A radio station on the surface of the earth radiates an EM-wave with an average total power of 50 kilo-watts. Assuming the transmitter radiates equally in all directions (not in real life), find the amplitudes Emax and Bmax detected by a satellite at a distance of 100km from the antenna.
To find the amplitudes Emax and Bmax detected by a satellite at a distance of 100km from the antenna, we can use the formula that relates power and electromagnetic field strength:
Power = 0.5 * ε₀ * c * Emax² * A,
where Power is the radiated power, ε₀ is the vacuum permittivity (also known as electric constant), c is the speed of light in vacuum, Emax is the maximum electric field strength, and A is the cross-sectional area of the sphere that the power is radiated into.
In this case, we are given the radiated power as 50 kilo-watts (50,000 watts), and we need to find the amplitudes Emax and Bmax at a distance of 100km from the antenna.
First, let's calculate the cross-sectional area of the sphere at a distance of 100km from the antenna. The formula for the surface area of a sphere is:
A = 4 * π * r²,
where r is the distance from the center of the sphere to a point on its surface. In this case, r is 100km (or 100,000 meters). Plugging in the values:
A = 4 * π * (100,000)².
Now, we can rearrange the power formula to solve for Emax:
Emax² = (2 * Power) / (ε₀ * c * A),
Emax² = (2 * 50,000) / (8.854 * 10⁻¹² * 3 * 10⁸ * 4 * π * (100,000)²).
Simplifying the equation:
Emax² = 3.61 * 10⁻⁸.
Taking the square root of both sides to find Emax:
Emax = √(3.61 * 10⁻⁸).
Now that we have Emax, we can use the relationship between the maximum electric field strength and the maximum magnetic field strength in an electromagnetic wave:
Emax = c * Bmax,
where c is the speed of light in vacuum, and Bmax is the maximum magnetic field strength. Rearranging the equation:
Bmax = Emax / c.
Plugging in the values:
Bmax = √(3.61 * 10⁻⁸) / 3 * 10⁸.
Calculating the value:
Bmax ≈ 6.45 * 10⁻¹⁷ Tesla.
Therefore, the amplitude Emax detected by the satellite is approximately √(3.61 * 10⁻⁸), and the amplitude Bmax is approximately 6.45 * 10⁻¹⁷ Tesla.