Problem 1 - Master Resource Book in JEE Main Physics

Question

A dentist has a small mirror of focal length \(16 \mathrm{~mm}\). He views the cavity in the tooth of a patient by holding the mirror at a distance of \(8 \mathrm{~mm}\) from the cavity. The magnification is (a] I (b) \(1.5\) (c) 2 (d) 3

Step-by-Step Solution

Answer

(c) 2

Step 1: Understand the Lens/Mirror Formula

The magnification formula for a concave mirror, such as the one used by the dentist, is related to the object distance (d_o), image distance (d_i), and focal length (f) of the mirror. The formula is:\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \] where f is the focal length, d_o is the object distance, and d_i is the image distance.

Step 2: Insert Known Values into the Formula

According to the problem statement, the focal length f is 16 mm and the object distance d_o is 8 mm. Substituting these values into the lens formula gives: \[ \frac{1}{16} = \frac{1}{8} + \frac{1}{d_i} \] We need to solve this for d_i.

Step 3: Solve for Image Distance

Rearrange the equation to solve for d_i:\[ \frac{1}{d_i} = \frac{1}{16} - \frac{1}{8} \] Calculate the right side:\[ \frac{1}{d_i} = \frac{1}{16} - \frac{2}{16} = -\frac{1}{16} \] Thus:\[ d_i = -16 \text{ mm} \] The negative sign indicates that the image is formed on the same side as the object, typical for a concave mirror.

Step 4: Calculate Magnification

The magnification M of the mirror is given by the formula:\[ M = \frac{d_i}{d_o} \] Substitute the values of d_i (-16 mm) and d_o (8 mm) into the equation:\[ M = \frac{-16}{8} = -2 \] The negative sign indicates the image is inverted, and the magnitude shows the image size is double that of the object.

Key Concepts

Concave Mirror

A concave mirror is a type of spherical mirror with a surface that curves inward, resembling the inside of a bowl. These mirrors converge light to a focal point, making them extremely useful for various applications, including in dentistry. When an object is placed in front of a concave mirror, the mirror reflects light in such a way that the reflected rays converge to form an image.

Here's how it works:

• The mirror's inner surface reflects the light, causing the rays to converge towards a focal point.
• If the object is placed closer to the mirror than the focal point, the reflected rays diverge, and the image formed is virtual, upright, and larger than the object.
• If the object is beyond the focal point, the image formed is real, inverted, and can be magnified or reduced based on the object's distance from the mirror.

Concave mirrors are favored in situations where magnification of the object is needed, such as for dentists examining teeth, because these mirrors can create enlarged images, making it easier to view small details.

Focal Length

The focal length of a mirror is the distance between the mirror's surface and its focal point, where parallel rays of light converge. In the case of the dentist's mirror mentioned, the focal length is 16 mm. This value is crucial because it determines the mirror's ability to magnify or reduce objects placed before it.

Understanding focal length helps in

• determining the type of image formed (real or virtual, magnified or diminished).
• predicting the behavior of light rays as they reflect off the mirror.

For mirrors, a shorter focal length means stronger magnification capabilities, which is why a short focal length mirror is ideal for detailed examinations, such as looking into the cavities of teeth. It is important to know that a negative focal length typically indicates a mirror, while a positive one is often used in the context of lenses.

Image Distance

Image distance (d_i) is the distance from the mirror to the image it forms. It's a critical component of the mirror equation: \[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]This formula links the object distance, image distance, and focal length. In our exercise, by knowing the focal length (16 mm) and the object distance (8 mm), we calculated the image distance as -16 mm.

The negative value of the image distance signifies that the image is formed on the same side as the object relative to the mirror. This is typical of concave mirrors when the object is closer to the mirror than its focal length, resulting in a virtual image.

Understanding image distance is crucial for:

• predicting where the image will appear relative to the mirror.
• determining whether the image is real or virtual.
• assessing the size and orientation of the image.

Object Distance

The object distance ( d_o ) is simply the distance from the object to the mirror. It plays a vital role in determining what kind of image a concave mirror will produce. In the dentist's scenario, the mirror is placed 8 mm away from the patient's teeth.

Object distance affects:

• the magnification of the image: As shown in our solution, a shorter object distance compared to the focal length results in a larger image.
• the nature of the image: If the object is within the focal length, the image is virtual and upright.
• position and orientation: It helps calculate where and how the image will appear, whether it will be real or inverted.

By using the relationship between object distance, focal length, and image distance, we can predict and manipulate the characteristics of the image formed on a concave mirror, which is useful in applications requiring detailed inspections, such as dentistry.