The position of an object placed 5 cm in front of concave mirror of radius of curvature 15 cm is

Options

A. 7.5 cm
B. 15 cm
C. 20 cm
D. 27.5 cm

Explanation

To find the position of the image formed by a concave mirror when an object is placed in front of it, we can use the mirror formula:

\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]

where:
- \( f \) is the focal length of the mirror,
- \( u \) is the object distance (always negative for concave mirrors),
- \( v \) is the image distance (which we need to find).

### Step 1: Determine the focal length of the concave mirror
The radius of curvature \( R \) is given as 15 cm. The focal length \( f \) is related to the radius of curvature by the formula:

\[ f = -\frac{R}{2} \]

Substituting the value of \( R \):

\[ f = -\frac{15 \, \text{cm}}{2} = -7.5 \, \text{cm} \]

### Step 2: Identify the object distance
The object distance \( u \) is given as 5 cm. Since the object is placed in front of the concave mirror, we take this distance as negative:

\[ u = -5 \, \text{cm} \]

### Step 3: Substitute values into the mirror formula
Now we can substitute the values of \( f \) and \( u \) into the mirror formula:

\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]

Substituting the known values:

\[ \frac{1}{-7.5} = \frac{1}{-5} + \frac{1}{v} \]

### Step 4: Rearranging the equation
Rearranging the equation to solve for \( \frac{1}{v} \):

\[ \frac{1}{v} = \frac{1}{-7.5} - \frac{1}{-5} \]

### Step 5: Finding a common denominator
The common denominator for 7.5 and 5 is 15. We can rewrite the fractions:

\[ \frac{1}{-7.5} = \frac{-2}{15}, \quad \frac{1}{-5} = \frac{-3}{15} \]

So,

\[ \frac{1}{v} = \frac{-2}{15} + \frac{3}{15} = \frac{1}{15} \]

### Step 6: Solve for \( v \)
Taking the reciprocal gives us:

\[ v = 15 \, \text{cm} \]

### Final Result
The position of the image is 15 cm in front of the concave mirror.

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