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Chapter 0: Problem 41

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? $$f=\frac{f_{1} f_{2}}{f_{1}+f_{2}} \text { for } f_{1}$$

Short Answer

Expert verified
The solved formula for \(f_{1}\) is \(f_{1}= \frac{ff_{2}}{f_{2}-f}\). The formula is used to calculate the combined focal length for two converging (or diverging) lenses in contact.

Step by step solution

01

Multiply both sides by denominator

Multiply both sides by \((f_{1}+f_{2})\), to get \(f(f_{1}+f_{2})=f_{1}f_{2}\). This process is designed to eliminate the denominator from the right hand side.
02

Expand the left hand side

Expand the left hand side, to get \(ff_{1}+ff_{2}=f_{1}f_{2}\).
03

Isolate terms with \(f_{1}\)

You want all terms involving \(f_{1}\) on one side and other terms on the opposite side. So, subtract \(ff_{1}\) from both sides, to get \(ff_{2}=f_{1}f_{2}-ff_{1}\).
04

Factor out \(f_{1}\)

Factor out \(f_{1}\) on the right side of the equation, result in \(ff_{2} = f_{1}(f_{2}-f)\). This process will allow us to get \(f_{1}\) by itself.
05

Solve for \(f_{1}\)

Solve the equation for \(f_{1}\), by dividing both sides by \((f_{2}-f)\). We get the final formula \(f_{1}= \frac{ff_{2}}{f_{2}-f}\)

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