A concave mirror forms an image of an object placed at 15 cm in front of it. If the image is formed at 30 cm in front of the mirror, what is the focal length of the mirror?

Understand the Problem

The question is asking us to determine the focal length of a concave mirror using the object distance and the image distance provided. We'll use the mirror formula, which relates the focal length (f), object distance (u), and image distance (v) as 1/f = 1/v + 1/u.

Answer

The focal length is $f \approx 6.67 \text{ cm}$.

Steps to Solve

Identify the mirror formula We will use the mirror formula, which states:

$$ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} $$

Where:

$f$ = focal length
$v$ = image distance
$u$ = object distance

Substitute the values for image distance and object distance Next, substitute the given values for $u$ and $v$ into the mirror formula. For example, if the object distance $u = 20 \text{ cm}$ and the image distance $v = 10 \text{ cm}$, we would have:

$$ \frac{1}{f} = \frac{1}{10} + \frac{1}{20} $$

Compute the right-hand side Calculate the right-hand side of the equation:

$$ \frac{1}{f} = \frac{2}{20} + \frac{1}{20} = \frac{3}{20} $$

Find the focal length To find the focal length $f$, take the reciprocal of both sides:

$$ f = \frac{20}{3} $$

Final calculation Now calculate the numerical value:

$$ f \approx 6.67 \text{ cm} $$

The focal length of the concave mirror is $f \approx 6.67 \text{ cm}$.

More Information

The focal length indicates how strongly the mirror converges or diverges light. In concave mirrors, the focal length is considered negative when using the sign conventions.

Tips

Incorrect sign conventions: It's important to use the sign conventions for concave mirrors, where the focal length is negative. Doing so can lead to errors in calculations.
Incorrectly adding fractions: Ensure you find a common denominator when adding fractions in the calculation step.

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