Question

A student performed the experiment of determination of focal length of a concave mirror by u-v method using an optical bench of length \(1.5\) meter. The foeal length of the mirror used is \(24 \mathrm{~cm}\). The maximum error in the location of the image can be \(0.2 \mathrm{~cm}\). The 5 sets of \((\mathrm{u}, \mathrm{v})\) values recorded by the student (in \(\mathrm{cm})\) are: \((42,56)\). \((48,48),(60,40),(66,33),(78,39) .\) The data set(s) that cannot come from experiment and is (are) incorrectly recorded, is (are) (A) \((42,56)\) (B) \((48,48)\) (C) \((66,33)\) (D) \((78,39)\)

Short answer

Data sets (A) \((42,56)\), (C) \((66,33)\) and (D) \((78,39)\) are incorrectly recorded.

Step-by-step solution

- Recall the mirror equation

Use the mirror equation, which relates the object distance (u), the image distance (v), and the focal length (f) of the mirror: \(\frac{1}{f} = \frac{1}{u} + \frac{1}{v}\). For a concave mirror, the object distance (u) is taken to be negative since the object is placed in front of the mirror, and the image distance (v) could be positive or negative depending on whether the image formed is real or virtual.

- Calculate the expected image distance for each object distance

Substitute the given focal length of \(f = -24~\mathrm{cm}\) (negative because the mirror is concave) into the mirror equation and solve for the image distance \(v\) for each given object distance \(u\).

- Determine the correctness of each data set

Compare the calculated image distances with the given image distances. If the absolute difference between the calculated and observed image distances is greater than the error threshold of \(0.2~\mathrm{cm}\), then the data set is considered incorrectly recorded.

- Identify the incorrect data set(s)

Analyze each set of \((u,v)\) values by applying the mirror equation and compare with the maximum permissible error. Mark any set that shows a deviation greater than \(0.2~\mathrm{cm}\) as incorrect.

Key concepts

Concave Mirror

A concave mirror is a spherical mirror with the reflective surface curved inward, resembling a portion of the interior of a sphere. This type of mirror converges light, producing either a real image that can be projected on a screen or a virtual image that cannot be projected depending on the position of the object.

When an object is placed at a distance greater than the focal length of a concave mirror, the image formed is real, inverted, and can be displayed on a screen. In contrast, when the object is within the focal length, the image is virtual, upright, and appears to be behind the mirror. These properties are essential in various applications, including telescopes, flashlights, and shaving mirrors.

Object Distance

In the context of spherical mirrors, object distance, often denoted by 'u', is the distance measured along the principal axis from the mirror's surface to the object. By convention, the object distance for a concave mirror is taken as negative because the object is placed in front of the mirror's reflecting surface.

Understanding object distance is crucial since it directly influences the characteristics of the image formed, such as size, orientation, and type (real or virtual). In mathematical expressions and experimentations involving mirrors, carefully considering the sign of the object distance is vital for accurate results.

Image Distance

Image distance, or 'v', is the distance from the mirror to the image formed by reflection of light. This value can be either positive or negative; for concave mirrors, a real image has a positive image distance, whereas a virtual image has a negative value to indicate that the image appears to be behind the mirror.

In experimental setups, determining the exact position of the image is critical, especially in the calculation of the mirror's focal length. The accuracy of measuring 'v' impacts the precision of studying optical phenomena and conducting related experiments, which is why attention to detail and proper measurement techniques are emphasized.

Focal Length Determination

The focal length of a concave mirror is the distance from the mirror's surface to its focal point, the point where parallel rays of light converge after reflection. To determine the focal length experimentally, one can use the mirror equation \(\frac{1}{f} = \frac{1}{u} + \frac{1}{v}\) with known object and image distances. By rearranging and solving this equation, the focal length can be deduced.

In classrooms and laboratories, this determination is a fundamental exercise in understanding optical principles. The accuracy of these determinations depends on precise measurements and careful consideration of experimental errors, as even small discrepancies can significantly affect the calculated focal length.

Optical Bench

An optical bench is a laboratory apparatus used to conduct experiments involving the principles of optics, such as the investigation of focal lengths for lenses and mirrors. It consists of a long, stable metal or wooden rod with a scale, upon which various optical components like lenses, mirrors, and light sources can be mounted and adjusted.

The precision of an optical bench determines the accuracy of the experimental findings. High-quality optical benches allow fine adjustments and minimal vibrations, contributing to reliable and reproducible results. The optical bench used in determining a concave mirror’s focal length is fundamental for aligning the object, mirror, and screen or detector for accurate distance measurements.

JEE Advanced Physics

JEE Advanced, an entrance examination for engineering programs in India, includes Physics as a critical component, with optics being a significant topic within the subject. The examination tests a student's understanding of concepts like the behavior of concave mirrors, focal length determination, and experimental skills using devices like the optical bench.

Candidates are expected to have a firm grasp of both theoretical concepts and practical skills. Problems similar to the focal length determination of concave mirrors using the u-v method are common, assessing a student’s ability to apply knowledge to practical and theoretical scenarios. Preparing for JEE Advanced Physics requires a balance of conceptual understanding and problem-solving expertise, often using textbook exercises and experiments to solidify knowledge in optics and other physics areas.