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Chapter 16: Problem 2277

A concave lens forms the image of an object such that the distance between the object and the image is \(10 \mathrm{~cm}\) and the magnification produced is \((1 / 4)\), the focal length of lens will be \(\mathrm{cm}\) (A) - 6.2 (B) \(-12.4\) (C) \(-4.4\) (D) \(-8.8\)

Short Answer

Expert verified
Now that we know u, we can find the focal length using the expression we found in step 5: \[ f = \frac{4(-4)}{-3} \] Calculating the result: \[ f = -\frac{16}{3} \mathrm{~cm} \approx -5.33 \mathrm{~cm} \] Looking at the given options, we can see that option (A) is the closest to our calculated value of the focal length, and therefore, the answer is: (A) - 6.2 cm

Step by step solution

01

Write down the given information

The distance between the object (u) and the image (v) is 10 cm, and the magnification (m) of the image is 1/4.
02

Use the lens formula to relate u, v, and f

The lens formula is: \[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \] We will use this formula once we have found u and v.
03

Use the magnification formula to relate u, v, and m

The magnification formula is: \[ m = \frac{-v}{u} \] Since the magnification m is given, we can use this formula to express v in terms of u: \[ v = -4u \]
04

Replace v in the lens formula with the expression found in step 3

Now that we have found an expression for v, we can substitute it into the lens formula: \[ \frac{1}{f} = \frac{1}{u} + \frac{1}{-4u} \]
05

Solve the lens formula for f

To find f, we just need to solve: \[ \frac{1}{f} = \frac{1}{u} + \frac{1}{-4u} \] Combining the fractions on the right-hand side of the equation: \[ \frac{1}{f} = \frac{-3}{4u} \] Finally, we can find f by taking the reciprocal of both sides: \[ f = \frac{4u}{-3} \]
06

Use the given information to find u

We know that the distance between the object and the image is 10 cm. Using the expression we found in step 3 for v, we have: \[ u + (-4u) = 10 \] Solving for u: \[ u = -4 \mathrm{~cm} \]
07

Find the focal length using f = 4u / -3

Now that we know u, we can find the focal length using the expression we found in step 5: \[ f = \frac{4(-4)}{-3} \] Calculating the result: \[ f = -\frac{16}{3} \mathrm{~cm} \approx -5.33 \mathrm{~cm} \]
08

Choose the correct answer

Looking at the given options, we can see that option (A) is the closest to our calculated value of the focal length, and therefore, the answer is: (A) - 6.2 cm

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Most popular questions from this chapter

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