Problem 71 - Master Resource Book in JEE Main Physics

Question

The focal lengths of the objective and eyelenses of a microscope are \(1.6 \mathrm{~cm}\) and \(2.5 \mathrm{~cm}\) respectively. The distance between the two lenses is \(21.7 \mathrm{~cm}\). If the final image is formed at infinity, the distance between the object the objeetive lens is (a) \(1.8 \mathrm{~cm}\) (b) \(1.70 \mathrm{~cm}\) (c) \(1.65 \mathrm{~cm}\) (d) \(1.75 \mathrm{~cm}\)

Step-by-Step Solution

Answer

The object distance is approximately \(1.75\, \mathrm{cm}\), so the answer is (d).

Step 1: Use Lens Formula for Objective

We need to use the lens formula for the objective lens. The formula is \( \frac{1}{f_o} = \frac{1}{v_o} - \frac{1}{u_o} \), where \( f_o = 1.6\, \text{cm} \) is the focal length of the objective lens, \( v_o \) is the image distance for the objective lens, and \( u_o \) is the object distance from the objective lens, which we need to find. The image formed by the objective lens acts as an object for the eyepiece.

Step 2: Define Image Distance for Objective

Since the final image is formed at infinity, the intermediate real image at the focal point of the eyepiece, which means \( v_e = f_e = 2.5\, \mathrm{cm} \). The image distance for the objective lens \( v_o \) should be such that the distance equation holds: \( v_o + f_e = 21.7\, \mathrm{cm} \). We solve for \( v_o \): \[ v_o = 21.7\, \mathrm{cm} - 2.5\, \mathrm{cm} = 19.2\, \mathrm{cm}. \]

Step 3: Calculate Object Distance for Objective

Substitute \( v_o = 19.2\, \mathrm{cm} \) and \( f_o = 1.6\, \mathrm{cm} \) into the lens formula: \[ \frac{1}{1.6} = \frac{1}{19.2} - \frac{1}{u_o}. \] Rearrange to solve for \( \frac{1}{u_o} \): \[ \frac{1}{u_o} = \frac{1}{19.2} - \frac{1}{1.6}. \] Calculate \( \frac{1}{u_o} \): \[ \frac{1}{u_o} = \frac{1}{19.2} - \frac{1}{1.6} = \frac{0.052 - 0.625}{1\, \mathrm{cm}^{-1}} = -0.573\, \mathrm{cm}^{-1}. \] Therefore, \[ u_o \approx -1.745\, \mathrm{cm}. \] The negative sign indicates the object is on the opposite side of the lens, but we are interested in the magnitude.

Step 4: Round Object Distance to Nearest Choice

The magnitude of the object distance is approximately \( 1.745\, \mathrm{cm} \). None of the answer choices match this number exactly, but rounding gives \( 1.75\, \mathrm{cm} \). Thus, the best match is option (d).

Key Concepts

Lens Formula

The lens formula is a fundamental equation in the study of lens optics, and it's crucial for calculating relationships between object distance (\( u \)), image distance (\( v \)), and the focal length (\( f \)) of a lens. The formula is expressed as:\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u}. \]This equation helps determine how light rays converge or diverge when passing through a lens, allowing us to find where an image will form depending on where an object is placed.
In contexts like microscopes or cameras, understanding how to manipulate this formula allows for precise control over the focus and magnification.

• \( f \): focal length – the distance from the lens to the focal point.
• \( v \): image distance – the distance from the lens to where the image is focused.
• \( u \): object distance – the distance from the lens to the object being viewed.

Knowing how to rearrange and solve this equation is essential for predicting the behavior of lenses in various optical devices.

Focal Length

The focal length of a lens is a key optical parameter that defines how strongly it converges or diverges light. For a converging lens (like the objective lens in a microscope), a shorter focal length indicates stronger light bending and higher magnification. In the given exercise, two focal lengths are provided:

• Objective lens: \( f_o = 1.6 \, \mathrm{cm} \)
• Eyepiece lens: \( f_e = 2.5 \, \mathrm{cm} \)

These focal lengths determine where the intermediate image (produced by the objective lens) must form so that the eyepiece can further magnify it for the viewer.
This is particularly crucial in composing compound optical systems like microscopes, where multiple lenses work together to achieve significant magnification and resolution enhancements.

Image Distance

Image distance (\( v \)) is where the image comes to focus after light passes through a lens. It is a crucial part of the lens formula and can drastically affect the clarity and magnification of the image seen through optical devices such as microscopes.
In the problem, we adjust the image distance for the objective lens to properly align the final image formation. For the microscope, the intermediate image distance for the objective lens is calculated to be \( v_o = 19.2 \, \mathrm{cm} \), marking the spot where the image from the objective lens focuses to be used as the object for the eyepiece lens.
Correctly calculating or adjusting the image distance allows for seamless progression of the light path, ensuring that the eyepiece can effectively present the magnified image to the user.

Object Distance

Object distance (\( u \)) is the distance from the lens to the object being observed. It affects how much the image is magnified and where the resulting image will appear. In the context of microscopes, calculating the object distance involves using the lens formula to ensure that the final image is coherent and magnified correctly.
In our example, the task was to determine the object distance for the microscope's objective lens. Solving the lens formula gives us an object distance of \( u_o \approx -1.745 \, \mathrm{cm} \). The negative value here indicates that the object lies on the opposite side from where the light hits the objective lens. However, for practical purposes, we focus on the magnitude, which rounds to \( 1.75 \, \mathrm{cm} \).
This calculation ensures that the intermediate image formed is in the right position for subsequent magnification by the eyepiece, crucial for efficient microscope operation.