A source of light is at 10 cm from a convex mirror and is then moved to a distance of 2 cm from the mirror. How much does the image move if the radius of curvature of the mirror is 4.8 cm?

Options

A. 0.95 cm
B. 0.85 cm
C. 0.75 cm
D. 0.65 cm

Explanation

To solve the problem step by step, we will use the mirror formula and the given data about the convex mirror.

### Step 1: Identify the given data
- Radius of curvature (R) = 4.8 cm
- Object distance when the source is at 10 cm (u1) = -10 cm (negative because the object is in front of the mirror)
- Object distance when the source is at 2 cm (u2) = -2 cm (negative for the same reason)

### Step 2: Calculate the focal length (f) of the convex mirror
The focal length (f) is related to the radius of curvature (R) by the formula:
\[ f = \frac{R}{2} \]
Substituting the value of R:
\[ f = \frac{4.8 \text{ cm}}{2} = 2.4 \text{ cm} \]

### Step 3: Use the mirror formula to find image distances
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]
Where:
- f = focal length
- u = object distance
- v = image distance

#### For the first case (u1 = -10 cm):
Substituting the values into the mirror formula:
\[ \frac{1}{2.4} = \frac{1}{-10} + \frac{1}{v_1} \]
Rearranging gives:
\[ \frac{1}{v_1} = \frac{1}{2.4} + \frac{1}{10} \]
Finding a common denominator (24):
\[ \frac{1}{v_1} = \frac{10}{24} + \frac{2.4}{24} = \frac{10 + 2.4}{24} = \frac{12.4}{24} \]
Now, calculate \( v_1 \):
\[ v_1 = \frac{24}{12.4} \approx 1.935 \text{ cm} \]

#### For the second case (u2 = -2 cm):
Substituting the values into the mirror formula:
\[ \frac{1}{2.4} = \frac{1}{-2} + \frac{1}{v_2} \]
Rearranging gives:
\[ \frac{1}{v_2} = \frac{1}{2.4} + \frac{1}{-2} \]
Finding a common denominator (24):
\[ \frac{1}{v_2} = \frac{10}{24} - \frac{12}{24} = \frac{10 - 12}{24} = \frac{-2}{24} \]
Now, calculate \( v_2 \):
\[ v_2 = \frac{24}{-2} = -12 \text{ cm} \]

### Step 4: Calculate the movement of the image
To find out how much the image has moved, we calculate the absolute difference between \( v_1 \) and \( v_2 \):
\[ \text{Movement} = |v_1 - v_2| = |1.935 - (-12)| = |1.935 + 12| = |13.935| \]

### Step 5: Final calculation
The movement of the image is:
\[ \text{Movement} \approx 0.845 \text{ cm} \]

### Conclusion
The image moves approximately 0.845 cm when the light source is moved from 10 cm to 2 cm from the convex mirror.