Determining the Focal Length of a Convex Mirror Using Object-Image Distance

To find the focal length of a convex mirror when an object is placed 25 cm in front of the mirror and its image is formed 14 cm behind the mirror, we can utilize the mirror formula. The mirror formula is given by:

Mirror Formula

The mirror formula is:

?1f 1u 1v?

where:

f is the focal length of the mirror u is the object distance, which is always taken as negative for distances in front of the mirror v is the image distance, which is positive for virtual images formed by convex mirrors (as the image is behind the mirror)

Given:

Object distance u -25 cm Image distance v 14 cm

Now, substituting these values into the mirror formula:

?1f 1-25 114?

?1-25 -0.04?

?114 ≈ 0.07142857?

Adding these values:

?1f -0.04 0.07142857 ≈ 0.03142857?

Taking the reciprocal to find f :

?f ≈ 1 / 0.03142857 ≈ 31.87 cm?

Thus, the focal length of the convex mirror is approximately 31.87 cm.

Generalizing the Approach

For a more generalized formula, let’s consider the situation where the object distance is u -x cm and the image distance is v y cm . The mirror formula would be:

?1f 1-x 1y?

Which simplifies to:

?1f -1x 1y?

Or:

?1f 1y - 1x?

And thus:

?f 1 / (1y - 1x)?

Example Calculation

For the scenario where u -25 cm and v 14 cm , substituting the values gives:

?f 1 / (1/14 - 1/25)?

?f 1 / (0.07142857 - 0.04)?

?f 1 / 0.03142857 ≈ 31.87 cm?

Therefore, the focal length of the convex mirror is approximately 31.87 cm.

Focusing on Further Concepts

The computation presented above involves understanding the sign conventions and applying the mirror formula accurately. These skills are crucial for solving related problems in optics. It’s important to note that in convex mirrors, the focal length is always positive, and the distances are treated as negative for objects in front of the mirror and positive for virtual images behind the mirror.

For further reading and a deeper understanding of mirror formulas and their applications, consider exploring more tutorials or consulting textbooks on basic optics and geometric optics.