The formula for calculating the number of images formed when two mirrors are placed at an angle $\alpha$ is $360/\alpha$. My question is how many images will be formed when it is a fraction? My teacher told to apply greatest integer function but in some places it says nearest even integer. So can someone please tell what is the correct formula?
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2 Answers
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Your answer is not completely correct. However, you are correct that we have to use the Floor Function (sometimes called the Greatest Integer Function) in case of a fractional value.
\begin{array} {|c|c|c|}\hline \rm Value\ of(\frac{360°}{\alpha}) & \rm Position\ of\ object & \rm No.\ of\ images\ formed \\ \hline \rm Even & \rm Symmetric & \frac{360°}{\alpha}-1 \\ \hline \rm Even & \rm Asymmetric & \frac{360°}{\alpha}-1 \\ \hline \rm Odd & \rm Symmetric & \frac{360°}{\alpha}-1 \\ \hline \rm Odd & \rm Asymmetric & \frac{360°}{\alpha} \\ \hline \end{array}
If $n=3.8$, so $3$ images will be formed. (Forget about decimal)
Similarly, $2$ images will be formed when the angle between the mirrors is $112.5°$.
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$\begingroup$ Thank you for your answer. Is there a proof for the above formulas? Or is it pattern recognition? Can you give any references for further read? $\endgroup$ Aug 19, 2020 at 5:49
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1$\begingroup$ @Vilvanesh, this results were given by my teacher. Unfortunately, he didn't have the proof. If you Google, you may find something helpful. $\endgroup$– SarGeAug 19, 2020 at 5:57
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When the value of $360/\alpha$ is a fraction, you are correct to presume that the number of images will be the nearest even integer. For example, if $360/\alpha = 3.8$, then the number of images that are completely formed will be $4$. However, instead of memorizing these long and uninteresting formulas, I would advise you to solve such problems fundamentally. You have to examine each event separately and then note the position of the image so formed after each event. Then, you have to consider the previous image as the virtual object of the next event and then analyze accordingly. I'm not going to add the complete solution here because solving these kind of problems using your fundamental concepts solely helps you tremendously in the long run. Cheerio!