The apparent thickness of a thick plano-convex lens is measured once with the plane face upward and then with the convex face upward. The value will be

Options

A. more in the firest case
B. same in the two cases
C. more in the second case
D. any of the above depending on the value of its actual thickness

Explanation

To solve the problem of finding the apparent thickness of a thick plano-convex lens when measured with the plane face upward and then with the convex face upward, we will follow these steps:

### Step 1: Define the Problem
We have a thick plano-convex lens, and we need to determine the apparent thickness when measured in two orientations:
1. Plane face upward
2. Convex face upward

### Step 2: Apparent Thickness with Plane Face Upward
When the plane face of the lens is facing upward, we can use the formula for apparent depth:
\[ \text{Apparent Thickness} = \frac{D}{\mu} \]
where \(D\) is the actual thickness of the lens and \(\mu\) is the refractive index of the lens material.

### Step 3: Apparent Thickness with Convex Face Upward
When the convex face is facing upward, we need to apply the formula for refraction at a curved surface. The formula for refraction at a curved surface is given by:
\[ \frac{\mu_2}{V} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R} \]
In this case:
- \(\mu_2 = 1\) (the refractive index of air)
- \(\mu_1 = \mu\) (the refractive index of the lens)
- \(u = -D\) (the object distance, taken as negative since it is on the same side as the incoming light)
- \(R = -D\) (the radius of curvature, taken as negative for the convex surface)

Substituting these values into the formula, we have:
\[ \frac{1}{V} - \frac{\mu}{-D} = \frac{1 - \mu}{-D} \]

### Step 4: Solve for \(V\)
Rearranging the equation gives:
\[ \frac{1}{V} + \frac{\mu}{D} = \frac{\mu - 1}{D} \]
\[ \frac{1}{V} = \frac{\mu - 1 - \mu}{D} = \frac{-1}{D} \]
Thus, we find:
\[ V = -D \]

### Step 5: Conclusion
From the calculations:
- Apparent thickness with plane face upward: \(\frac{D}{\mu}\)
- Apparent thickness with convex face upward: \(-D\)

Since \(-D\) indicates that the image appears at a distance \(D\) below the lens, we can conclude that the apparent thickness when the convex face is upward appears greater than when the plane face is upward.

### Final Answer
The apparent thickness of the lens is greater when measured with the convex face upward.

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