Problem 85 - Physics

Question

An insect subtends an angle of only \(4.0 \times 10^{-3} \mathrm{rad}\) at the unaided eye when placed at the near point. What is the angular size (magnitude only) when the insect is viewed through a microscope whose angular magnification has a magnitude of \(160 ?\)

Step-by-Step Solution

Answer

The angular size through the microscope is \(0.64\text{ rad}\).

Step 1: Understanding the Problem

We are tasked with finding the angular size of an insect when viewed through a microscope. We are given that its angular size at the near point is \(4.0 \times 10^{-3} \text{ rad}\), and the angular magnification of the microscope is 160.

Step 2: Angular Magnification Formula

The formula for angular magnification \(M\) is \( M = \frac{\text{angular size through the microscope}}{\text{angular size at the eye}} \). We can rearrange it to find the angular size through the microscope: \( \text{angular size through microscope} = M \times \text{angular size at the eye} \).

Step 3: Substitute the Values

Substitute the given values into the formula: \( \text{angular size through microscope} = 160 \times (4.0 \times 10^{-3} \text{ rad}) \).

Step 4: Calculate the Result

Perform the multiplication: \( 160 \times 4.0 \times 10^{-3} = 0.64 \text{ rad} \).

Key Concepts

angular size

Angular size refers to the apparent size of an object as seen from a particular vantage point. It is measured as the angle that the object subtends at the point of observation. Often, angular size is represented in units such as degrees, minutes, or radians. In the given problem, the insect subtends an angle of \(4.0 \times 10^{-3} \text{ rad}\) to the unaided eye.
The angular size is crucial in optics since it helps us understand how large an object appears, not necessarily its actual size. The concept is particularly important when studying distant astronomical objects or tiny microscopic subjects.

• Angular size can change based on distance; as an object moves closer, the angular size increases.
• Optical instruments like telescopes or microscopes often aim to increase the angular size to make objects more visible and detailed.

microscope

A microscope is an optical instrument that magnifies small objects. In essence, it enhances the angular size of an object, allowing observers to see details that would otherwise be impossible to detect with the unaided eye. A standard compound microscope, for instance, uses multiple lenses to achieve this magnification.
In the exercise, the microscope provides an angular magnification of 160. This means that it enlarges the insect's angular size 160 times. When optical lenses in the microscope focus light, they manipulate the path to enlarge the perceived image.

• Lenses are crucial components, typically consisting of eye lenses and objective lenses.
• The magnification determined by a microscope is often a product of the magnifications of its individual lenses.

rad radian

Radian is a unit of angular measure used extensively in mathematics and physics. It provides a natural way of describing angles compared to degrees. There are \(2\pi\) radians in a full circle, translating approximately to 360 degrees. Thus, one radian is equivalent to about 57.3 degrees.
The use of radians simplifies many equations in physics and engineering, particularly those dealing with angular motion and wave-related phenomena. In this scenario, the insect's initial angular size is given in radians, highlighting the precision and simplicity this unit offers in calculations.

• Radians are dimensionless and are often preferred in calculus due to their mathematical convenience.
• Simple trigonometric expressions can be derived more naturally using radians.

optics

Optics is the branch of physics that explores light properties and interactions. It covers a broad array of topics, including refraction, reflection, and diffraction. In practical applications, optics is the science that enables the functioning of instruments like microscopes, telescopes, and cameras.
Through understanding optics, scientists and engineers can manipulate light paths to achieve desired outcomes, such as making minute objects appear larger or enhancing image clarity.

• Key laws in optics include Snell's Law (refraction) and the Law of Reflection.
• Optical instruments use these principles to modify how light interacts with objects to alter their angular size effectively.