An object 5 cm tall is placed 1 m from a concave spherical mirror which has a radius of curvature of 20 cm. The size of the image is

Options

A. 0.11 cm
B. 0.5 cm
C. 0.55 cm
D. 0.60 cm

Explanation

To find the size of the image formed by a concave mirror, we can use the mirror formula and the magnification formula. Here’s a step-by-step solution:

### Step 1: Identify the given values
- Height of the object (h_o) = 5 cm
- Object distance (u) = 1 m = 100 cm (since 1 m = 100 cm)
- Radius of curvature (R) = 20 cm

### 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:
\[ f = \frac{20 \, \text{cm}}{2} = 10 \, \text{cm} \]

### Step 3: Use the sign convention
For a concave mirror:
- The focal length (f) is taken as positive: \( f = +10 \, \text{cm} \)
- The object distance (u) is also taken as positive: \( u = +100 \, \text{cm} \)

### Step 4: Use the magnification formula
The magnification (m) is given by:
\[ m = \frac{h_i}{h_o} = -\frac{f}{f - u} \]
Where:
- \( h_i \) = height of the image
- \( h_o \) = height of the object

### Step 5: Substitute the values into the magnification formula
We need to find \( h_i \):
\[ m = \frac{h_i}{5} = -\frac{10}{10 - 100} \]
\[ m = \frac{h_i}{5} = -\frac{10}{-90} \]
\[ m = \frac{h_i}{5} = \frac{1}{9} \]

### Step 6: Solve for the height of the image (h_i)
Now, we can find \( h_i \):
\[ h_i = m \cdot h_o \]
\[ h_i = \frac{1}{9} \cdot 5 \]
\[ h_i = \frac{5}{9} \, \text{cm} \]

### Step 7: Determine the sign of the image height
Since the magnification is positive, the image is upright. However, in the context of concave mirrors, we typically consider the image to be inverted. Thus:
\[ h_i = -\frac{5}{9} \, \text{cm} \]

### Final Result
The size of the image is approximately:
\[ h_i \approx -0.55 \, \text{cm} \]
This indicates that the image is inverted and smaller than the object.

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