What acn be the largest distance of an image of a real object from a convex mirror of radius of curvature 20 cm

Options

A. 10 cm
B. Infinity
C. 20 cm
D. None

Answer

To solve the problem of finding the largest distance of an image of a real object from a convex mirror with a radius of curvature of 20 cm, we can follow these steps:

### Step 1: Understand the relationship between radius of curvature and focal length
The focal length (f) of a convex mirror is given by the formula:
\[ f = \frac{R}{2} \]
where R is the radius of curvature. Given that R = 20 cm, we can calculate f:
\[ f = \frac{20 \, \text{cm}}{2} = 10 \, \text{cm} \]

**Hint:** Remember that for convex mirrors, the focal length is positive and is half of the radius of curvature.

### Step 2: Use the mirror formula
The mirror formula for curved mirrors is:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
where:
- \( f \) is the focal length,
- \( v \) is the image distance,
- \( u \) is the object distance (which is negative for real objects in the case of mirrors).

### Step 3: Substitute the known values into the mirror formula
Since we know \( f = 10 \, \text{cm} \), we can rewrite the formula:
\[ \frac{1}{10} = \frac{1}{v} + \frac{1}{u} \]

### Step 4: Express \( u \) in terms of \( v \)
Rearranging the formula gives:
\[ \frac{1}{v} = \frac{1}{10} - \frac{1}{u} \]
This can be rewritten as:
\[ \frac{1}{v} = \frac{u - 10}{10u} \]
Thus,
\[ v = \frac{10u}{u - 10} \]

### Step 5: Find the maximum value of \( v \)
To find the maximum value of \( v \), we can differentiate \( v \) with respect to \( u \) and set the derivative to zero:
Let:
\[ v = \frac{10u}{u - 10} \]

Differentiating \( v \) with respect to \( u \):
\[ \frac{dv}{du} = \frac{(u - 10)(10) - 10u(1)}{(u - 10)^2} = \frac{10u - 100}{(u - 10)^2} \]

Setting the numerator to zero for maximum:
\[ 10u - 100 = 0 \]
\[ u = 10 \, \text{cm} \]

### Step 6: Substitute \( u \) back to find \( v \)
Substituting \( u = 10 \, \text{cm} \) back into the equation for \( v \):
\[ v = \frac{10(10)}{10 - 10} \]
As \( u \) approaches infinity, \( v \) approaches the focal length:
\[ v \to 10 \, \text{cm} \]

### Conclusion
The largest distance of the image from the convex mirror is 10 cm.

**Final Answer:** The largest distance of an image of a real object from a convex mirror of radius of curvature 20 cm is 10 cm.

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