Question

A concave lens of focal length $\mathrm{f}$ forms an image which is n times the size of the object. What is the distance of the object from the lens? (A) $(1+\mathrm{n})\mathrm{f}$ (B) $(1-\mathrm{n})\mathrm{f}$ (C) $[(1-\mathrm{n})/\mathrm{n}]\mathrm{f}$ (D) $[(1+\mathrm{n})/\mathrm{n}]\mathrm{f}$

Short answer

The short answer is: $u=(1-n)f$, where u is the distance of the object from the concave lens, n is the magnification factor, and f is the focal length of the lens. The correct option is (B).

Step-by-step solution

Understanding magnification

In optics, the magnification of a lens describes how much an object is magnified (enlarged or diminished) when viewed through the lens. It is calculated as the ratio of the image height (h_i) to the object height (h_o). Magnification is also related to the object's distance (u) and the image's distance (v) from the lens. For lenses, the magnification formula is given by: $M=\frac{h_i}{h_o}=-\frac{v}{u}$ In this exercise, the magnification (M) is given as n times the size of the object, that is, M = n.

Lens formula

Now, we need to use the lens formula to relate the object's distance (u), image's distance (v), and focal length (f) of the lens. The lens formula is given by: $\frac{1}{f}=\frac{1}{u}+\frac{1}{v}$

Substituting magnification and solving for object distance

Since we have the lens formula and magnification formula, we can combine them to find the object's distance (u) in terms of the focal length (f) and magnification (n). First, rewrite the magnification formula in terms of v: $v=-\frac{u}{n}$ Now, substitute this expression for v into the lens formula: $\frac{1}{f}=\frac{1}{u}+\frac{1}{-\frac{u}{n}}$

Simplifying and solving

Now, simplify the equation and solve for u in terms of f and n: $\frac{1}{f}=\frac{1}{u}-\frac{n}{u}$ $\frac{1}{f}=\frac{1-n}{u}$ Now, solve for u: $u=\frac{1-n}{f}$ Since the options are given in a form where f is the numerator, we can rewrite the expression for u: $u=\frac{(1-n)}{1}f$ Now, compare this expression to the options given: (A) $(1+\mathrm{n})\mathrm{f}$ (B) $(1-\mathrm{n})\mathrm{f}$ (C) $[(1-\mathrm{n})/\mathrm{n}]\mathrm{f}$ (D) $[(1+\mathrm{n})/\mathrm{n}]\mathrm{f}$ The expression for the object's distance matches option (B), so the answer is: (B) $(1-\mathrm{n})\mathrm{f}$

Key concepts

Understanding the Lens Formula

A lens is a transparent object that refracts light to form an image. To understand how a lens works, we rely on the lens formula, which is a crucial concept in optics. This formula is essential for solving problems related to both concave and convex lenses. The lens formula is expressed as: $$\frac{1}{f}=\frac{1}{u}+\frac{1}{v}$$ Here,

• $f$ is the focal length of the lens, which is either positive for convex lenses or negative for concave lenses.
• $u$ is the object distance from the lens.
• $v$ is the image distance from the lens.

This equation lets us relate these three distances, enabling us to find one if the other two are known. In the case of concave lenses, where $f$ is negative, the lens formula helps determine how the lens impacts the path of light rays and the nature of the image formed.

Exploring Magnification in Optics

Magnification is a measure of how much larger or smaller an image is compared to the object itself. It's a key concept when dealing with lenses, concave or convex. Magnification $M$ is given by the formula: $$M=\frac{h_i}{h_o}=-\frac{v}{u}$$

• $h_i$ is the height of the image.
• $h_o$ is the height of the object.
• $v$ is the image distance, while $u$ is the object distance.

The negative sign indicates that the image formed by a concave lens is virtual and upright relative to the object. When the problem states that the image is $n$ times the size of the object, we understand it as: $$M=n$$ This information is used in combination with the lens formula to determine the other parameters like object distance, given the magnification and focal length.

Determining Object Distance

Object distance $u$ refers to how far the actual object is located from the lens. In the context of concave lenses, it's crucial to find object distance for solving many optics problems. Using both the lens formula and the magnification, we find the relation: $$-\frac{v}{u}=n$$ From this, we express $v$ in terms of $u$: $$v=-\frac{u}{n}$$ Substitute this in the lens formula: $$\frac{1}{f}=\frac{1}{u}+\frac{1}{(-\frac{u}{n})}$$ After some calculations and simplifying, this leads us to: $$u=\frac{(1-n)}{f}$$ This shows how the object distance relates to the focal length and the magnification $n$. By knowing these relationships, you can efficiently find where the object needs to be placed to achieve a specific magnification with a concave lens.

The Role of Focal Length

Focal length $f$ is a fundamental property of any lens, including concave lenses. It tells us how strongly the lens converges or diverges light. For concave lenses, the focal length is negative, indicating the lens' diverging nature. This characteristic affects how and where images are formed. The focal length can be determined physically by the curvature and refractive index of the lens material. It also plays a critical part in the lens formula: $$\frac{1}{f}=\frac{1}{u}+\frac{1}{v}$$ Here, focal length relates directly to the image and object distances. This relationship is useful not just mathematically but also in predicting how the lens will manipulate light. Understanding the focal length enables us to comprehend how lenses capture objects at various distances, adjusting your need for clarity and image size across different applications.