Two identical concave mirrors are set facing each other 1.0 m ...

Question

Two identical concave mirrors are set facing each other 1.0 m apart. A small lightbulb is placed halfway between the mirrors. A small piece of paper placed just to the left of the bulb prevents light from the bulb from directly shining on the left mirror, but light reflected from the right mirror still reaches the left mirror. A good image of the bulb appears on the left side of the piece of paper. What is the focal length of the mirrors?

Accepted Answer

Hello, fellow physicists today we're gonna solve for the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use. In order to solve this problem, a child is playing with a large perfectly spherical sapphire marble that has a refractive index of N equals 1.77 and a diameter of 8.0 centimeters. She places a small figurine 4.8 centimeters away from the edge of the marble and notices that an image of the figurine forms on the opposite side of the marble, consider what would happen if the marble was replaced with a thin lens that is placed where the center of the marble had been located originally. And if the image retains the same position, what would be the lenses focal length in this case. So that's our end goals. We're trying to figure out what the focal length would be in this particular situation where the marble is replaced with a thin lens. Awesome. We're also given some multiple choice answers. They're all in the same units of centimeters. So let's read them off to see what our final answer might be A is 4.0. B is 4.6 C is 4.8 and D is 5.6. OK. So first off, let us make the following assumptions that for first assumption, let's say that the marble will behave like a convex lens. Also let us assume that the medium surrounding the marble is air and air has a refractive index of 1.00. And let's also quickly note that this, that the refractive index of the sapphire is approximately equal to 1.77 as given to us in the prom itself. Now, if we were to draw an illustration of the ray tracing for this particular problem, we would get the following diagram, we would get that our marble, which is represented by a blue circle. It has an 8.0 centimeter diameter and we have our little gray arrow pointing upwards. And this gray arrow on the left pointing upwards represents our small figurine that's placed 4.8 centimeters away from the edge of the marble. We also know that we have our S one value between the marble and the first black arrow. And then we have our S one prime value which is between our object and the marble. And then our S two value is between our object and the end of our marble here. And our S two prime value is from the marble to where our image position is located. So we'll go more into depth about this ray tracing diagram in a minute, but to give us a little hint about what's gonna happen. So from this ray tracing method that we have just performed, it is hinting that the first image will be virtual and upright. So we need to prove with math that this is correct. So moving right along. So since the sapphire marble is spherical, the image produced due to refraction will be the result of the light refraction from the first surface being the marble's front side surface. And the second surface being the backside surface of the marble. So now we can determine the distance of the second image from the sphere which is S two prime by solving for S one prime first. So right now we need to determine the distance of the second image from the sphere which is denoted as S two prime. And to do this, we need to solve for S one prime first. And to do that, we need to use the following equation, we need to use that N one divided by S one plus N two, divided by S one prime is equal to N two minus N one all divided by capital R. So note that in this case, capital R is equal to positive 4.0 centimeters. Since the refraction from the first surface is positive 4.0 centimeters. And this is due to the fact that it's convex towards the lens. So with that in mind, we can now plug in our known variables. So when we do, we know that N one is equal to 1.0 which is the refractive index of air divided by 4.8 centimeters plus our N two value, which is 1.77 divided by our S one prime value, which is a value we do not know which is equal to 1.77 minus 1.0 all divided by 4.0 centimeters. So we need to now rearrange this equation to isolate and sulfur S one prime. So when we do that, we'll get that S one prime is equal to negative 1200 centimeters multiplied by 1.77 divided by 19, which is equal to negative 111.78 centimeters, which is approximately equal to negative 111.8 centimeters when you round to one decimal place. Also let us note that as expected using the ray tracing method as shown above, the first image will be virtual and upright. So now if we consider the refraction from the second surface, we can say that S two is equal to 118.8 centimeters plus or 111 my bad 111.8 centimeters plus 8.0 centimeters is equal to 119.8 centimeters. Awesome. And we'll note in this case, R equals negative 4.0 centimeters because it is concave towards the lens. Thus, we can go on to write that 1.77 divided by 119.8 centimeters plus 1.0 divided by S two prime is equal to 1.0 minus 1.77 divided by negative one, sorry, negative 4.0 centimeters. Awesome. So like before we need to simplify and rearrange this equation to isolate and solve for S two prime. So when we do that, we'll get that S two prime is equal to two, 39600 centimeters multiplied by 1.0 all divided by 425 83. So when we plug that into a calculator, we will get that S two prime is equal to 5.63 centimeters. Thus, the image is 5.63 centimeters away from the right edge of the marble. Thus 9.6 centimeters away from the center. Therefore, we can go ahead and write that one divided by S plus one, divided by S prime is equal to one divided by F. Now, we can plug in our known variables and start solving for F the focal length, which is our final answer. So one divided by F is equal to 4.8 centimeters plus 4.0 centimeters. So it's one divided by 4.8 centimeters plus 4.0 centimeters plus one divided by 5.63 centimeters plus 4.0 centimeters, which is equal to 0.217478 5235. So to solve for F all we have to do is take F is equal to one divided by 0.2. And don't forget this is in centimeters in verse one. So 0.21747852 52, 35 centimeters. So when we plug that into a calculator, we will get 4.59 centimeters, which is approximately equal to 4.6 centimeters. When we round to one decimal place, we did it, we found our final answer. So our focal length for the lens has to be approximately 4.6 centimeters. So looking at our multiple choice answers, the correct answer has to be the letter B 4.6 centimeters. Thank you so much for watching. Hopefully, that helped and I can't wait to see you in the next video. Bye.

Key Concepts

Concave Mirrors

Image Formation

Mirror Equation

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