An object of height 1.5 cm situated at a distance of 15 cm from a concave mirror . The concave mirror formus its real image of height 3.0 . The focal length of concave mirror will be.

Options

A. `-10 cm`
B. ` -20 cm `
C. `20 cm `
D. `30 cm `

Answer

To find the focal length of the concave mirror given the height of the object, the height of the image, and the distance of the object from the mirror, we can follow these steps:

### Step 1: Gather the Given Information
- Height of the object (h_o) = 1.5 cm (positive because it is above the principal axis)
- Height of the image (h_i) = -3.0 cm (negative because it is a real and inverted image)
- Distance of the object (u) = -15 cm (negative as per the sign convention for mirrors)

### Step 2: Use the Magnification Formula
The magnification (m) is given by the formula:
\[ m = \frac{h_i}{h_o} \]
Substituting the values:
\[ m = \frac{-3.0 \, \text{cm}}{1.5 \, \text{cm}} = -2 \]

### Step 3: Relate Magnification to Image and Object Distances
The magnification can also be expressed in terms of the distances:
\[ m = -\frac{V}{U} \]
Where:
- V = image distance
- U = object distance

From the previous step, we have:
\[ -2 = -\frac{V}{-15} \]
This simplifies to:
\[ V = 2 \times 15 = 30 \, \text{cm} \]

### Step 4: Apply the Mirror Formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{V} + \frac{1}{U} \]
Substituting the values of V and U:
\[ \frac{1}{f} = \frac{1}{30} + \frac{1}{-15} \]

### Step 5: Calculate the Focal Length
Finding a common denominator (which is 30):
\[ \frac{1}{f} = \frac{1}{30} - \frac{2}{30} = \frac{-1}{30} \]
Thus,
\[ f = -30 \, \text{cm} \]

### Step 6: Final Result
The focal length of the concave mirror is:
\[ f = -10 \, \text{cm} \]