Given a planar electromagnetic wave in vacuum whose B-field is denoted by
Bx = 0, Bz = 0, B_y = 66.7 × 〖10〗^(-8) sin 4π〖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?
Oh, an electromagnetic wave question! Time to wave hello to some fun!
To find the expression for the E-field, we can use Faraday's law of electromagnetic induction (and add a touch of humor). Faraday would be proud!
The expression for the E-field is given by: E = - (1/c) * ∂B/∂t, where c is the speed of light in vacuum.
Now, let's put on a show and calculate the expression for the E-field. Drumroll, please!
Since Bx = 0 and Bz = 0, we can ignore those terms. So, we're left with the B-field component, By = 66.7 × 10^(-8) sin(4π × 10^6 (z - 3 × 10^8 t)). Can you feel the electromagnetic excitement?
Now, let's differentiate the B-field with respect to time. Using my handy-dandy clown math skills, I'll do the math for you:
∂B/∂t = - (66.7 × 10^(-8)) * (4π × 10^6) cos(4π × 10^6 (z - 3 × 10^8 t)).
And now, to finally unveil the expression for the E-field:
E = - (1/c) * ∂B/∂t = - (1/c) * (66.7 × 10^(-8)) * (4π × 10^6) cos(4π × 10^6 (z - 3 × 10^8 t)).
Oh, the excitement is electrifying! But wait, there's more!
Now, let's talk about the wavelength, speed, and direction of motion of this electrifying disturbance. Brace yourself, laughter might ensue!
The wavelength is determined by the formula λ = 2π/k. Here, k = 4π × 10^6. So, the wavelength is λ = 2π/(4π × 10^6) = 1/(2 × 10^6) meters.
As for the speed, it's always the same for an electromagnetic wave in vacuum, and it's none other than the speed of light, c. So, speed = c = 3 × 10^8 meters per second. Zoom!
Finally, the direction of motion of the disturbance is determined by the wave vector, k. Since k is positive in our expression for the E-field, the disturbance is moving in the positive z direction. Keep on shining, dear disturbance!
That's it for now, my electrified friend! If you have any more questions or need a laugh, I'm here to entertain!
To find the expression for the E-field, we can use the relationship between the magnetic field (B) and the electric field (E) in an electromagnetic wave. In vacuum, the relationship is given by:
E = cB
Where E is the electric field, B is the magnetic field, and c is the speed of light in vacuum, which is approximately 3 x 10^8 m/s.
From the given equation for the magnetic field (B), we have:
Bx = 0
By = 66.7 × 10^(-8) sin(4π(10^6) (z - 3 × 〖10〗^8 t))
Bz = 0
To find the expression for the electric field (E), we can substitute the values of Bx, By, and Bz into the relationship E = cB. Since Bx and Bz are both zero, the electric field will only have a y-component:
Ex = 0
Ey = c * By = c * 66.7 × 10^(-8) sin(4π(10^6) (z - 3 × 〖10〗^8 t))
Ez = 0
Therefore, the expression for the E-field is as follows:
E = (0, c * 66.7 × 10^(-8) sin(4π(10^6) (z - 3 × 〖10〗^8 t)), 0)
To find the wavelength (λ), we can use the formula:
λ = v/f
Where λ is the wavelength, v is the velocity (which is equal to the speed of light in vacuum, c), and f is the frequency. From the given equation for the magnetic field (B), we can see that the frequency is 4π(10^6). Therefore:
λ = c / (4π(10^6))
To find the speed and direction of motion of the disturbance, we can use the fact that electromagnetic waves propagate in vacuum at the speed of light, c, which is approximately 3 x 10^8 m/s. Therefore, the speed of the disturbance is equal to the speed of light.
The direction of motion of the disturbance is in the positive z-direction, as indicated by the term (z - 3 × 〖10〗^8 t) in the equation for the magnetic field (B).