a sinusoidal electromagnetic wave has an electric field whose rms value is 100 V/m. What is the instantaneous rate S of energy flow for this wave?

The instantaneous rate of energy flow in an electromagnetic wave can be described by the Poynting vector, **S**, which is given by the cross product of the electric field (E) and the magnetic field (B) vectors. For sinusoidal waves in free space, where the impedances are purely real, the magnitude of the Poynting vector |S| representing the power per unit area is simply the product of the RMS (root mean square) values of the electric and magnetic fields, divided by the characteristic impedance (Zo) of free space.

This characteristic impedance of free space (Zo) is approximately 377 ohms (Ω). The formula for the average power per unit area (S_avg) transported by the wave is then given by:

S_avg = E_rms * B_rms / Zo

Now, to find B_rms (the RMS value of the magnetic field), we use the relationship between E and B in an electromagnetic wave in free space, which is given by:

E = B * c

where: E is the peak value of the electric field (not the RMS value), B is the peak value of the magnetic field (not the RMS value), c is the speed of light in a vacuum (~3 x 10^8 m/s).

For RMS values, this relationship becomes:

E_rms = B_rms * c

Therefore, we can solve for B_rms by rearranging the above equation:

B_rms = E_rms / c

Using the given value for E_rms (100 V/m), we can find B_rms:

B_rms = 100 V/m / (3 x 10^8 m/s)

Once we have calculated B_rms, we can then find the average power per unit area (S_avg):

S_avg = E_rms * B_rms / Zo

Let's do these calculations step-by-step:

1. Calculate B_rms: B_rms = 100 V/m / (3 x 10^8 m/s) B_rms = 3.333 x 10^(-7) T (Tesla)

2. Calculate S_avg: S_avg = 100 V/m * 3.333 x 10^(-7) T / 377 Ω S_avg ≈ (100 * 3.333 x 10^(-7)) / 377 W/m^2 S_avg ≈ 8.85 x 10^(-4) W/m^2

So, the instantaneous rate S of energy flow (averaged over one cycle of the wave) for this electromagnetic wave is approximately 8.85 x 10^(-4) watts per square meter.