Given a planar electromagnetic wave in vacuum whose B-field is denoted by Bx = 0, Bz = 0, B_y = 66.7 × ?10?^(-8) sin 4?p?10?^6 (z-3×?10?^8 t) Write an expression for the E-field. What are wavelength, speed and direction of motion of the disturbance?
If you are only studying St. Josef, you'll miss a lot in math, reading, writing, science, etc.
To write an expression for the E-field, we need to use Maxwell's equations, specifically Faraday's Law of electromagnetic induction. According to Faraday's Law, the rate of change of the magnetic field (B) with respect to time (t) induces an electric field (E), which is perpendicular to both the magnetic field and the direction of the change. In this case, the B-field is oscillating with time, so we differentiate it with respect to time to find the rate of change:
dB_y/dt = -66.7 × 10^(-8) cos(4π × 10^6 (z - 3 × 10^8 t)).
Since the E-field is perpendicular to the B-field, we consider the x-direction. Thus, the E-field will have components in the x and y directions. Using Faraday's Law, we can write the expression for the x-component of the E-field (Ex):
Ex = -dB_y/dt = 66.7 × 10^(-8) cos(4π × 10^6 (z - 3 × 10^8 t)).
Now, let's find the wavelength, speed, and direction of motion of the disturbance:
To find the wavelength (λ) of the electromagnetic wave, we know that the wave is described by sin(4π × 10^6 (z - 3 × 10^8 t)). The general form of a sinusoidal wave is sin(kx - ωt), where k is the wave number and ω is the angular frequency. Comparing the two forms, we can equate:
k = 4π × 10^6 and 2π/λ = k,
which allows us to solve for λ:
λ = 2π/k = 2π/(4π × 10^6) = 1/(2 × 10^6) = 0.5 × 10^(-6) = 0.5 μm.
The speed (v) of the electromagnetic wave in vacuum is given by the equation:
v = c,
where c is the speed of light, approximately equal to 3 × 10^8 m/s.
The direction of motion of the disturbance is determined by the wave vector k, which is perpendicular to the wavefront. From the given expression, we can see that the disturbance is traveling in the z-direction, as it is dependent on (z - 3 × 10^8 t).
Therefore, the expression for the E-field is Ex = 66.7 × 10^(-8) cos(4π × 10^6 (z - 3 × 10^8 t)), the wavelength is λ = 0.5 μm, the speed is v = 3 × 10^8 m/s, and the direction of motion is along the z-axis.