Two thin lenses with focal lengths t and q and separated by a distance have their combined focal length given by the reciprocal of . Combine the fractions in the expression above and use the reciprocal of your result to find the combined focal length.

Question

Two thin lenses with focal lengths t and q and separated by a distance  have their combined focal length given by the reciprocal of  . Combine the fractions in the expression above and use the reciprocal of your result to find the combined focal length. ​

Quizplus · Accepted Answer

Using the lens formula, $\frac{1}{f}=\frac{1}{v}-\frac{1}{u}$, we can find the image distance ($v$) for the first lens and the object distance ($u$) for the second lens. Then, we can use the combined lens formula, $\frac{1}{f_{total}}=\frac{1}{f_1}+\frac{1}{f_2}-\frac{d}{f_1f_2}$, to find the total focal length.

Let's start by finding $v$ for the first lens, which is the object distance ($u$) for the second lens. Using $f = t$:
$$\frac{1}{t}=\frac{1}{v}-\frac{1}{u_1}$$
Solving for $v$:
$$v = \frac{u_1 t}{u_1-t}$$
Now, let's find $u_2$ for the second lens by using the image distance $v$ from the first lens as the object distance ($u$) for the second lens. Using $f=q$:
$$\frac{1}{q}=\frac{1}{v}-\frac{1}{u_2}$$
Solving for $u_2$:
$$u_2 = \frac{qv}{q-v}$$
Now, we can use the combined lens formula to find $f_{total}$:
$$\frac{1}{f_{total}}=\frac{1}{t}+\frac{1}{q}-\frac{d}{tq}$$
Substituting $v$ and $u_2$:
$$\frac{1}{f_{total}}=\frac{1}{t}+\frac{1}{q}-\frac{d}{tq}=\frac{1}{t}+\frac{1}{q}-\frac{11eb0d3b\_977d\_12dd\_9790\_774bf2c59650\_TB1243\_11}{tq}$$
Multiplying both sides by $\frac{tq}{11eb0d3b\_977d\_12de\_9790\_e12de8e662d0\_TB1243\_11}$, which is the reciprocal of the combined focal length:
$$\frac{11eb0d3b\_977d\_12de\_9790\_e12de8e662d0\_TB1243\_11}{tqf_{total}}= \frac{11eb0d3b\_977d\_12de\_9790\_e12de8e662d0\_TB1243\_11}{tqf_{total}}$$
Simplifying:
$$\frac{11eb0d3b\_977d\_12de\_9790\_e12de8e662d0\_TB1243\_11}{f_{total}}= \frac{11eb0d3b\_977d\_12dd\_9790\_774bf2c59650\_TB1243\_11}{t} +\frac{11eb0d3b\_977d\_12de\_9790\_e12de8e662d0\_TB1243\_11}{q} -11eb0d3b\_977d\_12dd\_9790\_774bf2c59650\_TB1243\_11$$
Now, we can see that the combined focal length is the reciprocal of the coefficient of $f_{total}$:
$$f_{total} = \frac{1}{\frac{11eb0d3b\_977d\_12de\_9790\_e12de8e662d0\_TB1243\_11}{f_{total}}}$$
$$f_{total}=\boxed{11eb0d3b\_977d\_12e0\_9790\_33eb731fcf0c\_TB1243\_11}$$

Two Thin Lenses with Focal Lengths T and Q and Separated