed of light
• \( E \) is the electric field amplitude in volts per meter.

Substituting the known values, \( E = 1.35 \, \text{V/m} \), we compute:

\[ I = \frac{1}{2} (8.85 \times 10^{-12}) (3 \times 10^8) (1.35)^2 = 2.42 \, \text{W/m}^2\]This tells us that the wave's intensity is 2.42 watts per square meter. Intensity gives us insight into how much power the wave can transfer in an area over time.

Radiation Force

Radiation force is the average force exerted by an electromagnetic wave as it impinges on a surface. For a surface that fully absorbs the wave, this can be found by multiplying the wave's intensity by the area of the surface:

\[ F = I \times A\]where:

• \( I \) is the intensity of the wave, calculated previously
• \( A \) is the area of the surface in meters squared.

In this example, the area \( A \) is given as \( 0.240 \, \text{m}^2 \). Substitute these numbers into the formula:

\[ F = 2.42 \times 0.240 = 0.581 \, \text{N}\]This indicates that the electromagnetic wave exerts a force of \( 0.581 \, \text{newtons} \) on the absorbing surface. The concept of radiation force is useful in understanding how electromagnetic waves can transfer momentum to objects they interact with.