What is the focal length of a second lens that could be placed in contact with the first lens to provide an overall power of 30 D?

검증된 단계별 안내

Step 1: Recall the formula for the combined power of two lenses in contact: \( P_{total} = P_1 + P_2 \), where \( P_1 \) is the power of the first lens and \( P_2 \) is the power of the second lens.

Step 2: Rearrange the formula to solve for \( P_2 \): \( P_2 = P_{total} - P_1 \). Substitute \( P_{total} = 30 \, \text{D} \) and the given power of the first lens \( P_1 \) into the equation.

Step 3: Recall the relationship between the power of a lens and its focal length: \( P = \frac{100}{f} \), where \( P \) is the power in diopters and \( f \) is the focal length in centimeters.

Step 4: Rearrange the formula to solve for the focal length of the second lens: \( f_2 = \frac{100}{P_2} \). Substitute the value of \( P_2 \) obtained from Step 2 into this equation.

Step 5: Perform the substitution and simplify the expression to find the focal length of the second lens in centimeters. Ensure the units are consistent throughout the calculation.

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주요 개념

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Lens Power

The power of a lens is defined as the reciprocal of its focal length (in meters), expressed in diopters (D). It indicates how strongly the lens converges or diverges light. A positive power signifies a converging lens, while a negative power indicates a diverging lens. The overall power of a system of lenses in contact is the algebraic sum of their individual powers.

Focal Length

The focal length of a lens is the distance from the lens to the focal point, where parallel rays of light converge or appear to diverge. It is a critical parameter that determines the lens's ability to focus light. For thin lenses, the relationship between focal length (f) and power (P) is given by the formula P = 1/f, where f is in meters. Understanding this relationship is essential for calculating the required focal length of additional lenses.

Combining Lenses

When two lenses are placed in contact, their combined power can be calculated by simply adding their individual powers. This principle allows for the design of optical systems with desired focal lengths and powers. If the first lens has a known power, the second lens's power can be determined by rearranging the overall power equation to find the necessary focal length for the second lens to achieve the desired total power.