Find the distance of object from a concave mirror of focal length 10cm so that image size is four times the of the object.

To find the distance of the object from a concave mirror when the image size is four times that of the object, we can use the mirror formula and the magnification formula. Let's break this down step by step.

Understanding the Concepts

First, we need to recall a couple of important formulas related to mirrors:

• Mirror Formula: 1/f = 1/v + 1/u
• Magnification (m): m = -v/u

In these formulas:

• f is the focal length of the mirror.
• v is the image distance (the distance from the mirror to the image).
• u is the object distance (the distance from the mirror to the object).
• m is the magnification, which is the ratio of the height of the image to the height of the object.

Given Information

In this case:

• The focal length (f) of the concave mirror is -10 cm (negative because it’s a concave mirror).
• The magnification (m) is 4 (since the image size is four times that of the object).

Calculating the Object Distance

Since the magnification is given as 4, we can express this in terms of the image and object distances:

m = -v/u = 4

This implies:

-v = 4u

From this, we can express v in terms of u:

v = -4u

Now, we can substitute this value of v into the mirror formula:

1/f = 1/v + 1/u

Substituting the known values:

1/(-10) = 1/(-4u) + 1/u

To simplify this, we can find a common denominator, which is -4u:

1/(-10) = -1/(4u) + 4/(4u)

This simplifies to:

1/(-10) = 3/(4u)

Now, cross-multiplying gives us:

4u = -30

Solving for u, we find:

u = -30/4 = -7.5 cm

Interpreting the Result

The negative sign indicates that the object is located on the same side as the incoming light, which is typical for real objects in front of a concave mirror. Therefore, the object distance is 7.5 cm from the mirror.

Summary

In summary, to achieve an image size that is four times larger than the object size with a concave mirror of focal length 10 cm, the object must be placed at a distance of 7.5 cm from the mirror. This example illustrates how the relationships between object distance, image distance, and magnification work together in optics.