One hard and stormy night you find yourself lost in the forest when you come upon a small hut. Entering it you see a crooked old woman in the corner hunched over a crystal ball. You are about to make a hasty exit when you hear the howl of wolves outside. Taking another look at the gyspy you decide to take your chances with the wolves, but the door is jammed shut. Resigned to a bad situation you approach her slowly, wondering just what is the focal length of that nifty crystal ball.
If the crystal ball is 20 cm in diameter with R.I = 1.5 , the gypsy lady is 1.2m from the ball , where is the image of the gypsy in focus as you walk towards her?

Options

A. 6.9 cm from the crystal ball
B. 7.9 cm from the crystal ball
C. 8.9 cm from the crystal ball
D. None of these

Explanation

To solve the problem, we need to find the focal length of the crystal ball and then determine the position of the image of the gypsy lady based on her distance from the crystal ball.

### Step 1: Calculate the Focal Length of the Crystal Ball

1. **Identify the parameters**:
- Diameter of the crystal ball = 20 cm
- Radius of the crystal ball (R) = Diameter / 2 = 20 cm / 2 = 10 cm
- Refractive Index (μ) = 1.5

2. **Use the Lensmaker's Formula**:
The formula is given by:
\[ \frac{1}{f} = (\mu - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
Here, we consider:
- \( R_1 = +10 \) cm (the first surface, light travels towards it)
- \( R_2 = -10 \) cm (the second surface, light travels away from it)

3. **Substituting the values into the formula**:
\[ \frac{1}{f} = (1.5 - 1) \left( \frac{1}{10} - \left(-\frac{1}{10}\right) \right) \]
\[ \frac{1}{f} = 0.5 \left( \frac{1}{10} + \frac{1}{10} \right) = 0.5 \left( \frac{2}{10} \right) = 0.5 \times 0.2 = 0.1 \]
Therefore,
\[ f = \frac{1}{0.1} = 10 \text{ cm} \]

### Step 2: Calculate the Image Distance (V)

1. **Identify the object distance (U)**:
- Distance of the gypsy lady from the crystal ball = 1.2 m = 120 cm
- Since the object is on the same side as the incoming light, we take U as negative:
\[ U = -120 \text{ cm} \]

2. **Use the lens formula**:
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
Rearranging gives:
\[ \frac{1}{v} = \frac{1}{f} + \frac{1}{u} \]

3. **Substituting the known values**:
\[ \frac{1}{v} = \frac{1}{10} + \frac{1}{-120} \]
Finding a common denominator (120):
\[ \frac{1}{v} = \frac{12}{120} - \frac{1}{120} = \frac{11}{120} \]
Therefore,
\[ v = \frac{120}{11} \approx 10.91 \text{ cm} \]

### Step 3: Interpret the Result

The positive value of \( v \) indicates that the image is formed on the opposite side of the crystal ball from the gypsy lady. Thus, the image of the gypsy will be approximately 10.91 cm from the crystal ball towards the observer.

### Final Answer
The image of the gypsy will be in focus approximately **10.9 cm from the crystal ball towards the observer**.

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