Understanding the Concave Mirror Problem

This problem involves a concave mirror. We are given information about an image formed from a distant object and asked to find the characteristics (nature and magnification) of an image formed when a new object is placed at a specific distance.

Key Concepts for Concave Mirrors:

• A concave mirror is a mirror curved inwards.
• It can form both real, inverted images and virtual, erect images depending on the object's position.
• Real images formed by mirrors are typically inverted and appear in front of the mirror (on the same side as the object).
• Virtual images are typically erect and appear behind the mirror.
• The mirror formula relates object distance ($u$), image distance ($v$), and focal length ($f$): $\frac{1}{u} + \frac{1}{v} = \frac{1}{f}$.
• Magnification ($m$) describes the size and orientation of the image: $m = -\frac{v}{u}$. A negative $m$ indicates a real, inverted image, while a positive $m$ indicates a virtual, erect image.
• Sign Conventions (Cartesian): Distances are measured from the pole of the mirror. Distances measured in the direction of incident light are positive, and those measured against the direction are negative. For mirrors, objects are usually placed in front (negative $u$). Real images form in front (negative $v$), and virtual images form behind (positive $v$). Focal length ($f$) is negative for concave mirrors.

Step 1: Determine the Focal Length ($f$) of the Concave Mirror

We are told that the concave mirror forms a real and inverted image of a distant object. The image is formed at a distance of $15 \text{ cm}$ from the mirror. Let's apply the sign conventions:

• The object is "distant", which means the object distance $u \to -\infty$.
• The image is "real", so it forms in front of the mirror. According to the Cartesian sign convention, the image distance $v = -15 \text{ cm}$.
• Since it's a concave mirror, the focal length $f$ is negative.

Now, let's use the mirror formula: $ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} $ Substitute the values: $ \frac{1}{f} = \frac{1}{-\infty} + \frac{1}{-15 \text{ cm}} $ Since $\frac{1}{-\infty} = 0$, the equation becomes: $ \frac{1}{f} = 0 + \frac{1}{-15 \text{ cm}} $ $ \frac{1}{f} = -\frac{1}{15 \text{ cm}} $ Therefore, the focal length of the mirror is: $ f = -15 \text{ cm} $ This confirms the focal length is $-15 \text{ cm}$, consistent with a concave mirror.

Step 2: Calculate the Image Position ($v$) for the New Object

A new object is placed $20 \text{ cm}$ in front of this mirror. We need to find the nature and magnification of the image formed.

• Object distance $u = -20 \text{ cm}$ (negative as it's in front).
• Focal length $f = -15 \text{ cm}$ (determined in Step 1).

Using the mirror formula again:

The negative sign for $v$ indicates that the image is formed $60 \text{ cm}$ in front of the mirror. This means the image is real.

Step 3: Calculate the Magnification ($m$)

The magnification formula is $m = -\frac{v}{u}$.

• $v = -60 \text{ cm}$
• $u = -20 \text{ cm}$

Substitute these values into the magnification formula:

Interpreting the magnification ($m = -3$):

• The negative sign (-) indicates that the image is inverted.
• The magnitude (3) indicates that the image is magnified by a factor of 3.

Conclusion: Image Characteristics

Based on our calculations:

• The image distance $v = -60 \text{ cm}$ indicates a real image.
• The magnification $m = -3$ indicates the image is inverted and magnified by 3 times.

Therefore, the image formed is real, inverted, and magnified ($3 \times$).