a slab of thickness 5cm and refractive index 1.5 is held a few cm in ront of concave mirror of radius of curvature 40cm. THe faces of slab being perpendicular to principal axis of mirror. how far from mirror must a smaall object be placed if its reflected image coincides with the object?

To find out how far from the concave mirror a small object must be placed so that its reflected image coincides with the object, we need to consider the effects of the glass slab on the light rays and how they interact with the mirror. Let's break this down step by step.

Understanding the Setup

We have a concave mirror with a radius of curvature (R) of 40 cm. The focal length (f) of the mirror can be calculated using the formula:

• f = R/2 = 40 cm / 2 = 20 cm

This means that the focal point of the mirror is located 20 cm in front of the mirror.

Effect of the Glass Slab

The glass slab has a thickness (t) of 5 cm and a refractive index (n) of 1.5. When light passes through the slab, it slows down and bends due to refraction. The effective distance that light travels through the slab can be calculated using the formula:

• Effective thickness = t / n = 5 cm / 1.5 ≈ 3.33 cm

This means that when light travels through the slab, it behaves as if it has traveled approximately 3.33 cm instead of 5 cm.

Calculating the Object Distance

Now, we need to determine the distance (d) from the mirror where the object should be placed. Since we want the reflected image to coincide with the object, we can use the mirror formula:

• 1/f = 1/v + 1/u

Here, f is the focal length, v is the image distance, and u is the object distance. Since the image coincides with the object, we can say that:

• v = -u

Substituting this into the mirror formula gives us:

• 1/f = 1/(-u) + 1/u

This simplifies to:

• 1/f = 0

This indicates that the object must be placed at the focal point of the mirror, but we must also account for the effective thickness of the slab. Therefore, we need to adjust the object distance by the effective thickness we calculated earlier.

Final Calculation

Since the focal length is 20 cm, and we have an effective thickness of 3.33 cm, the object distance (u) from the mirror should be:

• u = f + effective thickness = 20 cm + 3.33 cm = 23.33 cm

Thus, the small object must be placed approximately 23.33 cm in front of the mirror for its reflected image to coincide with the object itself.

Summary

In summary, the object should be positioned about 23.33 cm away from the concave mirror, taking into account the effects of the glass slab. This ensures that the light rays reflect back to the same point where the object is located, creating the desired condition of coinciding image and object.