The radio telescope in Arecibo, Puerto Rico, has a diameter of 305 m . It can detect radio waves

step 1 The incident power is given by iA cosine theta equals 10 to the minus 26 watt per meter square t

The radio telescope in Arecibo, Puerto Rico, has a diameter...

The radio telescope in Arecibo, Puerto Rico, has a diameter of $305 \mathrm{m} .$ It can detect radio waves from space with intensities as small as $10^{-26} \mathrm{W} / \mathrm{m}^{2}$. (a) What is the average power incident on the telescope due to a wave at normal incidence with intensity $1.0 \times 10^{-26} \mathrm{W} / \mathrm{m}^{2} ?$ (b) What is the average power incident on Earth's surface? (c) What are the rms electric and magnetic fields?

The radio telescope in Arecibo, Puerto Rico, has a diameter of $305 \mathrm{m} .$ It can detect radio waves from space with intensities as small as $10^{-26} \mathrm{W} / \mathrm{m}^{2}$.
(a) What is the average power incident on the telescope due to a wave at normal incidence with intensity $1.0 \times 10^{-26} \mathrm{W} / \mathrm{m}^{2} ?$
(b) What is the average power incident on Earth's surface?
(c) What are the rms electric and magnetic fields?

Key Concepts

Aperture and Cross?Sectional Area

The area of a circular aperture, which is calculated using the formula A = ?r² (where r is the radius of the circle), is essential in determining the total power intercepted by an instrument such as a telescope or even an entire planet. By knowing the area, one can compute the product of this area and the wave's intensity to find the incident power. This principle is used in both evaluating the power received by a telescope and the power incident on Earth as a whole.

Poynting Vector and RMS Field Relationships

The Poynting vector in electromagnetic wave theory represents the directional energy flux, or power per unit area, of the wave. The intensity of an electromagnetic wave is related to the root-mean-square (rms) values of the electric and magnetic fields through formulas such as I = (1/2)c??E_rms², where c is the speed of light and ?? is the permittivity of free space. Moreover, the rms magnetic field can be found from the rms electric field by using the relation E_rms = cB_rms. These relationships bridge the gap between the macroscopic measurement of power and the microscopic field quantities responsible for energy transport.

Electromagnetic Wave Intensity

Intensity is defined as the amount of power carried by an electromagnetic wave per unit area perpendicular to the direction of propagation. It is a key quantity because it allows one to determine the total power intercepted by an object by multiplying the intensity by the area over which the wave is distributed. This concept is crucial for understanding how much energy from distant sources, such as astronomical objects, is received by detectors on Earth or by telescopes.

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