The focal length of a concave mirror is f

and the distance of the object from the

principal focus is a. The magnitude of

magnification obtained will be

To determine the magnitude of magnification obtained from a concave mirror when the object is placed at a distance \( a \) from the principal focus, we can use the mirror formula and the magnification formula. Let's break this down step by step.

Understanding the Mirror Formula

The mirror formula for a concave mirror is given by:

\(\frac{1}{f} = \frac{1}{v} + \frac{1}{u}\)

Where:

• f = focal length of the mirror
• v = image distance from the mirror
• u = object distance from the mirror (with a negative sign for concave mirrors)

Setting Up the Problem

In this scenario, the object is located at a distance \( a \) from the principal focus. Since the focal length \( f \) is positive for a concave mirror, the object distance \( u \) can be expressed as:

u = - (f + a)

This negative sign indicates that the object is in front of the mirror, which is the convention for concave mirrors.

Finding the Image Distance

Substituting \( u \) into the mirror formula, we have:

\(\frac{1}{f} = \frac{1}{v} - \frac{1}{(f + a)}\)

Rearranging this gives us:

\(\frac{1}{v} = \frac{1}{f} + \frac{1}{(f + a)}\)

To combine these fractions, we find a common denominator:

\(\frac{1}{v} = \frac{(f + a) + f}{f(f + a)} = \frac{2f + a}{f(f + a)}\)

Thus, the image distance \( v \) can be expressed as:

v = \frac{f(f + a)}{2f + a}

Calculating Magnification

The magnification \( m \) produced by a mirror is given by the formula:

m = -\frac{v}{u}

Substituting our expressions for \( v \) and \( u \), we get:

m = -\frac{\frac{f(f + a)}{2f + a}}{-(f + a)} = \frac{f(f + a)}{(2f + a)(f + a)}

Magnitude of Magnification

Since we are interested in the magnitude of magnification, we can express it as:

|m| = \frac{f(f + a)}{(2f + a)(f + a)}

This formula gives us the magnitude of magnification when the object is placed at a distance \( a \) from the principal focus of a concave mirror. The result indicates how much larger or smaller the image will be compared to the object, depending on the values of \( f \) and \( a \).

Example for Clarity

For instance, if the focal length \( f \) is 10 cm and the object is placed 5 cm from the principal focus (making \( a = 5 \) cm), we can substitute these values into our derived formula:

|m| = \frac{10(10 + 5)}{(2 \cdot 10 + 5)(10 + 5)} = \frac{10 \cdot 15}{(20 + 5)(15)} = \frac{150}{375} = 0.4

This means the image will be 0.4 times the size of the object, indicating a smaller image.

In summary, by using the mirror formula and magnification formula, we can effectively determine the magnitude of magnification for any given object distance relative to the focal length of a concave mirror.