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. 0.50 cm
C. 0.55 cm
D. 0.60 cm

Explanation

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

### Given Data:
- Height of the object (h_o) = 5 cm
- Object distance (u) = -1 m = -100 cm (negative because the object is in front of the mirror)
- Radius of curvature (R) = -20 cm (negative for concave mirror)

### Step 1: 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 2: Use the Mirror Formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Rearranging the formula to find v:
\[ \frac{1}{v} = \frac{1}{f} - \frac{1}{u} \]
Substituting the values of f and u:
\[ \frac{1}{v} = \frac{1}{-10} - \frac{1}{-100} \]
\[ \frac{1}{v} = -\frac{1}{10} + \frac{1}{100} \]
Finding a common denominator (100):
\[ \frac{1}{v} = -\frac{10}{100} + \frac{1}{100} = -\frac{9}{100} \]
Now, taking the reciprocal:
\[ v = -\frac{100}{9} \, \text{cm} \]

### Step 3: Calculate the Magnification (m)
The magnification (m) is given by the formula:
\[ m = \frac{h_i}{h_o} = -\frac{v}{u} \]
Substituting the values of v and u:
\[ m = -\left(-\frac{100}{9}\right) \div (-100) \]
\[ m = \frac{100}{9} \cdot \frac{1}{100} = \frac{1}{9} \]

### Step 4: Calculate the Height of the Image (h_i)
From the magnification formula:
\[ h_i = m \cdot h_o \]
Substituting the values:
\[ h_i = \frac{1}{9} \cdot 5 \, \text{cm} = \frac{5}{9} \, \text{cm} \approx 0.56 \, \text{cm} \]

### Step 5: Determine the Orientation of the Image
Since the magnification is positive, the image is upright. However, since we derived the value of v as negative, the image is inverted.

### Final Result
The size of the image is approximately \( 0.56 \, \text{cm} \) and it is inverted.

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