A lens forms an of an object. The object is 16.0 cm from the lens. The is 12.0 cm from the lens on the same side as the object. What is the focal length of the lens? Is the lens converging or diverging?

검증된 단계별 안내

Step 1: Begin by identifying the lens equation, which relates the object distance (d_o), image distance (d_i), and focal length (f). The equation is:

\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}

Step 2: Substitute the given values into the lens equation. The object distance (d_o) is 16.0 cm, and the image distance (d_i) is -12.0 cm (negative because the image is on the same side as the object, indicating a virtual image). The equation becomes:

\frac{1}{f} = \frac{1}{16.0} + \frac{1}{-12.0}

Step 3: Simplify the fractions to calculate the reciprocal of the focal length. Perform the addition of the two terms:

\frac{1}{f} = \frac{1}{16.0} + \frac{-1}{12.0}

. Combine the fractions by finding a common denominator.

Step 4: Once the reciprocal of the focal length is calculated, invert the result to find the focal length (f). Remember that the sign of the focal length will indicate the type of lens: a positive focal length corresponds to a converging lens, while a negative focal length corresponds to a diverging lens.

Step 5: Analyze the sign of the focal length obtained. If the focal length is positive, the lens is converging. If the focal length is negative, the lens is diverging. This conclusion is based on the properties of lenses and their ability to form real or virtual images.

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Lens Formula

The lens formula relates the object distance (u), image distance (v), and focal length (f) of a lens. It is expressed as 1/f = 1/v - 1/u. This formula is essential for determining the focal length of the lens when the object and image distances are known.

Converging and Diverging Lenses

Lenses can be classified as converging (convex) or diverging (concave). A converging lens brings parallel rays of light to a focus, resulting in a positive focal length, while a diverging lens causes parallel rays to spread out, resulting in a negative focal length. Identifying the type of lens is crucial for understanding its behavior with light.

Sign Convention in Optics

In optics, a sign convention is used to determine the signs of object distance, image distance, and focal length. Typically, distances measured in the direction of incoming light are positive, while those measured against it are negative. Understanding this convention is vital for correctly applying the lens formula and interpreting results.