The electric and magnetic fields of a plane electromagnetic wave are given by the formulas below, where and are both in meters and is in seconds. This wave is traveling through a medium whose index of refraction is 1.3.
Reminder: The "x.xxenn t" notation in the sine functions means " t".
= 2.8 z + 1.19e16 V/m
= z + 1.19e16 Tesla
(a) In what direction does the wave propagate?
Status: unsubmitted
(b) What is the magnitude of
unanswered
What is the direction of (assuming that the symbol is positive)?
Status: unsubmitted
(c) What is the wavelength of the wave (in meters)?
To determine the direction in which the wave propagates, we can examine the electric field equation:
E = 2.8 sin(10x) sin( t) V/m
Notice that the electric field has a sinusoidal dependence on the x-coordinate. This implies that the wave is propagating in the positive x-direction.
Now let's move on to the magnitude of the magnetic field:
B = sin(10x) sin( t) Tesla
To find the magnitude of B, we can simply calculate its maximum value. The maximum value of sin(x) is 1, so the maximum value of sin(10x) is also 1. Therefore, the maximum magnitude of B is:
B_max = 1 × 1 = 1 Tesla
For the direction of B, we need to examine the equation:
B = sin(10x) sin( t) Tesla
In this case, note that the sine function is symmetrical around the origin (x = 0). Thus, the direction of the magnetic field alternates as x increases or decreases. Assuming the value of is positive, the direction of B will be positive when x is positive, and negative when x is negative. Therefore, the direction of B is opposite to the direction of the x-coordinate.
Finally, to calculate the wavelength of the wave, we can refer to the general wave equation:
v = fλ
where v is the speed of light in the medium and λ is the wavelength.
Since the wave is traveling through a medium with an index of refraction of 1.3, the speed of light in the medium is given by:
v = (3 × 10^8 m/s) / 1.3 = 2.31 × 10^8 m/s
The frequency of the wave can be found from the argument of the sine functions in the electric and magnetic fields: 10x. Since the wave is propagating along the x-direction, the distance traveled in one full period of the wave is one wavelength. Hence, we can equate 10x to the full period which is 2π:
10x = 2π
Simplifying further, we calculate:
x = 0.2π
Thus, the wavelength λ is equal to twice the distance x:
λ = 2 × x = 2 × 0.2π ≈ 0.4π
So, the wavelength of the wave is approximately 0.4 times π meters.