A concave lens of focal length \(15 \mathrm{~cm}\) forms an image \(10 \mathrm{~cm}\) from the lens. How far is the object placed from the lens? Draw the ray diagram.

The lens formula is a mathematical representation used to relate the object distance \(u\), the image distance \(v\), and the focal length \(f\) of a lens. The formula is given by: \[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \].
For concave lenses, it's important to remember that both the focal length and the image distance are negative. This convention helps in solving the problems correctly.
When substituting values into the formula, always check the signs of the quantities. This ensures accurate calculation and interpretation of the results.
By rearranging and simplifying the equation, one can find the unknown distance - either the object distance \(u\) or the image distance \(v\) - when the other quantities are known.

Focal length \(f\) is a key characteristic of a lens. It is the distance between the lens and its focal point, where rays parallel to the principal axis converge (for convex lenses) or appear to diverge from (for concave lenses).
For a concave lens, the focal length is considered negative because the focal point is on the same side as the object.
In this exercise, the focal length of the concave lens is given as \(-15 \, \mathrm{cm}\).
This negative sign is crucial for solving problems, as it correctly directs the behavior of light and the formation of images by the concave lens.
The focal length is derived from the lens' curvature and the material's refractive index, determining how strongly the lens bends light rays.

A ray diagram visually represents how light rays interact with a lens to form an image. For a concave lens:

• Draw one ray parallel to the principal axis. It will diverge after passing through the lens, appearing to come from the focal point on the same side as the object.
• Draw another ray passing through the optical center of the lens. This ray travels straight without bending.

The point where these rays appear to intersect when extended backward, is the location of the virtual image.
Ray diagrams help in understanding the paths taken by the rays, providing a clear visualization of image formation.
They also aid in verifying calculated distances by visually representing the setup described in the problem.

A virtual image is formed when the diverging rays from a lens appear to originate from a common point behind the lens. Unlike real images, virtual images:

• Cannot be projected onto a screen.
• Are always upright relative to the object.
• Are formed on the same side of the lens as the object.

In a concave lens, virtual images are formed because the lens causes the light rays to diverge. These diverged rays, when traced backward, seem to meet at a point.
In the given exercise, the concave lens forms a virtual image that appears \(-10 \, \mathrm{cm}\) from the lens.
Understanding virtual images is crucial as they behave differently than the real images formed by convex lenses, making it important to follow the correct conventions and techniques while solving.