A point object is placed `33cm` from a convex mirror of curvature radius `=40cm` A glass plate of thickness `6cm` and refractive index `2.0` placed between the object and mirror,close to the mirror,find the distance of final image form the object?

Explanation

To solve the problem step by step, we will follow the outlined procedure:

### Step 1: Determine the Shift Due to the Glass Plate
The shift caused by the glass plate can be calculated using the formula:
\[ \text{Shift} = \text{Thickness} \times \left(1 - \frac{1}{\text{Refractive Index}}\right) \]

Given:
- Thickness of the glass plate = \(6 \, \text{cm}\)
- Refractive index of the glass plate = \(2.0\)

Calculating the shift:
\[ \text{Shift} = 6 \times \left(1 - \frac{1}{2}\right) = 6 \times \left(1 - 0.5\right) = 6 \times 0.5 = 3 \, \text{cm} \]

### Step 2: Adjust the Object Distance
The object distance \(U\) is given as \(33 \, \text{cm}\). Since the shift is in the direction of light travel, we subtract the shift from the object distance:
\[ U = - (33 - 3) = -30 \, \text{cm} \]

### Step 3: Calculate the Focal Length of the Convex Mirror
The focal length \(F\) of a convex mirror is given by:
\[ F = \frac{R}{2} \]
where \(R\) is the radius of curvature.

Given:
- Radius of curvature \(R = 40 \, \text{cm}\)

Calculating the focal length:
\[ F = \frac{40}{2} = 20 \, \text{cm} \]

### Step 4: Use the Mirror Formula to Find the Image Distance
The mirror formula is given by:
\[ \frac{1}{F} = \frac{1}{V} + \frac{1}{U} \]
Rearranging gives:
\[ \frac{1}{V} = \frac{1}{F} - \frac{1}{U} \]

Substituting the values:
\[ \frac{1}{V} = \frac{1}{20} - \frac{1}{-30} \]
Calculating the right side:
\[ \frac{1}{V} = \frac{1}{20} + \frac{1}{30} \]
Finding a common denominator (60):
\[ \frac{1}{V} = \frac{3}{60} + \frac{2}{60} = \frac{5}{60} = \frac{1}{12} \]
Thus,
\[ V = 12 \, \text{cm} \]

### Step 5: Adjust the Image Distance for the Shift
Since the shift is in the direction of light travel, we need to adjust the image distance:
\[ V' = V - \text{Shift} = 12 - 3 = 9 \, \text{cm} \]

### Step 6: Calculate the Distance Between the Object and the Image
The distance between the object and the image is given by:
\[ \text{Distance} = |U| + |V'| = 33 + 9 = 42 \, \text{cm} \]

### Final Answer
The distance of the final image from the object is \(42 \, \text{cm}\).

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