Which equation correctly describes the electromagnetic wave shown above Ex-Eo sin(kz wt) Ey E0 sin(kz-at) By-Bo sinikz-at) You currently have
O Briefly explain your reasoning
Which of the following actions will increase the energy carried by this wave?
O Increase E keeping w constant
O Increase w keeping E constant
O Both of the above actions will increase the energy of the wave
O Neither of the above actions will increase the energy of the wave
The correct answer and explanation is:
The correct equation for the electromagnetic wave is:
Ex=E0sin(kz−ωt)\text{E}_x = \text{E}_0 \sin(kz – \omega t)
Explanation:
An electromagnetic wave is composed of electric (E\mathbf{E}) and magnetic (B\mathbf{B}) fields that oscillate perpendicular to each other and to the direction of wave propagation. The general form of the wave equation for both fields in a plane wave is:
- E=E0sin(kz−ωt)\mathbf{E} = \mathbf{E}_0 \sin(kz – \omega t)
- B=B0sin(kz−ωt)\mathbf{B} = \mathbf{B}_0 \sin(kz – \omega t)
Where:
- E0\mathbf{E}_0 and B0\mathbf{B}_0 are the magnitudes of the electric and magnetic fields, respectively.
- kk is the wave number (related to the wavelength λ\lambda by k=2πλk = \frac{2\pi}{\lambda}).
- ω\omega is the angular frequency (related to the frequency ff by ω=2πf\omega = 2\pi f).
- tt is time, and zz is the spatial position in the direction of propagation.
This equation reflects the wave’s oscillations along the direction of propagation, which is typically chosen to be the zz-axis for simplicity.
Now, regarding the options for increasing the energy of the wave:
- Increase EE keeping ω\omega constant:
- The energy carried by an electromagnetic wave is directly proportional to the square of the amplitude of the electric field (E02E_0^2).
- Increasing E0E_0 (amplitude of the electric field) increases the energy carried by the wave because the energy density of an electromagnetic wave is proportional to E02E_0^2.
- Therefore, increasing E0E_0 while keeping ω\omega constant will increase the wave’s energy.
- Increase ω\omega keeping EE constant:
- The energy of an electromagnetic wave is also related to the frequency (ω\omega) through the photon energy, as E=hνE = h\nu, where ν=ω2π\nu = \frac{\omega}{2\pi} is the frequency, and hh is Planck’s constant.
- Increasing ω\omega increases the energy carried by the wave since higher frequency means more energy per photon.
So, the correct answer is:
Both of the above actions will increase the energy of the wave.
Explanation:
The energy carried by the wave depends on both the amplitude (E0E_0) and the frequency (ω\omega) of the wave. By either increasing the electric field amplitude E0E_0 (which increases the energy density) or increasing the frequency ω\omega (which increases the energy per photon), the overall energy carried by the wave is increased. Therefore, both actions will result in higher energy for the wave.