Problem 13 - An Introduction to Physical Science

Question

Sketch a ray diagram for a spherical convex lens with an object at \(D_{o}=2 f\) and describe the image characteristics.

Step-by-Step Solution

Answer

The image is real, inverted, and of the same size as the object when the object is at \(2f\).

Step 1: Identify Lens and Object Placement

We have a spherical convex lens which is commonly used to converge light. The object is placed at a distance \(D_{o} = 2f\), where \(f\) is the focal length of the lens. The position \(2f\) is significant because it affects how and where the rays converge to form an image.

Step 2: Draw the Principal Axis and Lens

Start by drawing a horizontal line which is the principal axis of the lens. Then, represent the lens symmetrically about this axis. Mark the focal points (\(F\)) on either side of the lens at equal distances from the lens.

Step 3: Position the Object

Mark the object on one side of the lens at a distance \(2f\) from the lens on the principal axis. Represent the object with an arrow perpendicular to the principal axis.

Step 4: Path of Ray 1: Parallel to Principal Axis

Draw a ray from the top of the object parallel to the principal axis. Upon reaching the lens, this ray refracts through the lens and passes through the focal point on the opposite side.

Step 5: Path of Ray 2: Through the Center of the Lens

Draw another ray from the top of the object straight through the center (optical center) of the lens. This ray will pass through without bending and continue straight in the same direction.

Step 6: Identify the Intersection Point

Observe where the two rays intersect on the other side of the lens. The point of intersection marks the top of the image.

Step 7: Determine Image Characteristics

As the rays intersect at \(2f\) on the opposite side of the lens from the object, the image is formed there. It is real, inverted compared to the object, and of the same size as the object.

Key Concepts

Convex Lens

A convex lens is a fundamental optical element used widely in various applications, from glasses to cameras. Its primary characteristic is its converging nature. Convex lenses are thicker in the middle than at the edges, which helps them bend, or refract, light rays towards a central point called the focal point.

• The center of a convex lens is known as the optical center.
• Convex lenses can focus parallel incoming light rays to a point on the opposite side.
• These lenses are used to correct farsightedness and are instrumental in devices needing image magnification.

To visualize how this lens works, ray diagrams are often used. When light rays pass through a convex lens, they refract and converge, demonstrating the lens's function and behavior. A key focus when drawing these diagrams is the relationship between the lens, object, and the formed image.

Focal Length

The focal length of a lens is the distance from its optical center to its focal point. This measurement is crucial because it determines how strongly the lens can converge (or diverge in the case of concave lenses) the incoming rays. For a convex lens, a shorter focal length means a stronger converging power.

• The symbol for focal length is usually represented by \( f \).
• In our scenario, the object is positioned at twice the focal length (\( 2f \)) from the lens.
• When the object is placed at \( 2f \), the image also forms at \( 2f \) on the opposite side of the lens, making it a unique case in optics.

Understanding focal length helps in calculating where an image will form and how large or small it will appear. It essentially defines the optical "strength" of the lens, shaping the way we perceive and manipulate light.

Image Characteristics

When an object is positioned at twice the focal length \( 2f \) from a convex lens, the image formed has distinct characteristics due to the specific arrangement of the object and lens. Knowing these characteristics helps in understanding practical optical applications like photography and vision correction.

• The image is located at \( 2f \) on the opposite side of the lens.
• It is real, meaning it can be projected onto a screen because the light rays physically converge at the image location.
• Inverted vertically compared to the object, which usually happens with real images.
• It is of the same size as the object, which occurs because both the object and image are equidistant from the lens.

This knowledge of image characteristics can help in predicting and analyzing where and how images form when lenses are used, which is crucial in fields like optics and photography.