Write down the electric field and associated magnetic field in vacuum for a traveling plane wave with the following properties. The amplitude of the electric vector is E0 and the frequency is ω. The radiation is linearly polarized in the y-z plane at an angle α of π/4 with respect to the y-axis, and it is traveling in the +x direction. The electric and magnetic field can be written in the following way
E = E0cos(ωt+(−1)bEkx)(cExˆ+dEyˆ+eEzˆ)
B = cos(ωt+(−1)bBkx)(cBxˆ+dByˆ+eBzˆ)
calculate :
dE=
eE=
dB = (Enter an expression in terms of E0, c)
eB = (Enter an expression in terms of E0, c)
any answer?
dB=-(E_0/c)*cos(pi/4)
eB=(E_0/c)*cos(pi/4)
remaining answers ??
anyone?
To calculate the values of dE, eE, dB, and eB, we need to use the given information about the amplitude of the electric vector (E0), the polarization angle (α), and the properties of the wave.
Let's start by breaking down the given expressions for the electric and magnetic fields:
E = E0cos(ωt + (-1)bEkx)(cExˆ + dEyˆ + eEzˆ)
B = cos(ωt + (-1)bBkx)(cBxˆ + dByˆ + eBzˆ)
Based on the given information, the wave is linearly polarized in the y-z plane at an angle α = π/4 with respect to the y-axis and is traveling in the +x direction. This means that the electric field vector lies in the y-z plane and has components in the y and z directions.
Given that the polarization angle is α = π/4, we can determine the coefficients dE and eE:
dE = E0cos(α) = E0cos(π/4) = E0/sqrt(2)
eE = E0sin(α) = E0sin(π/4) = E0/sqrt(2)
So, we have dE = E0/sqrt(2) and eE = E0/sqrt(2).
Next, let's determine the coefficients dB and eB in terms of E0 and c:
We know that the magnetic field in vacuum is related to the electric field by the speed of light (c) through the equation:
B = (1/c) * E
Comparing this equation to the given magnetic field expression:
B = cos(ωt + (-1)bBkx)(cBxˆ + dByˆ + eBzˆ)
We can conclude that:
cB = 1/c = 1/3 * 10^8 m/s
Therefore, dB = cB * E0 = (1/3 * 10^8 m/s) * E0
Similarly, for eB, we need to consider the polarization angle α and its relation to the cross product of the wave vector k and the electric field direction. Since the wave is linearly polarized in the y-z plane, the angle between k and the y-axis is α (π/4). We can use the cross product of unit vectors to find eB:
eB = -cB * E0 * sin(α) = -(1/3 * 10^8 m/s) * E0 * sin(π/4) = -(1/3 * 10^8 m/s) * E0 * (1/sqrt(2))
Therefore, eB = -(1/3 * 10^8 m/s) * E0 * (1/sqrt(2)).
In summary, the values of the coefficients are:
dE = E0/sqrt(2)
eE = E0/sqrt(2)
dB = (1/3 * 10^8 m/s) * E0
eB = -(1/3 * 10^8 m/s) * E0 * (1/sqrt(2))