Understanding Image Formation with a Concave Mirror

This question asks us to determine the position of the image formed by a concave mirror given its radius of curvature and the object distance. We will use the mirror formula and the relationship between the radius of curvature and the focal length of a spherical mirror.

Given Information

• Type of mirror: Concave mirror
• Radius of curvature (\(R\)): 50 cm
• Object distance from the mirror (\(u\)): 25 cm

Finding the Focal Length

For any spherical mirror, the focal length (\(f\)) is half of its radius of curvature (\(R\)). That is, \( f = \frac{R}{2} \). For a concave mirror, the focal length is considered negative as it is a real focus located in front of the mirror.

So, the magnitude of the focal length is:

\( |f| = \frac{50 \text{ cm}}{2} = 25 \text{ cm} \)

Applying the sign convention for a concave mirror, the focal length is:

\( f = -25 \text{ cm} \)

Applying the Mirror Formula

The mirror formula relates the focal length (\(f\)), the object distance (\(u\)), and the image distance (\(v\)):

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

We need to use the correct sign conventions for the object distance as well. The object is placed 25 cm from the mirror, and for a real object in front of the mirror, the object distance (\(u\)) is taken as negative.

So, \( u = -25 \text{ cm} \).

Now, substitute the values of \( f \) and \( u \) into the mirror formula:

\( \frac{1}{-25} = \frac{1}{-25} + \frac{1}{v} \)

Calculating the Image Position

We want to find the image distance \( v \). Let's rearrange the equation to solve for \( \frac{1}{v} \):

\( \frac{1}{v} = \frac{1}{-25} - \frac{1}{-25} \)

\( \frac{1}{v} = \frac{-1}{25} - \left( \frac{-1}{25} \right) \)

\( \frac{1}{v} = \frac{-1}{25} + \frac{1}{25} \)

\( \frac{1}{v} = 0 \)

To find \( v \), we take the reciprocal of both sides:

\( v = \frac{1}{0} \)

This indicates that the image is formed at infinity.

Interpretation of the Result

The object distance (\( u = 25 \text{ cm} \)) is equal to the magnitude of the focal length (\( |f| = 25 \text{ cm} \)). When an object is placed exactly at the focal point of a concave mirror, the rays of light from the object, after reflection from the mirror, become parallel to the principal axis. Parallel rays are considered to meet at infinity, thus forming an image at infinity.

Revision Table: Concave Mirror Key Points

Additional Information: Image Formation at Infinity

The case where the object is placed at the focal point (\( u = f \)) and the image is formed at infinity (\( v = \infty \)) is a specific and important scenario for concave mirrors. This principle is used in devices like searchlights, headlights, and flashlights. A light source (like a bulb) is placed at the focal point of a concave reflector. The light rays diverging from the source hit the reflector and are reflected as a strong, parallel beam of light that travels over a long distance, effectively forming an image at infinity.

Conversely, if the object is at infinity (like distant stars), the image is formed at the focal point. This is why telescopes use concave mirrors to form images of distant objects at their focal plane.