An electromagnetic wave in a vacuum traveling in the +x direction generated by a variable source initially has a wavelength of 2.65 x 10^-4 m and a maximum electric field Emax in the +y direction of 8.20×10^−3 V/m. What are the coefficients in the equation for the magnetic field component of the wave after the period increases by a factor of 1.90?
Bmax = ?
k = ?
w = ?
So far I have Bmax = 2.733 x 10^-11 T, k = 1.102 x 10^-8 1/m, w = 3.744 x 10^ -14 rad/s
I know at least one of those is wrong, but I don't know why.
To solve this problem, we can use the following equations relating the wavelength (λ), wave number (k), angular frequency (ω), speed of light (c), and period (T):
λ = c / f (1)
k = 2π / λ (2)
ω = 2πf (3)
c = λf (4)
T = 1 / f (5)
Given that the initial wavelength is λ_0 = 2.65 x 10^-4 m and the final wavelength is λ = 1.90 * λ_0, we can find the factor by which the period increases.
1. Calculate the initial frequency (f_0):
Using equation (1):
f_0 = c / λ_0
2. Calculate the final frequency (f):
Since the speed of light remains constant, we can use equation (1) with the new wavelength to find the final frequency:
f = c / λ
3. Find the factor by which the period increases:
The period (T) is the reciprocal of the frequency (f), so we have:
T = 1 / f_0
T' = 1 / f
The factor by which the period increases is T' / T.
4. Calculate the final period (T'):
Using the factor calculated above:
T' = T * (T' / T)
Now, let's find the coefficients in the equation for the magnetic field component of the wave after the period increases.
Given that Emax = 8.20 × 10^−3 V/m, we know that Emax is related to Bmax through the equation:
Emax = c * Bmax
Using equation (4), we can rewrite it as:
Bmax = Emax / c
Now, let's calculate the values:
1. Calculate the factor by which the period increases:
T = 1 / f_0
T' = T * (T' / T)
2. Calculate the final period (T'):
3. Calculate the final frequency (f):
f = c / λ
4. Calculate the final angular frequency (ω):
Using equation (3):
ω = 2πf
5. Calculate the final wave number (k):
Using equation (2):
k = 2π / λ
6. Calculate the final magnetic field (Bmax):
Using equation Bmax = Emax / c:
Now, to check if your values are correct, compare them with the ones you mentioned: Bmax = 2.733 x 10^-11 T, k = 1.102 x 10^-8 1/m, and ω = 3.744 x 10^ -14 rad/s. If they match, then you have the correct values. Otherwise, there may be an error in your calculations.
To determine the coefficients in the equation for the magnetic field component of the wave after the period increases by a factor of 1.90, we can use the relationship between the electric and magnetic fields in an electromagnetic wave.
The equation for the magnetic field component of an electromagnetic wave is given by:
B = (Emax/c) * sin(kx - wt)
Where:
B is the magnetic field component
Emax is the maximum electric field amplitude
c is the speed of light in vacuum
k is the wave number
x is the position
w is the angular frequency
Given:
Initial wavelength, λ = 2.65 x 10^-4 m
Maximum electric field amplitude, Emax = 8.20×10^−3 V/m
Factor by which the period increases, f = 1.90
We can find the wave number, k, using the relation k = 2π / λ.
k = 2π / (2.65 x 10^-4) ≈ 23,674.70 rad/m
Next, we need to find the angular frequency, w, using the relation w = 2πf.
w = 2πf = 2π / T, where T is the initial period.
We can find the initial period, T₀, using the relation T₀ = λ / c.
Using the speed of light in vacuum, c ≈ 3 x 10^8 m/s:
T₀ = (2.65 x 10^-4) / (3 x 10^8) ≈ 8.83 x 10^-13 seconds
w = 2π / (8.83 x 10^-13) ≈ 7.13 x 10^12 rad/s
Now, let's calculate the new period, T:
T = f * T₀ = 1.90 * (8.83 x 10^-13) ≈ 1.68 x 10^-12 seconds
Finally, we can find the new angular frequency, w, using the relation w = 2π / T:
w = 2π / (1.68 x 10^-12) ≈ 3.74 x 10^12 rad/s
Now, let's calculate the coefficients:
Bmax = Emax / c
Bmax = 8.20×10^−3 / (3 x 10^8)
Bmax ≈ 2.73 x 10^-11 T
k remains the same: k ≈ 23,674.70 rad/m
w = 3.74 x 10^12 rad/s
Therefore, the correct coefficients are:
Bmax ≈ 2.73 x 10^-11 T
k ≈ 23,674.70 rad/m
w ≈ 3.74 x 10^12 rad/s