Cosmic microwave background radiation fills all space with an average energy density of 4 x 10^-14 j/m^3.(a) Find the rms value of electric field associated with this radiation.(b) How far from a 7.5 kW radio transmitter emitting uniformly in all direction would you find a comparable value?
There is a formula that relates radiation energy density to the E-field strength.
Get to know it.
Energy density = (1/2) epsilono * E^2
For an electromagnetic wave, there is an equal amount of energy in the magnetic field, as I recall.
For the E-field energy density at a distance R from that transmitter,
divide the radiated power by 4 pi R^2. That is the rate that energy is radiated away. Half of that power is due to the E-field. Divide by c for the E-field energy density. Set equal to the part (a) value and solve for R
To find the rms value of the electric field associated with cosmic microwave background radiation, we can use the formula:
E = √(2ε₀c^2U)
where E is the rms electric field, ε₀ is the vacuum permittivity (8.854 x 10^-12 F/m), c is the speed of light (3 x 10^8 m/s), and U is the energy density of the radiation.
(a) Substituting the given values into the formula, we have:
E = √(2 * 8.854 x 10^-12 F/m * (3 x 10^8 m/s)^2 * 4 x 10^-14 J/m³)
Evaluating this expression, we find:
E ≈ 1.77 x 10^-6 V/m
Therefore, the rms value of the electric field associated with cosmic microwave background radiation is approximately 1.77 x 10^-6 V/m.
(b) To find how far from a 7.5 kW radio transmitter we would find a comparable value, we can use the formula for the power density of a uniformly radiating source:
P = (cε₀/2) * E^2
where P is the power density, ε₀ is the vacuum permittivity, c is the speed of light, and E is the electric field.
We need to solve this equation for distance (r):
P_r = P_t / (4πr²)
where P_r is the power density at radius r, P_t is the power output of the transmitter, and r is the distance from the transmitter.
Setting the expressions for power density equal to each other and solving for r, we have:
(cε₀/2) * E^2 = P_t / (4πr²)
Simplifying and rearranging the equation, we get:
r = √(P_t / (2πE^2 * cε₀))
Substituting the given values into the equation, we have:
r = √(7500 W / (2π(1.77 x 10^-6 V/m)^2 * (3 x 10^8 m/s) * (8.854 x 10^-12 F/m)))
Evaluating this expression, we find:
r ≈ 3.68 x 10^6 m
Therefore, to find a comparable value, you would need to be approximately 3.68 x 10^6 meters (or 3.68 million meters) away from a 7.5 kW radio transmitter emitting uniformly in all directions.
To find the rms value of the electric field associated with the cosmic microwave background radiation, we can use the formula:
E_rms = sqrt(2 * energy density * vacuum impedance)
Given that the average energy density of the cosmic microwave background radiation is 4 x 10^-14 J/m^3, and the vacuum impedance is approximately 376.73 Ω (ohms), we can calculate the rms value of the electric field as follows:
(a) Electric Field Calculation:
E_rms = sqrt(2 * 4 x 10^-14 J/m^3 * 376.73 Ω)
≈ sqrt(3.009 x 10^-11 V^2/m^2)
≈ 5.485 x 10^-6 V/m
So, the rms value of the electric field associated with the cosmic microwave background radiation is approximately 5.485 x 10^-6 V/m.
To find the distance from a 7.5 kW radio transmitter where the electric field is comparable to the cosmic microwave background radiation, we can use the formula:
Electric Field = sqrt(Power / (4π * distance^2))
Given that the power of the radio transmitter is 7.5 kW (or 7.5 x 10^3 W), and we want to find the distance, we can rearrange the formula as:
distance = sqrt(Power / (4π * Electric Field))
Substituting the values into the formula, we get:
(b) Distance Calculation:
distance = sqrt(7.5 x 10^3 W / (4π * 5.485 x 10^-6 V/m))
≈ sqrt(2.719 x 10^9 m^2)
≈ 52,155.71 m
So, you would find a comparable value to the rms electric field of the cosmic microwave background radiation at a distance of approximately 52,155.71 meters (or 52.16 km) from the 7.5 kW radio transmitter emitting uniformly in all directions.