Problem 43 - Master Resource Book in JEE Main Physics

Question

A convex lens of foeal length \(f\) produces a virtual image \(n\) times the size of the object. Then the distance of the object from the lens is (a) \((n-1) f\) (b) \((n+1) f\) (c) \(\left(\frac{n-1}{n}\right) f\) (d) \(\left(\frac{n+1}{n}\right) f\)

Step-by-Step Solution

Answer

The distance of the object from the lens is \( \left(\frac{n-1}{n}\right)f \).

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 of the lens, \( v \) is the image distance, and \( u \) is the object distance. For a virtual image produced by a convex lens, \( v \) is negative.

Step 2: Relate magnification to the distances

The magnification \( m \) is given by \( m = - \frac{v}{u} \). For a virtual image, \( m = n \) and is positive, implying \( v = -nu \).

Step 3: Substitute magnification equation into lens formula

Substituting \( v = -nu \) into the lens formula, we have \( \frac{1}{f} = \frac{1}{-nu} - \frac{1}{u} \).

Step 4: Simplify the equation

Combine the terms: \( \frac{1}{f} = \frac{-1 + n}{nu} \).

Step 5: Solve for object distance \(u\)

Rearrange the equation to solve for \( u \): \( u = \left(\frac{n-1}{n}\right)f \).

Key Concepts

Lens Formula

The lens formula is a central tool in optics, connecting the focal length of the lens to the object and image distances. It is presented as:\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]Here:

• \(f\) represents the focal length, which is the distance from the lens to its focus.
• \(v\) is the image distance. This is the distance from the lens to where the image forms.
• \(u\) refers to the object distance, the space between the lens and the object.

A convex lens, such as the one in our problem, can produce both real and virtual images. Notably, when dealing with convex lenses creating a virtual image, the image distance \(v\) becomes negative. Using the lens formula allows us to solve for unknown variables in these optics problems by substituting known values like the focal length and image distance.

Magnification in Optics

Magnification gives us a sense of how much bigger or smaller the image is compared to the object. The formula for magnification \(m\) in lenses is:\[ m = - \frac{v}{u} \]This equation highlights:

• The negative sign indicates the inversion of the image, turning it upside down in real image scenarios.
• For virtual images produced by a convex lens, the magnification \(m\) is both positive and equals \(n\), where \(n\) is the factor by which the image size is larger than the object.

In the context of our exercise, we see that \(v = -nu\). This is critical to substitute into the lens formula and helps to connect the lens and magnification formulas so that we can determine the object's placement for a specific image size.

Virtual Image Formation

Virtual images are a fascinating aspect of lenses, formed where the light rays from the object appear to diverge after passing through the lens. They are different from real images:

• Virtual images cannot be projected on a screen because the light doesn't actually converge to form the image at a real location.
• They are often upright, as opposed to the inverted nature of real images.

Convex lenses can create virtual images when the object is positioned within the focal length, closer to the lens. The image appears on the same side as the object and is larger than the object (magnified). In our problem, the given magnification indicates how much larger the virtual image is compared to the object. This understanding leads us to derive the position of the object using the lens and magnification formulas, factoring in the negative image distance for virtual image formation.