A candle and a screen are 73 cm apart. Find two points between candle and screen where you coul

A candle and a screen are 73 cm apart. Find two points...

Key Concepts

Thin Lens Equation

The thin lens equation (1/f = 1/u + 1/v) is fundamental in geometrical optics and describes the relationship between the focal length (f) of a lens and the distances from the lens to the object (u) and to the image (v). This equation allows one to predict the position of the image given the object distance and the lens's focal length, which is essential when designing optical systems that require precise image formation.

Object-Image Distance Constraint

When an object and its image are fixed at a certain distance apart, the distances from the lens to the object and image are constrained by their sum. In practical applications, this constraint combined with the thin lens equation often leads to a quadratic equation whose two solutions represent the two distinct lens positions that produce a sharp image of the object.

Conjugate Points in Optical Imaging

The concept of conjugate points refers to pairs of object and image points that are interrelated through the optical system such that one point is imaged clearly to the other. In the context of lens imaging, for a fixed separation between the object and the image, there are typically two sets of conjugate points (or lens positions) that satisfy the thin lens equation, illustrating the dual nature of image formation in converging lenses.

Transcript

00:01 In problem 49, it says a candle and a screen are 73 centimeters apart.

00:06 We're supposed to find the two points between the candle and the screen where you could put a convex lens with a 17 centimeter focal length to give a sharp image of the candle on the screen.

00:17 So 17 centimeters is given, and we know that the total distance between the candle and the screen is 73 centimeters.

00:31 So we're going to use the lens.

00:34 Equation which is 1 over f equals 1 over the object distance plus the one over the image distance but we're also going to use the fact and i'll put these in red and then i'll go back to black the object distance which is from the candle to the lens plus the image distance which is from the lens to the screen oops i did not mean to put a subscript there so the object distance is s and the image distance is s prime those two add to be 73 centimeters.

01:12 So we could replace s prime, for example, with 73 minus s.

01:20 Okay? so i just need to go back to black and try for it.

01:25 So we're going to use the lens equation, but with that substitution.

01:29 So i have 1 over 17, which is one over my focal length.

01:33 Everything's in centimeters, so i can leave the units off for now.

01:37 One over 17 is equal to 1 over s.

01:41 Plus 1 over s prime, which is 73 minus s.

01:49 So i need to combine the terms on my right -hand side.

01:53 So making a common denominator, i can do 73 minus s over s times 73 minus s plus 1, i'm sorry, plus s because i need to multiply by the thing it doesn't have.

02:12 So 73 minus s.

02:15 That's my right -hand side.

02:16 My left -hand side is still 1 over 17.

02:19 And then i have like terms because both denominators are the same.

02:23 So 73 minus s plus s is just 73 in the numerator.

02:30 And then i, of course, i still have the same 73 minus s in the denominator.

02:36 So 1 over 17 is equal to 73 over the quantity s times 73 minus s.

02:43 So we can just do a little bit of, cross multiplication.

02:48 17 times 73 is 1 ,241.

02:54 And then s times 73 minus s times 1 is just s times 73 minus s.

03:03 So i'm going to have a quadratic because i'll have an s term and an s squared term.

03:07 So let's just expand the right hand side.