Concave Mirror Image Calculation
This solution explains how to determine the characteristics of an image formed by a concave mirror using the mirror formula.
Given Information
- Object distance, \(u\) = 15 cm. Since the object is in front of the mirror, by sign convention, \(u = -15\) cm.
- Radius of curvature, \(R\) = 20 cm. For a concave mirror, the radius of curvature is negative, so \(R = -20\) cm.
Calculations
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Calculate Focal Length (\(f\)):
The focal length of a mirror is half its radius of curvature. \(f = \frac{R}{2}\) \(f = \frac{-20 \text{ cm}}{2}\) \(f = -10 \text{ cm}\) The negative focal length confirms it's a concave mirror.
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Apply the Mirror Formula:
The mirror formula relates object distance (\(u\)), image distance (\(v\)), and focal length (\(f\)): \(\frac{1}{f} = \frac{1}{v} + \frac{1}{u}\)
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Solve for Image Distance (\(v\)):
Substitute the known values of \(f\) and \(u\) into the formula: \(\frac{1}{-10} = \frac{1}{v} + \frac{1}{-15}\) Rearrange the formula to solve for \(\frac{1}{v}\): \(\frac{1}{v} = \frac{1}{-10} - \frac{1}{-15}\) \(\frac{1}{v} = -\frac{1}{10} + \frac{1}{15}\) To subtract the fractions, find a common denominator, which is 30: \(\frac{1}{v} = -\frac{3}{30} + \frac{2}{30}\) \(\frac{1}{v} = \frac{-3 + 2}{30}\) \(\frac{1}{v} = -\frac{1}{30}\) Invert both sides to find \(v\): \(v = -30 \text{ cm}\)
Image Characteristics
The calculated image distance is \(v = -30\) cm.
- The negative sign for \(v\) indicates that the image is formed on the same side as the object (in front of the mirror).
- The distance of the image from the mirror is 30 cm.
Therefore, the correct statement is that the image is 30 cm from the mirror on the same side as the object.
