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Step-by-Step Solution
Step 1: Understand the Problem
We have a convex lens. When an object is placed at 10 cm or at 20 cm from the lens, the size of the formed image is the same. We need to find the focal length of the lens.
Step 2: Write the Lens Formula
The lens formula is given by:
\frac{1}{v} - \frac{1}{u} = \frac{1}{f}
where:
u is the object distance.
v is the image distance.
f is the focal length of the lens.
Step 3: Express the Magnification
The linear magnification m produced by a thin lens is given by:
m = \frac{v}{u}.
Using the lens formula, another way to write m is:
m = \frac{f}{f + u}
(after substituting for v from the lens formula and simplifying).
Step 4: Apply the Given Condition of Equal Image Sizes
The magnifications for the two object distances (10 cm and 20 cm) must have the same magnitude but opposite signs (one image could be inverted, the other could be erect), so we set:
m_{1} = -\, m_{2}.
Here, let u_{1} = 10 cm and u_{2} = 20 cm. Then,
\frac{f}{f - 10} = - \,\frac{f}{f - 20}.
Step 5: Solve for Focal Length
From the above equation:
\frac{f}{f - 10} = -\frac{f}{f - 20},
we can cross-multiply:
f \,(f - 20) = -f \,(f - 10).
Simplify:
f - 20 = - f + 10 \quad \text{(after canceling f on both sides, if non-zero)},
f + f = 20 + 10,
2f = 30,
f = 15 \,\text{cm}.
Step 6: Conclusion
The focal length of the convex lens is \boxed{15 \text{ cm}} .
