A candle flame 0.5 cm high is kept between a wall and a concave mirror of focal length 1.5 m such that its image 1.5 cm high is formed on the wall. Find the distance of candle from the wall and also the distance between the wall and the mirror.

Options

A. 4 m, 6 m
B. 6 m, 7 m
C. 6 m, 8 m
D. 9 m, 7 m

Answer

To solve the problem step by step, we will use the mirror formula and the magnification formula for concave mirrors.

### Step 1: Understand the given data
- Height of the candle (object height, \( h_o \)) = 0.5 cm
- Height of the image (\( h_i \)) = -1.5 cm (negative because the image is inverted)
- Focal length of the concave mirror (\( f \)) = -1.5 m (concave mirrors have a negative focal length)

### Step 2: Use the magnification formula
The magnification (\( m \)) of a mirror is given by the formula:
\[ m = \frac{h_i}{h_o} = -\frac{v}{u} \]
Where:
- \( v \) = image distance from the mirror
- \( u \) = object distance from the mirror

Substituting the values we have:
\[ m = \frac{-1.5 \text{ cm}}{0.5 \text{ cm}} = -3 \]
This means:
\[ -3 = -\frac{v}{u} \implies v = 3u \]

### Step 3: Use the mirror formula
The mirror formula is:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting \( v = 3u \) into the mirror formula:
\[ \frac{1}{-1.5} = \frac{1}{3u} + \frac{1}{u} \]

### Step 4: Simplify the equation
To simplify, we first find a common denominator:
\[ \frac{1}{-1.5} = \frac{1 + 3}{3u} = \frac{4}{3u} \]
Thus, we have:
\[ -\frac{2}{3} = \frac{4}{3u} \]

### Step 5: Solve for \( u \)
Cross-multiplying gives:
\[ -2 \cdot 3u = 4 \implies -6u = 4 \implies u = -\frac{4}{6} = -\frac{2}{3} \text{ m} \]
Since \( u \) is negative, it indicates the object (candle) is on the same side as the incoming light.

### Step 6: Calculate \( v \)
Using \( v = 3u \):
\[ v = 3 \left(-\frac{2}{3}\right) = -2 \text{ m} \]

### Step 7: Find distances
- The distance of the candle from the wall is the distance from the wall to the mirror plus the distance from the mirror to the candle.
- The distance from the wall to the mirror is \( |v| = 2 \text{ m} \).
- The distance of the candle from the wall is \( |v| - |u| = 2 \text{ m} - \frac{2}{3} \text{ m} = \frac{6}{3} - \frac{2}{3} = \frac{4}{3} \text{ m} \).

### Final Answers
- Distance of the candle from the wall = \( \frac{4}{3} \text{ m} \)
- Distance between the wall and the mirror = \( 2 \text{ m} \)