A concave mirror having a radius of curvature 40 cm is placed in front of an illuminated point source at a distance of 30 cm from it. Find the location of the image.

Explanation

To find the location of the image formed by a concave mirror, we can follow these steps:

### Step 1: Identify the given values
- Radius of curvature (R) = 40 cm
- Object distance (u) = 30 cm (since the object is in front of the mirror, we will take it as negative)

### Step 2: Calculate the focal length (f)
The focal length (f) of a concave mirror is given by the formula:
\[ f = \frac{R}{2} \]
Substituting the value of R:
\[ f = \frac{40 \, \text{cm}}{2} = 20 \, \text{cm} \]
Since it's a concave mirror, we take the focal length as negative:
\[ f = -20 \, \text{cm} \]

### Step 3: Use the mirror formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]
Where:
- \( f \) = focal length
- \( u \) = object distance
- \( v \) = image distance

Substituting the known values into the mirror formula:
\[ \frac{1}{-20} = \frac{1}{-30} + \frac{1}{v} \]

### Step 4: Solve for image distance (v)
Rearranging the equation to solve for \( \frac{1}{v} \):
\[ \frac{1}{v} = \frac{1}{-20} - \frac{1}{-30} \]

Finding a common denominator (which is 60):
\[ \frac{1}{v} = \frac{-3}{60} + \frac{2}{60} \]
\[ \frac{1}{v} = \frac{-1}{60} \]

Now, taking the reciprocal to find \( v \):
\[ v = -60 \, \text{cm} \]

### Conclusion
The image is formed at a distance of 60 cm in front of the mirror (the negative sign indicates that the image is real and located on the same side as the object).

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