Правка раздела «Solution»
en/13.4.13.md
+1 −1
| ### Statement | |||
| $13.4.13.$ In front of a spherical mirror of radius of curvature $R$, the focus of which is a point light source $S$, at a height $h$ above the axis of the mirror $SO$, a small plate is placed, the plane of which is perpendicular to the axis. Find the ratio of illuminations of the left and right sides of the plate, if $h\ll R,l$. | |||
| ### Solution | |||
|  | |||
| Suppose the power of the light source is $P$. Then the intensity of the light falling on the right side of the plate is the same as that at the mirror, i.e. $P/(4\pi f^2)=P/(\pi R^2)$, because light rays reflected from the mirror are parallel to the principal axis as the source is at the focal point. On the other hand, the intensity of the light falling on the left side of the plate is | |||
| @@ -10,8 +10,8 @@Solution | |||
| \[\frac{P\cos\theta}{4\pi(l^2+h^2)}=\frac{Pl}{4\pi(l^2+h^2)^{3/2}}\approx\frac{P}{\pi(2l)^2},\] | |||
| − | where $\theta$ is the angle between the principal axis and the line joining the source and the plate. ($h\ll l$ but is nonzero, because the plate must not obstruct the light th | ||
| + | where $\theta$ is the angle between the principal axis and the line joining the source and the plate. ($h\ll l$ but is nonzero, because the plate must not obstruct the light that has to travel to the mirror and be reflected onto the right side of the plate.) Thus, the ratio of the intensities on the left and the right sides of the plate is $(R/(2l))^2$. | ||
| #### Answer | |||
| $(R/(2l))^2$ | |||
| ### Statement | ### Statement | ||
| $13.4.13.$ In front of a spherical mirror of radius of curvature $R$, the focus of which is a point light source $S$, at a height $h$ above the axis of the mirror $SO$, a small plate is placed, the plane of which is perpendicular to the axis. Find the ratio of illuminations of the left and right sides of the plate, if $h\ll R,l$. | $13.4.13.$ In front of a spherical mirror of radius of curvature $R$, the focus of which is a point light source $S$, at a height $h$ above the axis of the mirror $SO$, a small plate is placed, the plane of which is perpendicular to the axis. Find the ratio of illuminations of the left and right sides of the plate, if $h\ll R,l$. | ||
| ### Solution | ### Solution | ||
|  |  | ||
| Suppose the power of the light source is $P$. Then the intensity of the light falling on the right side of the plate is the same as that at the mirror, i.e. $P/(4\pi f^2)=P/(\pi R^2)$, because light rays reflected from the mirror are parallel to the principal axis as the source is at the focal point. On the other hand, the intensity of the light falling on the left side of the plate is | Suppose the power of the light source is $P$. Then the intensity of the light falling on the right side of the plate is the same as that at the mirror, i.e. $P/(4\pi f^2)=P/(\pi R^2)$, because light rays reflected from the mirror are parallel to the principal axis as the source is at the focal point. On the other hand, the intensity of the light falling on the left side of the plate is | ||
| @@ -10,8 +10,8 @@Solution | |||
| \[\frac{P\cos\theta}{4\pi(l^2+h^2)}=\frac{Pl}{4\pi(l^2+h^2)^{3/2}}\approx\frac{P}{\pi(2l)^2},\] | \[\frac{P\cos\theta}{4\pi(l^2+h^2)}=\frac{Pl}{4\pi(l^2+h^2)^{3/2}}\approx\frac{P}{\pi(2l)^2},\] | ||
| where $\theta$ is the angle between the principal axis and the line joining the source and the plate. ($h\ll l$ but is nonzero, because the plate must not obstruct the light th |
where $\theta$ is the angle between the principal axis and the line joining the source and the plate. ($h\ll l$ but is nonzero, because the plate must not obstruct the light that has to travel to the mirror and be reflected onto the right side of the plate.) Thus, the ratio of the intensities on the left and the right sides of the plate is $(R/(2l))^2$. | ||
| #### Answer | #### Answer | ||
| $(R/(2l))^2$ | $(R/(2l))^2$ | ||