Determination of Object Distance and Magnification in a Convex Mirror

A convex mirror is a curved surface that creates a virtual, upright, and reduced image. This article will guide you through the steps to determine the object distance and magnification when given the image distance and radius of curvature of a convex mirror.

Understanding the Problem

Let's consider the scenario where the image of an object in a convex mirror is located 4 cm away from the mirror, and the mirror has a radius of curvature of 24 cm.

Step-by-Step Solution

Step 1: Calculate the Focal Length of the Convex Mirror

The focal length f of a mirror is given by the formula:

f frac{R}{2}

Given: Radius of curvature R 24 ,text{cm}

Substituting the value of R:

f frac{24 ,text{cm}}{2} 12 ,text{cm}

Step 2: Use the Mirror Formula to Find the Object Distance

The mirror formula is given by:

frac{1}{f} frac{1}{u} frac{1}{v}

Where:

f is the focal length of the mirror. u is the object distance. v is the image distance.

Given: Image distance v 4 ,text{cm} (positive since the image is virtual and formed behind the mirror).

Rearranging the formula to find u:

frac{1}{u} frac{1}{f} - frac{1}{v}

Substituting the values of f and v:

frac{1}{u} frac{1}{12} - frac{1}{4}

Calculating the right side:

frac{1}{u} frac{1}{12} - frac{3}{12} -frac{2}{12} -frac{1}{6}

Thus:

u -6 ,text{cm}

The negative sign indicates that the object is located in front of the mirror.

Step 3: Calculate the Magnification of the Image

The magnification m is given by the formula:

m -frac{v}{u}

Substituting the values of v and u:

m -frac{4}{-6} frac{4}{6} frac{2}{3}

Summary of Results:

Object Position u: -6 cm (the object is 6 cm in front of the mirror). Magnification m: frac{2}{3} (the image is smaller than the object).

Thus, the object is positioned 6 cm in front of the mirror, and the magnification is frac{2}{3}

Understanding the Convex Mirror's Behavior

For a convex mirror, all positions behind the mirror are considered negative. This means:

R -24 ,text{cm} (behind the mirror radius of curvature)

f R/2 -12 ,text{cm} (behind the mirror focal length)

q -4 ,text{cm} (behind the mirror image position)

p in front of mirror object position

The mirror equation:

frac{1}{p} - frac{1}{q} frac{1}{f}

Solving for p the only unknown:

We know:

a convex mirror always produces a virtual image, so v -4 ,text{cm}. r 24 ,text{cm}.

Applying the spherical mirror equation:

frac{1}{v} - frac{1}{u} frac{2}{r} frac{2}{24} frac{1}{12}

Rearranging to find u:

frac{1}{u} frac{1}{12} - frac{1}{4} frac{1}{12} - frac{3}{12} -frac{2}{12} -frac{1}{6}

Thus:

u -6 ,text{cm}

The object needs to be placed at a position 6 cm in front of the convex mirror.

Conclusion

A convex mirror produces a virtual, upright, and reduced image. By using the mirror formula and magnification formula, we can determine the object distance and magnification given specific conditions. This knowledge is crucial for understanding the behavior of convex mirrors in various applications such as rearview mirrors in vehicles and security systems.