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Chapter 18: Problem 43

The relation between lateral magnification \(m\), object distance \(u\), and focal length \(f\) of a spherical mirror is (A) \(m=\frac{f-u}{f}\) (B) \(m=\frac{f}{f+u}\) (C) \(m=\frac{f+u}{f}\) (D) \(m=\frac{f}{f-u}\)

Short Answer

Expert verified
The correct relation between lateral magnification (m), object distance (u), and focal length (f) of a spherical mirror is (A) \( m = \frac{f-u}{f} \).

Step by step solution

01

Write down the mirror formula and the lateral magnification definition.

We have the mirror formula for a spherical mirror: \( \frac{1}{u}+\frac{1}{v}=\frac{1}{f} \) And the definition of lateral magnification (m): \( m = - \frac{v}{u} \)
02

Isolate v in the mirror formula.

To find the relation between m, u, and f, we will first isolate v in the mirror formula to get v in terms of u and f. \( \frac{1}{v} = \frac{1}{f} - \frac{1}{u} \) Now, to get v, we'll take the reciprocal of both sides: \( v = \frac{1}{\frac{1}{f} - \frac{1}{u}} \)
03

Use the lateral magnification definition to eliminate v.

We will use the definition of lateral magnification, \( m = - \frac{v}{u} \), to eliminate v and find the required relation. First, isolate v in the lateral magnification formula: \( v = -mu \) Now, substitute this expression for v into the expression for v obtained in step 2: \( -mu = \frac{1}{\frac{1}{f} - \frac{1}{u}} \)
04

Simplify the expression for m:

Now, our goal is to simplify this expression to find the correct relation for m. First, multiply both sides by -u: \( mu = -u(\frac{1}{f - \frac{u}{u}}) \) Simplify the terms in the brackets on the right-hand side: \( mu = -u(\frac{1}{f - 1}) \) Finally, divide both sides by -u and rearrange to get the expression for m: \( m = \frac{-u}{f-u} \) Comparing the obtained relation with the given options, the answer is: (A) \( m=\frac{f-u}{f} \)

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