Answer to Question #140640 in Optics for Sharfa

Question #140640
When a converging lens is placed between an object and a screen , an image of height 4 cm is produced on the screen. Without changing the distance between the object and the screen , another image of height 16 cm is produced on the screen at another position of the lens. Find the height of the object.
1
Expert's answer
2020-10-29T07:02:19-0400

Assume the distance between the object and screen is x, then object distance (between object and lens) plus image distance (between screen and lens) equals x:


x=i1+o1=i2+o2   (0)x=i_1+o_1=i_2+o_2\space\space\space(0)


for both cases.

Also, we know that magnification is the image distance divided by object distanceб or image height divided by object height:

m1=i1o1, m1=hi1ho=4hoi1o1=4h. m2=i2o2, m2=hi2ho=16hoi2o2=16h.m_1=\frac{i_1}{o_1},\space m_1=\frac{h_{i1}}{h_o}=\frac{4}{h_o}\rightarrow\frac{i_1}{o_1}=\frac{4}{h}.\\\space\\ m_2=\frac{i_2}{o_2},\space m_2=\frac{h_{i2}}{h_o}=\frac{16}{h_o}\rightarrow\frac{i_2}{o_2}=\frac{16}{h}.


Divide one by another:


i1o2i2o1=14i1o1=i24o2. (1)\frac{i_1o_2}{i_2o_1}=\frac{1}{4}\rightleftarrows \frac{i_1}{o_1}=\frac{i_2}{4o_2}.\space(1)


Multiply one by another:


i1i2o1o2=64h2. (2)\frac{i_1i_2}{o_1o_2}=\frac{64}{h^2}.\space(2)

Moreover, according to thin lens equation, we have


1f=1i1+1o1=1i2+1o2, or i1+o1i1o1=i2+o2i2o2i1o2=i2o1. (3)\frac{1}{f}=\frac{1}{i_1}+\frac{1}{o_1}=\frac{1}{i_2}+\frac{1}{o_2},\\\space\\ \text{or}\\\space\\ \frac{i_1+o_1}{i_1o_1}=\frac{i_2+o_2}{i_2o_2}\rightleftarrows \frac{i_1}{o_2}=\frac{i_2}{o_1}.\space(3)

Plug equation (1) in (2):


i224o22=64h2. (4)\frac{i_2^2}{4o_2^2}=\frac{64}{h^2}.\space(4)

Plug equation (3) in (2):


i22o12=64h2. (5)\frac{i_2^2}{o_1^2}=\frac{64}{h^2}.\space(5)

Since right parts of (4) and (5) are equal, the left pars are equal as well:

i224o22=i22o12o1=2o2.\frac{i_2^2}{4o_2^2}=\frac{i_2^2}{o_1^2}\rightarrow o_1=2o_2.

Now plug this o-equation in (3):

i1o2=i22o2i1=i22. (6)\frac{i_1}{o_2}=\frac{i_2}{2o_2}\rightarrow i_1=\frac{i_2}{2}.\space(6)

Now look at the equation (0) in the very beginning and plug (6) into it:


i22+2o2=i2+o2i2=2o2. (7)\frac{i_2}{2}+2o_2=i_2+o_2\rightarrow i_2=2o_2.\space(7)


Finally, it's time to plug (7) in (4):


(2o2)24o22=64h2, 1=64h2h=8 cm.\frac{(2o_2)^2}{4o_2^2}=\frac{64}{h^2},\\\space\\ 1=\frac{64}{h^2}\rightarrow h=8\text{ cm}.

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