1. Write an expression for a R - state lightwave of angular frequency and amplitude E0 propagating in the positive x-direction such that at t=0 and x=0 the E-field points in the negative z-direction.
r statelight wave of frequencyw propagating in positivex direction
r statelight
To write an expression for a R-state lightwave in the given scenario, we can use the standard form for a general electromagnetic wave, which is represented as:
E(x, t) = E0 cos(kx - ωt + φ)
In this equation:
- E(x, t) represents the electric field vector at a particular position (x) and time (t).
- E0 is the amplitude of the wave.
- k is the wave vector, and it represents the spatial frequency of the wave.
- ω is the angular frequency of the wave.
- φ is the phase constant, which determines the initial phase of the wave.
Based on the given information:
- The wave is an R-state lightwave.
- The E-field points in the negative z-direction when t=0 and x=0.
To match these specifications, we need the initial phase (φ) to be 90 degrees (π/2 radians) because a negative z-direction corresponds to a phase shift of π/2 relative to a positive x-direction.
With these considerations, the expression for the R-state lightwave becomes:
E(x, t) = E0 cos(kx - ωt + π/2)
Note: If you have specific values for the wave vector (k), angular frequency (ω), and amplitude (E0), you can substitute them directly into the equation.
To express the electric field (E) of a lightwave in the positive x-direction at a given time and position, we can use the general equation of a sinusoidal wave:
E = E0 * sin(kx - ωt + φ)
Here's how to derive the expression:
1. Given the required properties:
- At t=0 and x=0, the E-field points in the negative z-direction.
- The angular frequency of the wave is ω.
2. The wave is propagating in the positive x-direction, so we know that the wave vector k is positive.
3. To determine the sign of the phase constant φ:
- At t=0 and x=0, the E-field points in the negative z-direction.
- We can represent the negative z-direction as a unit vector −ẑ.
- The E-field is perpendicular to the direction of propagation, so it lies in the yz-plane.
- Therefore, the E-field vector lies along the negative y-axis at t=0 and x=0.
- This implies that the phase constant φ is equal to -π/2 because the sine function has a negative value in the negative y-axis.
4. Plugging in the given values into the equation, we get:
E = E0 * sin(kx - ωt - π/2)
And that's the expression for the R-state lightwave with angular frequency ω and amplitude E0 propagating in the positive x-direction, such that at t=0 and x=0 the E-field points in the negative z-direction.