Spherical mirrors and lenses Questions

Solution

Option (c) is correct.

Explanation:

Given:

• Focal length, \( f = -10 \) cm (concave lens is negative)
• Image distance, \( v = -5 \) cm (image on the same side as the object)

Using the lens formula: \[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \] Substituting the known values: \[ \frac{1}{-10} = \frac{1}{-5} + \frac{1}{u} \] Calculating \( \frac{1}{u} \): \[ \frac{1}{u} = \frac{1}{-10} - \frac{1}{-5} \] \[ \frac{1}{u} = -0.1 + 0.2 = 0.1 \implies u = 10 \text{ cm} \] Now, use the magnification formula: \[ M = \frac{-v}{u} \] Substituting the values: \[ M = \frac{-(-5)}{10} = \frac{5}{10} = 0.5 \] Thus, the magnification produced is \( 0.5 \).

Correct Answer

📖 Governing Law / Core Concept

Concave lenses are diverging lenses that only produce virtual images, where the rays appear to diverge from a common point behind the lens.

🧪 Step-by-Step Breakdown

• A convex lens can produce both real and virtual images depending on object placement.
• A concave lens only produces virtual images, always smaller than the object.
• A convex lens can magnify (greater than object size), demagnify (smaller than object size), and create equal size images.
• Concave lens behavior:
  • Real images: Never formed
  • Virtual images: Always formed between the focal point and the lens

🔍 Option Analysis

• Option A: True – Convex lenses can generate various types of images.
• Option B: False – Concave lenses cannot create real images.
• Option C: True – Convex lenses can vary image sizes based on object distance.
• Option D: True – Concave lenses produce images smaller than the objects.

⚡ Mnemonic / Speed-Run

Remember: "Concave Can't Create Real" for understanding image formation in concave lenses.

6. Visual Suggestion

📖 Factual Verification & Reference

Correct Answer

📖 Governing Law / Core Concept

The spherical mirror formula is given by: Where:

• \( f \) = focal length (m)
• \( u \) = object distance (m)
• \( v \) = image distance (m)

In the case of a plane mirror, the focal length approaches infinity, leading to an equal distance of the object and image from the mirror surface.

🧪 Step-by-Step Breakdown

For a plane mirror, we consider the limit as follows: As \( f \to \infty \): Substituting in the mirror formula, we have: This leads us to \( v = -u \), indicating that distances are equal and both are on opposite sides of the mirror.

🔍 Option Analysis

- **B**: Focal length goes to zero - Incorrect, as this would imply the mirror does not converge rays. - **C**: Image distance goes to zero - Incorrect, as it would mean the image is at the mirror surface. - **D**: Image distance goes to infinity - Incorrect, as an infinite image distance is contradictory for a plane mirror.

⚡ Mnemonic / Speed-Run

Remember that **Plane Mirror = Infinite Focal Length**. It helps to recall that light reflects off a plane mirror without convergence.

6. Visual Suggestion

📖 Factual Verification & Reference

Correct Answer

📖 Governing Law / Core Concept

The power \( P \) of a lens is given by the formula: where \( P \) is the power in Dioptres (D) and \( f \) is the focal length in meters (m).

🧪 Step-by-Step Breakdown

Given \( f = 25 \) cm, convert to meters: Substituting into the power formula: Calculating:

🔍 Option Analysis

- Option (a): +2.5 Dioptre - Incorrect, as \( P = 4 \). - Option (b): +3 Dioptre - Incorrect, as \( P = 4 \). - Option (d): +5 Dioptre - Incorrect, as \( P = 4 \).

⚡ Mnemonic / Speed-Run

To remember the power formula, think: **"Power equals one over focal!"**

6. Visual Suggestion

📖 Factual Verification & Reference

Correct Answer

📖 Governing Law / Core Concept

According to the New Cartesian Sign Convention, the formula for mirrors is given by: \[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \] where: - \( f \) = focal length (meters), - \( v \) = image distance (meters), - \( u \) = object distance (meters).

🧪 Step-by-Step Breakdown

For spherical mirrors: - **Sign Convention**: - Focal length (\( f \)) is taken positive for concave mirrors. - Object distance (\( u \)) is negative when the object is in front of the mirror. - Image distance (\( v \)) is positive if the image is formed on the same side as the object. Thus, the relationship holds for spherical mirrors only.

🔍 Option Analysis

- **Option B**: It applies only to spherical lenses. Incorrect because the formula is not valid for lenses. - **Option C**: It applies to spherical mirrors as well as spherical lenses. Incorrect because lenses have different conventions. - **Option D**: It is an invalid formula. Incorrect as it is valid for spherical mirrors.

⚡ Mnemonic / Speed-Run

Remember: "M" for Mirrors, "L" for Lenses. Use "Molar" to recall that mirrors follow the Cartesian convention.

6. Visual Suggestion

📖 Factual Verification & Reference

Correct Answer

📖 Governing Law / Core Concept

The power \( P \) of a lens is defined as the reciprocal of its focal length \( f \) (in meters):

🧪 Step-by-Step Breakdown

Given the focal length of the concave lens:

The value of \( f = -0.5 \, m \) (negative, as it is a concave lens).

Substituting into the power formula:

🔍 Option Analysis

• Option A: Incorrect; the power should be negative.
• Option B: Incorrect; the magnitude is correct, but the sign is wrong.
• Option C: Incorrect; positive power does not apply to concave lenses.

⚡ Mnemonic / Speed-Run

Recall that for concave lenses, \( P = \frac{1}{f} \) where \( f \) is negative.

6. Visual Suggestion

Correct Answer

📖 Governing Law / Core Concept

The power of a lens is given by:

where \( P \) is the power in diopters (D) and \( f \) is the focal length in meters (m). The total power of a combination of lenses is the algebraic sum of their individual powers.

🧪 Step-by-Step Breakdown

Given:

• Power of first lens, \( P_1 = +2.5 \, \text{D} \)
• Power of second lens, \( P_2 = -2.0 \, \text{D} \)

Calculating total power:

Now calculating the focal length:

📌 Hints / Properties Used:

• Lens formula: \( P = \frac{1}{f} \)
• Additive property of lens power: \( P = P_1 + P_2 \)

📊 Visual Diagram Suggestion:

A ray diagram illustrating the combination of the two lenses showing parallel incoming rays converging at a focal point, demonstrating the effective focal length of +2.0 m.

Correct Answer

📖 Governing Law / Core Concept

The formation of images in concave mirrors is governed by the mirror formula:

Where:

• f = focal length (m)
• u = object distance (m)
• v = image distance (m)

🧪 Step-by-Step Breakdown

For a concave mirror, the center of curvature (C) is at a distance equal to twice the focal length (2f). When an object is placed at the center of curvature:

(negative due to the convention, where \( R \) is the radius of curvature).

Using the mirror formula:

This shows that the image is located at the center of curvature.

🔍 Option Analysis

• Option A: At infinity - Incorrect, as the image forms at a finite distance.
• Option C: At the focal point - Incorrect, the image is not formed here when the object is at C.
• Option D: Between the focal point and the center of curvature - Incorrect for this configuration.

⚡ Mnemonic / Speed-Run

Remember: "C is Real and Inverted". At the center of curvature, the image formed is real, inverted, and of the same size.

Correct Answer

📖 Governing Law / Core Concept

The ray diagram principle for thin lenses states that when an object is placed at a distance equal to \( 2f \) (twice the focal length), the image formed is real, inverted, and of the same size as the object.

🧪 Step-by-Step Breakdown

If an object is situated at a distance equal to \( 2f \), the following conditions hold:

Thus, the image is formed at \( 2f \) on the opposite side, indicating it's real and inverted.

🔍 Option Analysis

• Option (a): Incorrect. Depicts a virtual image.
• Option (b): Incorrect. Shows an image formation at a location closer than \( 2f \).
• Option (c): Incorrect. Displays a case where the object is between \( f \) and \( 2f \).

⚡ Mnemonic / Speed-Run

Focal length (f), 2 times the object distance gives real images. Remember: For \( u = 2f \), \( v = 2f \) (Inverted & same size).