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. (a) What is the focal length of the lens? Is the lens converging or diverging?

Verified step by step guidance

1. Identify the type of image formed. Since the image is on the same side as the object, it is a virtual image, which is typically formed by a diverging lens.
2. Use the lens formula \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\), where \(f\) is the focal length, \(d_o\) is the object distance, and \(d_i\) is the image distance. Note that for virtual images, \(d_i\) is considered negative.
3. Substitute the given values into the lens formula. Here, \(d_o = 16.0\,\text{cm}\) and \(d_i = -12.0\,\text{cm}\) (negative because the image is virtual).
4. Solve the lens formula for \(f\) to find the focal length of the lens.
5. Determine the type of lens based on the sign of the focal length. If \(f\) is negative, the lens is diverging; if positive, it is converging.

Key Concepts

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). Converging lenses focus parallel rays of light to a point, resulting in a positive focal length, while diverging lenses spread light rays outward, leading to 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. This convention is vital for correctly applying the lens formula and interpreting the results.