Chapter 23, Problem 91GP

To determine

Part (a)To Show:

The formula of focal length of combination of two thin lenses is represented by $f_T=f_1{f}_2/\left(f_1+f_2\right)$

Answer to Problem 91GP

Solution:

$f_T=f_1{f}_2/\left(f_1+f_2\right)$

Explanation of Solution

Given:

focal length of two lenses f₁ and f₂, combined focal length f and combined power P, and power of both the lenses P₁ and P₂.

Formula used:

lens formula for focal length

$1/f=1/v–1/u$

Calculation:

If two thin lenses of focal length f₁ and f₂ are placed in contact with each other both on same principle axis as shown in diagram

If a point object is placed on the principle axis at a distance u from the first lens L₁. Its image would be formed by lens L₁ at I' at a distance v' from lens L_1.

Then, from the lens formula, we have

$1/v\text{'}–1/u=1/f_1 \dots \dots \dots \dots .\left(1\right)$

I' will be used as a virtual object for second lens L₂ which forms a final image I at a distance v from it.

Then, we have

$1/v-1/v\text{'}=1/f_2 \dots \dots \dots \dots \mathrm{..}\left(2\right)$

From equation (1)and (2)by adding it we get

$1/v–1/u=1/f_1+1/f_2{}_{ } \dots \dots \dots \dots .\left(3\right)$

If we replace both the lens by a single lens L which forms the image of an object at a distance u from it, at a distance v, then focal length of the equivalent lens would be given by

$1/v-1/u=1/f_T \dots \dots \dots \dots \dots \left(\text{4}\right)$

From equation (3)and (4)we get

$\begin{array}{l}1/f_T=1/f_1+1/f_2 \dots \dots \dots \dots .\left(\text{5}\right)\\ 1/f_T=\left(f_2+f_1\right)/f_1{f}_2\\ f_T=f_1{f}_2/\left(f_1+f_2\right) \end{array}$

To determine

Part (b)To Show:

power of combination of lenses is equal to the sum of their separate powers, $P=P_1+P_2$_.

Answer to Problem 91GP

Solution:

$P=P_1+P_2$

Explanation of Solution

Given:

focal length of two lenses f₁ and f₂, combined focal length f and combined power P, and power of both the lenses P₁ and P₂.

Formula used:

power is given as P = 1/f

Calculation:

$1/f_T=1/f_1+1/f_2 \dots \dots \dots \dots .\left(\text{5}\right)$

From equation (5), we get

$P_T=P_1+P_2$

That means total power is $P_T=P_1+P_2$_.