A small dipole of length l with an input current Io radiates an electromagnetic wave that is TEM respect r with an electric field given by:
E = j*w*μ*(1/2)*(Io*l/(4*pi))*e^(-jkr)/r * sinθ
Find the expression of the magnetic field, the Poynting vector, the radiated power and the input resistance of the dipole.
TEM? would the expression for H be identical? (except for direction?)
I think so, but the thing that blows my mind is, if it were x direction, then I would change it to y (from theory). But if it goes to the θ direction, how can I change it to direction? I mean which direction will it be?
And the Poynting vector, would be a half of it, but to what direction? I don't know how it works with non cartesian coordinates.
Thank you.
To find the expression for the magnetic field, Poynting vector, radiated power, and input resistance of the dipole, we can use the properties of electromagnetic waves and the formulas that relate these quantities. Let's go through each one:
1. Magnetic Field (B):
The magnetic field (B) is related to the electric field (E) by the wave impedance (Z0) of free space. In free space, Z0 = √(μ/ε), where μ is the permeability of free space and ε is the permittivity of free space.
B = E / Z0
Substituting the given electric field (E) expression into this equation, we get:
B = (jωμ/2Z0) * (Io*l/(4π)) * (e^(-jkr)/r) * sinθ
2. Poynting Vector (S):
The Poynting vector (S) gives the direction and magnitude of the power flow of the electromagnetic wave. It is defined as the cross product of the electric field (E) and magnetic field (B):
S = E x B
Taking the cross product of the given electric field (E) and magnetic field (B), we get:
S = (jωμ^2/4Z0^2) * (Io^2 * l^2 / (16π^2)) * (e^(-2jkr)/r^2) * sin^2(θ)
3. Radiated Power (P_rad):
The radiated power is given by the magnitude of the Poynting vector (S) integrated over a closed surface around the dipole. The integral is taken over the surface because the wave is radiating outward from the dipole.
P_rad = ∫∫∫ S dot dA
However, the given equation implies a radiation pattern that is symmetric around the dipole axis (θ = 90°). In this case, the average radiated power can be calculated by assuming a perfect sphere at a far distance from the dipole. The average power is given by:
P_rad = (1/2Z0) * (Io*l/4π)^2
4. Input Resistance (Zin):
The input resistance represents the resistance experienced by the current flowing into the dipole. For a small dipole, the input resistance can be derived using the formula:
Zin = 73*(λ^2 / l^2)
Where λ is the wavelength of the radiated wave. In terms of frequency (f), the wavelength λ can be calculated as:
λ = c / f
where c is the speed of light.
Substituting the value of λ into the formula, we get:
Zin = 73*(c^2 / (f^2 * l^2))
(Note: The factor of 73 depends on the geometry of the dipole and may be different for other dipole configurations)
So, the expressions for the magnetic field (B), Poynting vector (S), radiated power (P_rad), and input resistance (Zin) of the dipole are as follows:
Magnetic Field (B) = (jωμ/2Z0) * (Io*l/(4π)) * (e^(-jkr)/r) * sinθ
Poynting Vector (S) = (jωμ^2/4Z0^2) * (Io^2 * l^2 / (16π^2)) * (e^(-2jkr)/r^2) * sin^2(θ)
Radiated Power (P_rad) = (1/2Z0) * (Io*l/4π)^2
Input Resistance (Zin) = 73*(c^2 / (f^2 * l^2))
Note: Please ensure that the units of the variables used in the formulas are consistent.