Problem 44 - Master Resource Book in JEE Main Physics

Question

A concave lens of focal length \(20 \mathrm{~cm}\) produces an image half in size of the real object. The distance of the real object is (a) \(20 \mathrm{~cm}\) (b) \(30 \mathrm{~cm}\) (c) \(10 \mathrm{~cm}\) (d) \(60 \mathrm{~cm}\)

Step-by-Step Solution

Answer

The distance of the real object is 60 cm.

Step 1: Understand the Lens Formula

The lens formula is given by \( \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \), where \( f \) is the focal length, \( v \) is the image distance, and \( u \) is the object distance. For a concave lens, the focal length \( f \) is negative.

Step 2: Identify Conditions for the Problem

Given that the image is half the size of the object, the magnification \( m \) is -0.5 (negative because the image is virtual and upright in a concave lens). Remember, \( m = \frac{v}{u} \), so we have \( -0.5 = \frac{v}{u} \), or \( v = -0.5u \).

Step 3: Substitute Known Values

We are given \( f = -20 \text{ cm} \) and from Step 2, \( v = -0.5u \). We substitute these into the lens formula: \( \frac{1}{-20} = \frac{1}{-0.5u} - \frac{1}{u} \).

Step 4: Solve the Equation for Object Distance

Rearrange and solve the equation: \[-\frac{1}{20} = -\frac{2}{u} - \frac{1}{u}\] \[-\frac{1}{20} = -\frac{3}{u}\] and thus, \( u = 60 \text{ cm} \).

Step 5: Confirm with the Set of Choices

Compare \( u = 60 \text{ cm} \) with the given options: (a) 20 cm, (b) 30 cm, (c) 10 cm, (d) 60 cm. Option (d) is correct.

Key Concepts

Lens Formula

The lens formula is a fundamental equation used to relate the focal length of a lens with the object and image distances. It is expressed as:\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]Here,

• f is the focal length of the lens.
• v is the image distance from the lens.
• u is the object distance from the lens.

For concave lenses, the focal length is considered negative, as it diverges light rays. When applying the lens formula, special attention is needed for sign conventions:

• Distances measured in the same direction as the incident light are positive.
• Distances against the direction of the light are negative.
• Focal lengths for concave lenses are negative.

Understanding and correctly applying this formula helps in determining unknown distances when two are known, providing a complete picture of the optical system in question.

Focal Length

The focal length is a crucial parameter of a lens, denoting the distance from the lens to the point where rays converge to form an image. For a concave lens, the focal length ( f) is negative, indicating that it does not converge light but instead makes it diverge. This divergence gives an impression that the light is originating from a point on the same side of the lens as the object, outside the lens.

Key Characteristics of Focal Length

• Concave lenses are also known as diverging lenses because of their ability to spread out incoming light rays.
• The focal point, shown virtually on the same side as the light source, indicates the illusionary origin of light after passing through the lens.
• The shorter the focal length, the stronger the diverging power of the lens.

These properties are essential for solving problems related to image forming by lenses and understanding how they manipulate light paths to produce images.

Magnification

Magnification is a measure of how much larger or smaller an image is compared to the object itself. For lenses, magnification (m) is calculated using:\[ m = \frac{v}{u} \]Here,

• v is the image distance.
• u is the object distance.

In the context of a concave lens, images formed are virtual, upright, and reduced in size. This results in negative magnification, as seen in the example with a magnification of -0.5, signaling that the image is upright and half the size of the object.

Important Points about Magnification

• A negative magnification value implies an upright and virtual image.
• In concave lenses, magnification values range between 0 and -1, indicating diminished image sizes.
• This helps predict not only how much smaller the image appears but also confirms the virtual nature of the image produced by concave lenses.

Magnification is essential in predicting and understanding the properties of the images produced by lenses, particularly in medical imaging, precision optical equipment, and everyday optics in cameras and corrective lenses.