Optics

The formula $\frac{1}{p}+\frac{1}{i}=\frac{1}{f}$ is called the Gaussian form of the thin-lens formula. another form of this formula, the Newtonian form, is obtained by considering the distance $x$ from the object to the first focal point and the distance $x'$ from the second focal point to the image. I am going to show that $xx'=f^{2}$ is the Newtonian form of the thin-lens formula.

In this situation, my diagram is going to look as follows below:

Starting with the thin lens formula, $\frac{1}{p}+\frac{1}{i}=\frac{1}{f}$ , I note that the object distance is $x+f$ , and the image distance is $x'+f$ , where the focal distances for the two lenses are equivalent. After making these substitutions into the thin lens formula, and solving for $f$ I obtain the equation $f=\frac{xx'+fx+fx'+f^{2}}{x'+x+2f}$ . Now I can multiply each side of the equation by $x'+x+2f$ and solve for $f^{2}$ to obtain the final solution $f^{2}=xx'$ , which is the Newtonian form of the thin lens equation.

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A luminous point is moving at speed $V_0$ toward a spherical mirror with radius of curvature $r$ , along a central axis of the mirror. I am gong to prove that the image of this point is moving at speed $V_{I}=-(\frac{r}{2p-r})^2 V_{0}$

Since it is a spherical mirror I am going to start with a given equation whose derivation I will not provide. This equation relates the focal length, image distance, and object distance of a spherical mirror and that equation is $\frac{1}{p}+\frac{1}{i}=\frac{1}{f}$ . Since I am trying to find the speed of the image produced by the spherical mirror, I will solve for $i$ , which is the image distance. After doing this, I obtain the following equation $i=\frac{pf}{p-f}$ . Since $f$ is related to $r$ by the equation $f=\frac{r}{2}$ , I can plug this into my equation for $i$ , which gives me $i=\frac{pr}{2p-r}$ . Since $i$ is a “position”, if i take the first time derivative of it, I will obtain the image velocity or $V_{I}$ . After taking this derivative by applying the quotient rule, and treating $r$ , the distance to the center of curvature, as a constant, I obtain the following formula: $\frac{\frac{dp}{dt}(2p-r)+2\frac{dp}{dt}pr}{(2p-r)^2}$ . Here, I notice that $\frac{dp}{dt}$ is simply the velocity of the object. Since the two velocities will have different directions, I let the object distance be negative. After making the substitution $V_{0}=-\frac{dp}{dt}$ , I obtain the following equation $\frac{-V_{0}(2p-r)+2V_{0}pr}{(2p-r)^2}$ . Now I can distribute $-V_{0}$ , and cancel certain like terms. After doing so the final equation becomes $V_{I}=-(\frac{r}{2p-r})^2 V_{0}$ which is what I sought to prove.

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Newtonian Form of the Thin-Lens Formula

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