The distance of a lightbulb from a large plane mirror is...

The distance of a lightbulb from a large plane mirror is twice the distance of a person from the plane mirror. Light from the lightbulb reaches the person by two paths. It travels to the mirror at an angle of incidence $\theta$ and reflects from the mirror to the person. It also travels directly to the person without reflecting off the mirror. The total distance traveled by the light in the first case is twice the distance traveled by the light in the second case. Find the value of the angle $\theta .$

Key Concepts

Law of Reflection

The law of reflection states that the angle of incidence is equal to the angle of reflection. This principle is fundamental in the study of light and optics, as it determines how light rays behave when they strike reflective surfaces such as mirrors.

Image Formation in Plane Mirrors

In plane mirror optics, the image of an object appears to be located the same perpendicular distance behind the mirror as the object is in front of it. This concept is crucial for understanding how light paths are related via reflection and for setting up geometric relationships in optical problems.

Geometric Analysis of Optical Paths

Using geometric principles to analyze the paths traveled by light involves relating distances and angles within a system. This technique allows one to compare different routes, such as direct versus reflected paths, by establishing relationships between distances traveled and angles of incidence, which is essential for solving problems in geometrical optics.

Transcript

00:01 So this question is going to require us to go all the way back to geometry class.

00:06 It is actually quite difficult if you do not remember a certain rule called the law of cosines.

00:12 So the first thing we're going to do is we're going to draw this situation.

00:22 So we have our observer.

00:27 No, we have our light here, right? this is our light.

00:30 Our light bulb.

00:31 I'll denote this in yellow.

00:36 And then we have our person right here.

00:39 So there is a distance between them.

00:42 Let's call this c.

00:43 So we are told that this person is, or the light bulb, is twice as far away from the mirror as the person.

00:56 So we're going to call this the distance between the person and the wall x and the distance between the light and the wall 2x.

01:04 And we're going to call this distance from the light to the point at which it is reflected, a, and the distance from which it is reflected to the person, b.

01:19 So we are told that the distance from the light bulb to the mirror to the person is twice the distance of the exact distance between the light bulb and the person directly.

01:40 So that means that 2c is equal to a plus b.

01:47 Next, we're going to note that if we were to bisect this angle, right, we were to bisect this angle by making this line perpendicular to the plane mirror, we would have theta here and theta prime here.

02:05 And as we know, theta is equal to theta prime.

02:11 So the next thing we're going to do is to take into account that these two angles, we're going to call this phi and this one phi.

02:20 And since these two angles are the same, these ones are going to be as well.

02:25 So since these two angles are the same, that means that this angle is going to be 90 minus this angle.

02:32 It's going to be the same for this case.

02:33 So we'll note that these have all three of the same angles.

02:41 So we have a 90 degree angle.

02:42 We have phi and this angle.

02:44 They're all the same in both of these triangles, which means that these are both similar triangles.

02:49 So now that we've noticed that these are similar triangles, we can identify that b is the hypotenuse of this triangle and a is the hypotenuse of this triangle.

02:59 So since this has a bottom of x, right, x is to 2x, we have b is to 2, 2.

03:20 So that means that the ratio of b to a is going to be the same.

03:27 A is going to be to b...

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