Question

In an experiment to determine the focal length \((f)\) of a concave mirror by the \(u-v\) method, a student places the object pin \(\mathrm{A}\) on the principal axis at a distance \(x\) from the pole P. The student looks at the pin and its inverted image from a distance keeping his/her eye in line with PA. When the student shifts his/her eye towards left, the image appears to the right of the object pin. Then, (A) \(x2 f\)

Short answer

B) \(f<x<2f\)

Step-by-step solution

Evaluate the behavior of the image in a concave mirror

When an object is placed between the focal point (F) and the center of curvature (C) of a concave mirror, the image formed is virtual, erect, and magnified, and it would appear behind the mirror. However, when the object is shifted outwards of the focal length, the image becomes real, inverted, and is on the same side as the object but beyond the center of curvature. Once the object is beyond the center of curvature, the image will be between F and C. When the eye moves to the left (towards the direction of the object), a real image that is to the right of the line of sight would indicate that the image lies between C and F.

Determine the possible positions of object pin A

Given that when the student shifts his/her eye towards left, the image appears to the right of the object pin, this phenomenon indicates that the image formed by the concave mirror is real and inverted and lies between the focal point F and twice the focal length (2f), which is the center of curvature C. This means the object must be located beyond the focal length F but still within twice the focal length (2f).

Match the observation with the given options

From the previous analysis, we can deduce that the object must be placed somewhere between the focal length and twice the focal length of the concave mirror to match the student's observation. Out of the provided options, (B) \(f<x<2f\) aligns with our evaluation and is the correct answer.

Key concepts

u-v Method

The u-v method is a practical approach used in optics to determine the focal length of a mirror or a lens. It involves placing an object at various positions (usually denoted as 'u' for object distance) and measuring the corresponding image distance (denoted as 'v'). The focal length (f) can be calculated using the mirror equation \( \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \).

In the context of the textbook exercise, the student used this method by strategically moving the object pin (A) along the principal axis and observing the relative position of its image. The observation regarding the direction in which the image shifts relative to the object pin provides crucial information to infer the range of the object's distance from the mirror's pole (P). By applying the logic behind the u-v method, we deduce that for the given scenario, the object pin is located at a distance greater than the focal length but less than twice the focal length.

Image Formation

Understanding how image formation occurs with concave mirrors helps in determining the position of the object in relation to the mirror. The rule of thumb is: if the object is placed closer to the mirror than the focal length, the image will be virtual and magnified. If the object is beyond the focal length, the image becomes real and inverted.

A key aspect of identifying image properties is parallax, which arises when the observer's eye position changes. In our case, the image is seen to move in the opposite direction to the observer's eye movement, a tell-tale sign that the image is real and inverted. Linking this behavior to the various zones of image formation (within focal length, between focal length and center of curvature, beyond center of curvature), it becomes clear that the student's observations are consistent with the object being between the focal length and twice the focal length of the mirror.