For each thin lens shown in Fig. E34.37, calculate the location of the image of an object that is 18.0 cm to the left of the lens. The lens material has a refractive index of 1.50, and the radii of curvature shown are only the magnitudes.

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Identify the lens type and use the lens maker's formula to calculate the focal length of the lens. The lens maker's formula is:

), where n is the refractive index of the lens material, R1 is the radius of curvature of the first surface, and R2 is the radius of curvature of the second surface.

Substitute the given values into the lens maker's formula. For this problem, n = 1.50, and the radii of curvature (R1 and R2) are provided in the figure. Be sure to assign the correct signs to R1 and R2 based on the lens's geometry (convex surfaces are positive, concave surfaces are negative).

Once the focal length (f) is determined, use the thin lens equation to find the image location. The thin lens equation is:

\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}, where d_o is the object distance (18.0 cm in this case) and d_i is the image distance.

Rearrange the thin lens equation to solve for the image distance (d_i):

\frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o} . Substitute the calculated focal length (f) and the given object distance (d_o = 18.0 cm) into this equation.

Simplify the equation to find the value of

d_i

, which represents the location of the image. Ensure the sign of d_i is interpreted correctly to determine whether the image is real or virtual and on which side of the lens it is located.

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주요 개념

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Thin Lens Formula

The thin lens formula relates the object distance (do), image distance (di), and focal length (f) of a lens. It is expressed as 1/f = 1/do + 1/di. This formula is essential for determining the position of the image formed by the lens based on the position of the object and the lens's focal length.

Refractive Index

The refractive index (n) of a material indicates how much light slows down when passing through it compared to a vacuum. For a lens, the refractive index affects the focal length, which can be calculated using the lensmaker's equation. In this case, the refractive index of 1.50 suggests that light travels slower in the lens material than in air.

Lensmaker's Equation

The lensmaker's equation provides a relationship between the focal length of a lens, its radii of curvature, and its refractive index. It is given by 1/f = (n - 1)(1/R1 - 1/R2), where R1 and R2 are the radii of curvature of the lens surfaces. This equation is crucial for calculating the focal length needed to apply the thin lens formula.