To determine the image distance for an illuminated object placed in front of a converging lens (convex lens), we can use the lens formula:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
Where:
- \(f\) is the focal length of the lens,
- \(v\) is the image distance from the lens,
- \(u\) is the object distance from the lens.
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Focal Length: The focal length given is \(f = 30 , \text{cm}\) (positive for a converging lens).
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Object Distance: The object distance is given as \(u = -7 , \text{vm} = -700 , \text{cm}\) (negative as per the sign convention that measures distances against the direction of the incoming light).
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Substituting values into the lens formula: Now we can substitute the values into the lens formula:
\[ \frac{1}{30} = \frac{1}{v} - \frac{1}{-700} \]
This simplifies to:
\[ \frac{1}{30} = \frac{1}{v} + \frac{1}{700} \]
Rearranging gives:
\[ \frac{1}{v} = \frac{1}{30} - \frac{1}{700} \]
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Finding a common denominator: The least common multiple of 30 and 700 is 2100. We can rewrite the fractions:
\[ \frac{1}{30} = \frac{70}{2100} \quad \text{and} \quad \frac{1}{700} = \frac{3}{2100} \]
Thus:
\[ \frac{1}{v} = \frac{70}{2100} - \frac{3}{2100} = \frac{67}{2100} \]
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Calculating \(v\): Inverting gives:
\[ v = \frac{2100}{67} \approx 31.34 , \text{cm} \]
Thus, the image distance \(v\) from the lens is approximately 31.34 cm.
The positive value indicates that the image is formed on the opposite side of the lens from the object, which is typical for a converging lens with a real object.