## 物理代写|几何光学代写Geometrical Optics代考|PHYS201

2023年1月2日

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## 物理代写|几何光学代写Geometrical Optics代考|Classical Model for Electrical Conduction

In the following, we will describe a classical model of electrical conduction in conductors. Note that Paul Drude first proposed that model in 1900. The model explains Ohm’s law, and it shows that resistivity can be related to the motion of electrons in a conductor. Drude model has its limitations because it is a classical model; however, it introduces concepts that can be applied to more sophisticated models.

We will consider a conductor formed by atoms positioned in a regular array and a set of electrons that can freely move, which are called conduction electrons. If the atoms are not part of a solid, the conduction electrons are bound to atoms and are not free to move; however, when atoms condense into a solid, then conduction electrons become mobile. In there is no external electric field, the conduction electrons move in random directions in all the conductors with average speeds of the order of $10^6 \mathrm{~m} / \mathrm{s}$, similar to the motion of molecules in gas inside a container. Often, the conduction electrons in a conductor are assumed to form an electron gas. In the absence of the electric field, the drift velocity of the free electrons is zero, and hence there is no current, as presented in Fig. 5.6. In other words, on average, the same number electrons move in one direction as in the opposite direction, and thus the net charge flow is zero.

When an electric field is applied, as shown Fig. 5.7, the free electrons drift gently in the opposite direction of the electric field. Besides, the free electrons still undergo a random motion, as described in Fig. 5.6. Now, the average drift speed $v_d$ is much smaller (typically $10^{-4} \mathrm{~m} / \mathrm{s}$ ) than average speed between collisions (typically $10^6$ $\mathrm{m} / \mathrm{s})$. Therefore, the electric field $\mathbf{E}$ modified the random motion and made the electrons to drift in the opposite direction to the field. It is important to emphasize that there is slight curvature in the trajectories of the free electrons, as indicated in Fig. 5.7 because of the acceleration of the electrons between collisions. That is because the electric field applies a force on the free electrons.

In the classical model, we assume that the motion of an electron after a collision is independent of its motion before the collision. Besides, the excess energy gained by the electrons in the electric field is transferred to the atoms of the conductor during their collision. The energy transferred to the atoms increases the vibration energy of atoms, and therefore, the temperature of the conductor increases. Note that the temperature increase of a conductor because of the resistance can be used efficiently, such as in electric toasters and other familiar appliances. To derive a mathematical model, we will consider a free electron of mass $m_e$ and charge $q(-e)$ in an electric field $\mathbf{E}$.

## 物理代写|几何光学代写Geometrical Optics代考|Resistance and Temperature

It is found that the resistivity of a metal varies approximately linearly with temperature in a limited range of temperatures as follows:
$$\rho=\rho_0\left(1+\alpha\left(T-T_0\right)\right)$$

where $\rho$ is the resistivity at some temperature $T$ (in degrees Celsius), $\rho_0$ is the resistivity at some reference temperature $T_0$ (usually taken to be $20^{\circ} \mathrm{C}$ ), and $\alpha$ is the temperature coefficient of resistivity. It is easy to obtain the temperature coefficient of resistivity as
$$\alpha=\frac{1}{\rho_0} \frac{\rho-\rho_0}{T-T_0}=\frac{1}{\rho_0} \frac{\Delta \rho}{\Delta T}$$
The unit for $\alpha$ is degrees Celsius ${ }^{-1}\left[\left({ }^{\circ} \mathrm{C}\right)^{-1}\right]$. Because resistance is proportional to resistivity, we can write the variation of resistance as
$$R=R_0\left(1+\alpha\left(T-T_0\right)\right)$$

There exists a class of materials whose resistance decreases to zero below a specific temperature of $T_c$, known as the critical temperature. These materials are known as superconductors. If we would plot the resistance as a function of temperature for a superconductor, it follows that a superconductor behaves like a standard metal for $T>T_c$, and for $T \leq T_c$ its resistivity suddenly becomes zero. That was discovered by the Dutch physicist Heike Kamerlingh-Onnes (1853-1926), in 1911, working with mercury (a superconductor material below $4.2 \mathrm{~K}$ ). Recently, it has been shown that the resistivity of superconductors for $T<T_c$ is less than $4 \times 10^{-25} \Omega \cdot \mathrm{m}$; that is, around $10^{17}$ times lower than the resistivity of copper metal, which can practically be considered zero. There are thousands of superconductors with critical temperatures that are substantially higher than initially thought possible. Because of low values of resistivity of superconductors, once a current set up in a superconductor wire, it will persist without any applied potential difference. Note that, already, steady currents are observed to persist in superconducting loops for several years with no apparent decay.

# 几何光学代考

## 物理代写|几何光学代写Geometrical Optics代考|Resistance and Temperature

$$\rho=\rho_0\left(1+\alpha\left(T-T_0\right)\right)$$

$$\alpha=\frac{1}{\rho_0} \frac{\rho-\rho_0}{T-T_0}=\frac{1}{\rho_0} \frac{\Delta \rho}{\Delta T}$$

$$R=R_0\left(1+\alpha\left(T-T_0\right)\right)$$

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## MATLAB代写

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