A converging lens with a focal length of 9.00 cm forms an of a 4.00-mm-tall real object that is to the left of the lens. The is 1.30 cm tall and erect. Where are the object and located? Is the real or virtual?

Verified step by step guidance

1. Identify the given values: focal length of the lens (f) = 9.00 cm, height of the object (h_o) = 4.00 mm, and height of the image (h_i) = 1.30 cm. Note that the image height is positive, indicating an erect image.
2. Convert all measurements to the same unit for consistency. Convert the object height from mm to cm: h_o = 4.00 mm = 0.40 cm.
3. Use the magnification formula for lenses, which is given by \( m = \frac{h_i}{h_o} = -\frac{d_i}{d_o} \), where m is the magnification, h_i is the image height, h_o is the object height, d_i is the image distance, and d_o is the object distance. Calculate the magnification by substituting the known values of h_i and h_o.
4. Substitute the magnification value into the magnification formula to solve for the ratio \( \frac{d_i}{d_o} \). Use this ratio and the lens formula \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \) to form two equations with two unknowns (d_o and d_i).
5. Solve the system of equations to find the values of d_o and d_i. Determine the nature of the image (real or virtual) by the sign of d_i. If d_i is positive, the image is real; if d_i is negative, the image is virtual.

Key Concepts

Converging Lens

A converging lens, or convex lens, is a transparent optical device that bends light rays inward to a focal point. It has a positive focal length, meaning it can form real images when the object is placed outside its focal length. The behavior of light through a converging lens is governed by the lens formula, which relates the object distance, image distance, and focal length.

Magnification

Magnification is the ratio of the height of the image to the height of the object, and it also relates to the distances of the object and image from the lens. It can be calculated using the formula: magnification (m) = height of image (h') / height of object (h) = - (image distance (v) / object distance (u)). A positive magnification indicates an erect image, while a negative value indicates an inverted image.

Real and Virtual Images

Real images are formed when light rays converge and can be projected onto a screen, appearing inverted. In contrast, virtual images occur when light rays diverge, and they cannot be projected onto a screen; they appear upright. The nature of the image (real or virtual) depends on the position of the object relative to the focal length of the lens.