Total Magnification of Two Converging Lenses

[aChordate]

Homework Statement

Two identical converging lenses have focal lengths of 15 cm and are aligned on the
same principal axis. They are separated by a distance of 120 cm. An object sits a distance
of 21 cm to the left of the left lens. What is the total magnification of the two lenses?

Homework Equations

Thin lens equation
1/f=1/do+1/di

magnification equation
m=di/do

The Attempt at a Solution

Lens 1:
1/.15m=1/.21+1/di

d-i=0.525m

m=0.525m/0.21m=2.5

Lens 2:
1/.15=1/(.21+1.20m)+1/di

d-i=0.168m

m=0.80


m-total=0.80+2.5 ?

 

[collinsmark]

Homework Statement

Two identical converging lenses have focal lengths of 15 cm and are aligned on the
same principal axis. They are separated by a distance of 120 cm. An object sits a distance
of 21 cm to the left of the left lens. What is the total magnification of the two lenses?

Homework Equations

Thin lens equation
1/f=1/do+1/di

That's the right equation.

magnification equation
m=di/do

You're missing a negative sign in there somewhere.

The Attempt at a Solution

Lens 1:
1/.15m=1/.21+1/di

d-i=0.525m

I think you mean "di" or d_i instead of "d-i". But whatever the case, your number is correct.

m=0.525m/0.21m=2.5

You're missing a negative sign somewhere.

Lens 2:
1/.15=1/(.21+1.20m)+1/di

Here is where things start to go south (read: here is where things start to go wrong).

You don't want to use the actual distance to the original object when evaluating the second lens. Instead, when evaluating the second lens, treat the image produced by the first lens as the "object."

It might help to draw a diagram.

Draw the first lens and the original object. Then draw the image created by the first lens. That image is the "object" that you will be using for the second lens.

d-i=0.168m

m=0.80

Of course you'll have to redo the image produced by the second lens. Use the image produced by the first lens as the second lens' object.

m-total=0.80+2.5 ?

When it does come time to combine the magnifications, you'll need to multiply the respective magnifications from the individual lenses, not add them.

 

[aChordate]

On my second attempt I got an mtotal of 0.357

 

[collinsmark]

On my second attempt I got an mtotal of 0.357

I arrived at a different answer.

Could you show how you got your answer? Maybe I can help then.

 

[aChordate]

Lens 1:
1/.15m=1/.21+1/di

d-i=-0.525m

m=-0.525m/0.21m=-2.5

Lens 2:
1/.15=1/(.21+1.20m)+1/di

d-i=0.171m

m=-0.171/1.2m=-0.143


m-total=-2.5*-.143=0.357

 

[collinsmark]

Lens 1:
1/.15m=1/.21+1/di

d-i=-0.525m

m=-0.525m/0.21m=-2.5

Yes, so far so good.

Lens 2:
1/.15=1/(.21+1.20m)+1/di

Here's the problem. For the second lens, you are using the total distance all the way back to the original object for d_O. Essentially, this is as through the first lens isn't even there!

Instead, you need to use the image produced by the first lens as the second's lens' object. You've already calculated that it lies 52.5 cm to the right of the first lens. This image of the first lens is what you want to use as the object of the second lens.

 

[aChordate]

Yes, so far so good.


Here's the problem. For the second lens, you are using the total distance all the way back to the original object for d_O. Essentially, this is as through the first lens isn't even there!

Instead, you need to use the image produced by the first lens as the second's lens' object. You've already calculated that it lies 52.5 cm to the right of the first lens. This image of the first lens is what you want to use as the object of the second lens.

Oops, I meant to correct that on the last attempt. Let me try again.

Lens 1:
1/.15m=1/.21+1/di

d-i=-0.525m

m=-0.525m/0.21m=-2.5

Lens 2:

do=1.20m-.525m=0.675m

1/.15=1/0.675m+1/di

d-i=0.193

m=-0.193m/1.2m=-0.161


m-total=-2.5*-0.161=0.402


I hope that's correct!

 

[collinsmark]

Oops, I meant to correct that on the last attempt. Let me try again.

Lens 1:
1/.15m=1/.21+1/di

d-i=-0.525m

m=-0.525m/0.21m=-2.5

So far so good, again.

Lens 2:

do=1.20m-.525m=0.675m

1/.15=1/0.675m+1/di

d-i=0.193

Also good!

m=-0.193m/1.2m=-0.161

Instead of using 120 cm as the distance to the object, use the distance of the first lens' image to the second lens. (Hint: you've already calculated this. This is the second lens' "object" distance. )

 

[aChordate]

m=-0.193m/0.675m=-0.286

mtotal=-2.5*-.286=.715

thank you!

 

[collinsmark]

m=-0.193m/0.675m=-0.286

mtotal=-2.5*-.286=.715

thank you!

There you go.

(If you use more precision in your intermediate calculations, you might obtain a slightly better answer due to rounding; but your above answer is roughly correct.)