A 2.0-cm-tall object is 15 cm in front of a plano-convex polystyrene plastic lens that has a 13 cm radius of curvature. What are the (a) position and (b) height of the image?

Verified step by step guidance

1. Determine the focal length of the plano-convex lens using the lens maker's formula: $f=\frac{1}{n-1}\cdot \frac{1}{R}$, where $n$ is the refractive index of polystyrene (approximately 1.59) and $R$ is the radius of curvature (13 cm).
2. Use the thin lens equation to find the image position: $\frac{1}{f}=\frac{1}{d}$_o+1d_i, where $d$_o is the object distance (15 cm), $d$_i is the image distance, and $f$ is the focal length calculated in step 1.
3. Rearrange the thin lens equation to solve for the image distance $d$_i: $d$_i=11f−1d_o.
4. Calculate the magnification of the image using the magnification formula: $m=-\frac{d}{}$_id_o, where $d$_i is the image distance and $d$_o is the object distance.
5. Determine the height of the image using the magnification: $h$_i=m⋅h_o, where $h$_o is the object height (2.0 cm) and $m$ is the magnification calculated in step 4.

Key Concepts

Lens Formula

The lens formula relates the object distance (u), image distance (v), and focal length (f) of a lens. It is given by the equation 1/f = 1/v - 1/u. For a plano-convex lens, the focal length can be determined using the radius of curvature (R) with the formula f = R/2. Understanding this relationship is crucial for finding the position of the image.

Magnification

Magnification (M) is the ratio of the height of the image (h') to the height of the object (h), expressed as M = h'/h = -v/u. It indicates how much larger or smaller the image is compared to the object. This concept is essential for determining the height of the image once the image distance is calculated.

Sign Convention in Optics

In optics, a sign convention is used to determine the signs of distances and heights. Typically, distances measured in the direction of the incoming light are negative, while those in the opposite direction are positive. This convention helps in correctly applying the lens formula and calculating magnification, ensuring accurate results for image position and height.