(II) A diverging lens is placed next to a converging lens of focal length ƒ_C , as in Fig. 33–14. If ƒ_T represents the focal length of the combination, show that the focal length of the diverging lens, ƒ_D , is given by

1/ƒ_D = (1/ƒ_T) - (1/ƒ_C)

검증된 단계별 안내

Step 1: Understand the problem setup. The system consists of two lenses: a converging lens with focal length ƒ_C and a diverging lens with focal length ƒ_D. The combination of these lenses results in an effective focal length ƒ_T. The goal is to derive the relationship between ƒ_D, ƒ_C, and ƒ_T.

Step 2: Recall the formula for the effective focal length of a combination of two lenses placed in contact. The formula is given by:

\frac{1}{ƒ} = \frac{1}{ƒ} + \frac{1}{ƒ}

, where ƒ is the effective focal length of the system.

Step 3: Substitute the effective focal length ƒ_T for ƒ in the formula. This gives:

\frac{1}{ƒ} = \frac{1}{ƒ} + \frac{1}{ƒ}

Step 4: Rearrange the equation to isolate the term involving ƒ_D. Subtract

\frac{1}{ƒ}

from both sides:

\frac{1}{ƒ} = \frac{1}{ƒ} - \frac{1}{ƒ}

Step 5: Interpret the result. The derived equation shows that the reciprocal of the focal length of the diverging lens, ƒ_D, is equal to the difference between the reciprocals of the effective focal length of the combination, ƒ_T, and the focal length of the converging lens, ƒ_C. This relationship can be used to calculate ƒ_D if ƒ_T and ƒ_C are known.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Lens Formula

The lens formula relates the focal length (f), object distance (u), and image distance (v) for lenses. It is expressed as 1/f = 1/v - 1/u. This formula is fundamental in optics, allowing us to understand how lenses form images based on their focal lengths and the positions of objects.

Focal Length of Lenses

The focal length of a lens is the distance from the lens to the focal point, where parallel rays of light converge (for converging lenses) or appear to diverge (for diverging lenses). The sign of the focal length indicates the type of lens: positive for converging lenses and negative for diverging lenses, which is crucial for calculations involving multiple lenses.

Combination of Lenses

When multiple lenses are used together, the effective focal length of the combination can be determined using the formula 1/ƒ_T = 1/ƒ_C + 1/ƒ_D. This relationship shows how the focal lengths of individual lenses interact, allowing for the design of optical systems with desired properties by combining different types of lenses.