# Energy conservation: electromagnetic wave in matter

• happyparticle
In summary, the conversation discusses the concept of a damped driven oscillator and how it can be used to model the oscillation of electrons inside a conductor. The equation of motion and solution for a damped driven oscillator are provided, and it is mentioned that by solving this equation, the amplitude of oscillation of the electrons can be found. The maximum kinetic energy of the electrons is also mentioned, and the conversation ends with a request for help.
happyparticle
Homework Statement
An electromagnetic wave passes through a material with conductivity ##\sigma##. The wave is attenuated, and the amplitude exponentially decreases with the distance traveled. Show that the total energy is conserved and what happen with the energy lost by the wave.
Relevant Equations
$$u = u_e + e_m$$
$$\tilde{E} (z,t) = \hat{E}_0 e^{i(\tilde{k}z - \omega t)} \hat{x}$$
$$\tilde{B} (z,t) = \hat{E}_0 \frac{\tilde{k}}{\omega}e^{i(\tilde{k}z - \omega t)} \hat{y}$$
Hi,
I completely failed this homework. I mean I think I know what happen, but I don't know how to show it mathematically. The energy lost by the wave is used to oscillate the electrons inside the conductor. Thus, the electrons acts like some damped driven oscillators.
I guess I have to find ##e_m, u_e##, but I don't know with what to compare. That's pretty all I know.
Any help will me appreciate, thanks.

The energy lost by the wave is used to oscillate the electrons inside the conductor, so we can model the system as a damped driven oscillator. The equation of motion for a damped driven oscillator is given by:$$\ddot{x} + 2 \beta \dot{x} + \omega_0^2 x = F_0 \cos(\omega t)$$ where $\beta$ is the damping coefficient, $\omega_0$ is the natural frequency of the oscillator and $F_0$ is the amplitude of the driving force. The solution to this equation is given by: $$x(t) = A_1 e^{-(\beta + i \omega_d)t} + A_2 e^{-(\beta - i \omega_d)t} + \frac{F_0}{m(\omega_0^2 - \omega^2 + i 2 \beta \omega)} \cos(\omega t)$$where $\omega_d = \sqrt{\omega_0^2 - \beta^2}$. The constants $A_1$ and $A_2$ are determined by the initial conditions of the system. By solving this equation you can find the amplitude of oscillation of the electrons in the conductor. The maximum kinetic energy of the electrons is given by $e_m = \frac{1}{2}mv_e^2$, where $v_e$ is the peak velocity of the electrons. Hope this helps.

## 1. What is energy conservation?

Energy conservation is the principle that states energy cannot be created or destroyed, but can only be converted from one form to another. This means that the total amount of energy in a closed system remains constant.

## 2. What is an electromagnetic wave?

An electromagnetic wave is a type of energy that is created by the movement of electrically charged particles. It consists of oscillating electric and magnetic fields that travel through space at the speed of light.

## 3. How does energy conservation apply to electromagnetic waves in matter?

Energy conservation applies to electromagnetic waves in matter because these waves are a form of energy that can be transferred to matter and converted into other forms of energy, such as heat or light.

## 4. What factors affect the conservation of energy in electromagnetic waves?

The conservation of energy in electromagnetic waves can be affected by factors such as the type of matter the wave is passing through, the wavelength and frequency of the wave, and the temperature and pressure of the medium.

## 5. How can we conserve energy in electromagnetic waves?

To conserve energy in electromagnetic waves, we can use materials that are good conductors and reflectors of these waves, such as metals, to minimize energy loss. We can also use technologies like solar panels to capture and convert electromagnetic energy into usable forms.

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