Edits to “Statement”, “Solution”, “Answer”
en/13.1.14.md
+12 −5
| @@ -1,13 +1,20 @@ | |||
| ### Statement | |||
| − | $13.1.14.$ | ||
| + | $13.1.14.$ Show that if the distances from the subject and image to the focus of a concave | ||
| + | mirror are $l_1$ and $l_2$, then $$l_1 \cdot l_2 = f^2$$ where f is the focal length of the mirror. | ||
| ### Solution | |||
| − | |||
|  | |||
| + | We are going to use the formula for a thin lens that relates the object distance $a$ image distance $b$ and focal length $f$. Look at the picture above as a referance: $$\frac{1}{f} = \frac{1}{a} + \frac{1}{b}$$ | ||
| + | We also know that $a = l_1 + f$ and $b = l_2 + f$. Putting this into our formula: | ||
| + | $$\frac{1}{f} = \frac{1}{l_1 + f} + \frac{1}{l_2 + f}$$ | ||
| + | after simplifying: | ||
| + | $$f = \frac{(l_1 + f)(l_2 + f)}{l_1 + l_2 + 2f}$$ | ||
| + | Multiplying both sides by $l_1 + l_2 + 2f$ gives us: | ||
| + | $$fl_1 + fl_2 + 2f^2 = l_1l_2 + l_1f + l_2f + f^2$$ | ||
| + | $$2f^2 = l_1l_2 + f^2$$ | ||
| + | $$f^2 = l_1l_2$$ | ||
| − | |||
| − | |||
| #### Answer | |||
| − | [Insert a concise answer or boxed result] | ||
| + | [$f^2 = l_1l_2$] | ||
| @@ -1,13 +1,20 @@ | |||
| ### Statement | ### Statement | ||
| $13.1.14.$ |
$13.1.14.$ Show that if the distances from the subject and image to the focus of a concave | ||
| mirror are $l_1$ and $l_2$, then $$l_1 \cdot l_2 = f^2$$ where f is the focal length of the mirror. | |||
| ### Solution | ### Solution | ||
|  |  | ||
| We are going to use the formula for a thin lens that relates the object distance $a$ image distance $b$ and focal length $f$. Look at the picture above as a referance: $$\frac{1}{f} = \frac{1}{a} + \frac{1}{b}$$ | |||
| We also know that $a = l_1 + f$ and $b = l_2 + f$. Putting this into our formula: | |||
| $$\frac{1}{f} = \frac{1}{l_1 + f} + \frac{1}{l_2 + f}$$ | |||
| after simplifying: | |||
| $$f = \frac{(l_1 + f)(l_2 + f)}{l_1 + l_2 + 2f}$$ | |||
| Multiplying both sides by $l_1 + l_2 + 2f$ gives us: | |||
| $$fl_1 + fl_2 + 2f^2 = l_1l_2 + l_1f + l_2f + f^2$$ | |||
| $$2f^2 = l_1l_2 + f^2$$ | |||
| $$f^2 = l_1l_2$$ | |||
| #### Answer | #### Answer | ||
| [Insert a concise answer or boxed result] | [$f^2 = l_1l_2$] | ||