Problem 45 - Physics

Question

A future space station in orbit about the earth is being powered by an electromagnetic beam from the earth. The beam has a cross-sectional area of \(135 \mathrm{~m}^{2}\) and transmits an average power of \(1.20 \times 10^{4} \mathrm{~W}\). What are the rms values of the (a) electric and (b) magnetic fields?

Step-by-Step Solution

Answer

The rms electric field is approximately 57.7 V/m, and the rms magnetic field is approximately 1.92 × 10⁻⁷ T.

Step 1: Understand the Given Values

We are given that the total average power transmitted by the electromagnetic beam is \( P = 1.20 \times 10^4 \mathrm{~W} \) and the cross-sectional area of the beam is \( A = 135 \mathrm{~m}^2 \). We need to find the rms values of the electric and magnetic fields.

Step 2: Calculate Intensity of the Beam

The intensity \( I \) of the beam, which is the power per unit area, can be calculated using the formula \( I = \frac{P}{A} \). Here, \( P = 1.20 \times 10^4 \mathrm{~W} \) and \( A = 135 \mathrm{~m}^2 \), so \( I = \frac{1.20 \times 10^4}{135} \mathrm{~W/m}^2 \).

Step 3: Use Intensity to Find Electric Field

The intensity of an electromagnetic wave is given by \( I = \frac{c \varepsilon_0}{2} E_{rms}^2 \), where \( c \) is the speed of light (\( 3 \times 10^8 \mathrm{~m/s} \)) and \( \varepsilon_0 \) is the permittivity of free space (\( 8.85 \times 10^{-12} \mathrm{~F/m} \)). Rearranging for \( E_{rms} \), we have: \( E_{rms} = \sqrt{\frac{2I}{c\varepsilon_0}} \). Substitute \( I \) to find \( E_{rms} \).

Step 4: Calculate RMS Electric Field Value

Substituting the value from Step 2 into the equation \( E_{rms} = \sqrt{\frac{2 \times 88.89}{3 \times 10^8 \times 8.85 \times 10^{-12}}} \), calculate \( E_{rms} \). The result is \( E_{rms} \approx 57.7 \mathrm{~V/m} \).

Step 5: Use Electric Field to Find Magnetic Field

The rms value of the magnetic field \( B_{rms} \) can be found using the relationship between the electric and magnetic fields in an electromagnetic wave: \( B_{rms} = \frac{E_{rms}}{c} \). Substitute the value calculated for \( E_{rms} \) and \( c = 3 \times 10^8 \) to find \( B_{rms} \).

Step 6: Calculate RMS Magnetic Field Value

Calculate \( B_{rms} = \frac{57.7}{3 \times 10^8} \). The result is \( B_{rms} \approx 1.92 \times 10^{-7} \mathrm{~T} \).

Key Concepts

Intensity

Intensity is a measure of the power of an electromagnetic wave per unit area, effectively describing how much energy is transported by the wave. It's instrumental in understanding the strength of waves like those that power remote stations. The formula for intensity \( I \) is \( I = \frac{P}{A} \), where \( P \) is the power in Watts and \( A \) is the area in square meters through which the power flows.

In our example, the beam's power is \( 1.20 \times 10^4 \) Watts across an area of \( 135 \; \mathrm{m}^2 \). Substituting these values into the formula gives \( I = \frac{1.20 \times 10^4}{135} \approx 88.89 \; \mathrm{W/m}^2 \). This intensity value is key to determining both the electric and magnetic fields of the beam.

Electric Field

The electric field in an electromagnetic wave is directly related to its intensity. Its Root Mean Square (RMS) value \( E_{rms} \) offers a way to express the effective strength of the alternating field. For electromagnetic waves, the intensity is related to the electric field by the equation \( I = \frac{c \varepsilon_0}{2} E_{rms}^2 \). Here, \( c \) is the speed of light at \( 3 \times 10^8 \; \mathrm{m/s} \) and \( \varepsilon_0 \) is the permittivity of free space, valued at \( 8.85 \times 10^{-12} \; \mathrm{F/m} \).

By rearranging the formula, \( E_{rms} = \sqrt{\frac{2I}{c\varepsilon_0}} \), we can substitute \( I = 88.89 \; \mathrm{W/m}^2 \) to find \( E_{rms} \). This calculation yields \( E_{rms} \approx 57.7 \; \mathrm{V/m} \), reflecting the effective electric field strength across the beam.

Magnetic Field

Once the electric field \( E_{rms} \) of an electromagnetic wave is known, the magnetic field \( B_{rms} \) can be easily determined since both fields are intrinsically linked in these waves. The relationship between them is captured by the formula: \( B_{rms} = \frac{E_{rms}}{c} \), where \( c \) is the speed of light.

This formula shows that the magnetic field is typically much smaller than the electric field due to the high speed of light. Given \( E_{rms} = 57.7 \; \mathrm{V/m} \), \( B_{rms} \) is calculated as \( B_{rms} = \frac{57.7}{3 \times 10^8} \approx 1.92 \times 10^{-7} \; \mathrm{T} \). This value represents the effective magnetic field as experienced along the wave.

Permittivity of Free Space

The permittivity of free space \( \varepsilon_0 \) is a fundamental physical constant crucial in electromagnetism. It characterizes how electric fields behave in a vacuum, influencing both wave propagation and the interaction between electric charges. Its value, \( \varepsilon_0 = 8.85 \times 10^{-12} \; \mathrm{F/m} \), frequently appears in formulas related to electric fields and capacities.

This constant plays a pivotal role when calculating the electric field's RMS value, linking the electric field strength to the intensity of the wave. It's an essential component of the equation \( I = \frac{c \varepsilon_0}{2} E_{rms}^2 \), highlighting its importance in various electromagnetic calculations.

RMS Values

Root Mean Square (RMS) values provide a useful way of expressing the effective strength of alternating electric and magnetic fields in EM waves. Unlike peak values, RMS values account for variations in the fields over time, offering a more consistent measurement.

The concept emerges from the square root of the average of squares, which is particularly applicable in AC circuit analysis and electromagnetic wave evaluations. For electric fields, \( E_{rms} = \sqrt{\frac{2I}{c\varepsilon_0}} \) gives a measure that reflects its average impact rather than transient peaks.

RMS values ensure calculations align closer to the wave's practical influence, particularly crucial in real-world applications where consistency matters more than peak readings.