# 物理代写|力学代写mechanics代考|CIVL2210

## 物理代写|力学代写mechanics代考|The Ellipsoid Mirror

Consider an axisymmetric ellipsoid mirror with semiaxes $a$ and $b$ along the $z$ and $r$ axes, respectively, illuminated by a point light source $S$ placed along its axis of symmetry $z$ at a distance $A$ from the $r$-axis (Fig. 6.2). A reference screen is placed at a distance $z_0$ from the $r$-axis. The equation of the surface $z=f(x, y)$ of the mirror referred to the system $\operatorname{Orz}$ is
$$z=f(x, y)=\frac{a}{b}\left(b^2-r^2\right)^{1 / 2}$$
The equation of the caustic on the screen is given by Eq. (6.11) with
$$\tan \alpha=\frac{\mathrm{d} z}{\mathrm{~d} r}, \quad \tan \varphi=\frac{r}{A+z}$$
Using Eq. (6.13) we obtain the following equation for the initial curve of the caustic

$$\frac{z_0}{b}=\frac{B_1\left[1-\left(\frac{r}{b}\right)^2\right]^{1 / 2}+B_2}{\Delta_1\left[1-\left(\frac{r}{b}\right)^2\right]^{1 / 2}+\Delta_2}$$
with
\begin{aligned} B_1=& 2\left(\frac{a}{b}\right)^2\left{1+\left[\left(\frac{a}{b}\right)^2-1\right]\left(\frac{r}{b}\right)^2-\left(\frac{a}{b}\right)^2\right} \ &+\left{1-\left[\left(\frac{a}{b}\right)^2+1\right]\left(\frac{r}{b}\right)^2-2\left(\frac{a}{b}\right)^2\right}\left(\frac{\mathrm{A}}{b}\right)^2 \ B_2=\left(\frac{A}{b}\right)\left(\frac{a}{b}\right)\left(3\left{1+\left[\left(\frac{a}{b}\right)^2-1\right]\left(\frac{r}{b}\right)^2\right}-4\left(\frac{a}{b}\right)^2\right) \ \Delta_1 &=\left(\frac{A}{b}\right)\left{1+\left[\left(\frac{a}{b}\right)^2-1\right]\left(\frac{r}{b}\right)^2-4\left(\frac{a}{b}\right)^2\right} \ \Delta_2=&\left(\frac{a}{b}\right)\left(\left{1+\left[\left(\frac{a}{b}\right)^2-1\right]\left(\frac{r}{b}\right)^2\right}\right.\ &\left.-2\left{\left(\frac{a}{b}\right)^2-\left[\left(\frac{a}{b}\right)^2-1\right]\left(\frac{r}{b}\right)^2\right}-2\left(\frac{A}{b}\right)^2\right) \end{aligned}

## 物理代写|力学代写mechanics代考|Intensity Distribution of Light Rays Reflected

The intensity of light reflected from the front or the rear face of the specimen is the same at each reflection; let $\beta$ be the reduction ratio of the intensity of reflected rays relative to the incident rays $(\beta<1)$. The reduction ratio for the rays that refract at the two faces differs for the rays that enter or emerge from the specimen. Let $\alpha$ and $\alpha^{\prime}$ be the corresponding reduction ratios.

If $I$ is the intensity of the incident light ray, the intensity of the first reflected ray is $\beta I$ and the intensity of the first refracted ray is $\alpha I$. This last ray emerges from the rear face of the specimen with intensity $\alpha \alpha^{\prime}$ I. At reflection from rear face, the ray passes through the thickness of the specimen with intensity $\alpha \beta 1$. The intensity of light for the first three rays that emerge from the front face and the first two rays that emerge from the rear face of the specimen is shown in Fig. 6.5. According to the law of conservation of energy and assuming that the specimen does not absorb light energy we have
$$I=I_f+I_r$$
where $I_f$ and $I_r$ are the total intensity of light that emerges from the front and rear faces of the specimen, respectively.
We obtain for the intensities $I_f$ and $I_r$
$$\begin{gathered} I_f=\beta I+\alpha \alpha^{\prime} \beta\left(1+\beta^2+\beta^4+\beta^6+\cdots\right) I=\beta\left(1+\frac{\alpha \alpha^{\prime}}{1-\beta^2}\right) I, \quad \beta<1 \ I_r=\alpha \alpha^{\prime}\left(1+\beta^2+\cdots\right) I=\frac{\alpha \alpha^{\prime} I}{1-\beta^2} \end{gathered}$$
Introducing the values of $I_f$ and $I_r$ into Eq. (6.23) we obtain
$$I=\beta\left(1+\frac{\alpha \alpha^{\prime}}{1-\beta^2}\right) I+\frac{\alpha \alpha^{\prime} I}{1-\beta^2}$$
from which it follows that
$$\alpha \alpha^{\prime}=(1-\beta)^2$$

## 物理代写|力学代写mechanics代考|The Ellipsoid Mirror

$$z=f(x, y)=\frac{a}{b}\left(b^2-r^2\right)^{1 / 2}$$

$$\tan \alpha=\frac{\mathrm{d} z}{\mathrm{~d} r}, \quad \tan \varphi=\frac{r}{A+z}$$

$$\frac{z_0}{b}=\frac{B_1\left[1-\left(\frac{r}{b}\right)^2\right]^{1 / 2}+B_2}{\Delta_1\left[1-\left(\frac{r}{b}\right)^2\right]^{1 / 2}+\Delta_2}$$

## 物理代写|力学代写mechanics代考|Intensity Distribution of Light Rays Reflected

$$I=I_f+I_r$$

$$I_f=\beta I+\alpha \alpha^{\prime} \beta\left(1+\beta^2+\beta^4+\beta^6+\cdots\right) I=\beta\left(1+\frac{\alpha \alpha^{\prime}}{1-\beta^2}\right) I, \quad \beta<1 I_r=\alpha \alpha^{\prime}\left(1+\beta^2+\cdots\right) I$$

$$I=\beta\left(1+\frac{\alpha \alpha^{\prime}}{1-\beta^2}\right) I+\frac{\alpha \alpha^{\prime} I}{1-\beta^2}$$

$$\alpha \alpha^{\prime}=(1-\beta)^2$$

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