The human eye can be regarded as a single spherical refractive surface of curvature of cornea 7.8 mm. If a parallel beam of light comes to focus at 3.075 cm behind the refractive surface, the refractive index of the eye is

Options

A. 1.34
B. 1.72
C. 1.5
D. 1.61

Explanation

To find the refractive index of the human eye, we can use the lens maker's formula, which relates the refractive index of a lens (or spherical refractive surface) to its focal length and radius of curvature. The formula is given by:

\[ \frac{\mu_2 - \mu_1}{R} = \frac{1}{f} \]

Where:
- \(\mu_1\) is the refractive index of air (approximately 1.0),
- \(\mu_2\) is the refractive index of the eye,
- \(R\) is the radius of curvature,
- \(f\) is the focal length.

### Step 1: Identify the given values
- Curvature of the cornea (radius of curvature, \(R\)) = 7.8 mm = 0.78 cm (conversion from mm to cm)
- Focal length (\(f\)) = 3.075 cm
- Refractive index of air (\(\mu_1\)) = 1.0

### Step 2: Rearrange the lens maker's formula to solve for \(\mu_2\)
We can rearrange the formula to find \(\mu_2\):

\[ \mu_2 = \mu_1 + \frac{R}{f}(\mu_2 - \mu_1) \]

### Step 3: Substitute the known values into the equation
Substituting the known values into the rearranged formula:

\[ \mu_2 = 1 + \frac{0.78}{3.075}(\mu_2 - 1) \]

### Step 4: Solve for \(\mu_2\)
Now, we need to solve for \(\mu_2\). First, let's simplify the equation:

\[ \mu_2 = 1 + \frac{0.78}{3.075} \mu_2 - \frac{0.78}{3.075} \]

Now, combine like terms:

\[ \mu_2 - \frac{0.78}{3.075} \mu_2 = 1 - \frac{0.78}{3.075} \]

Factoring out \(\mu_2\):

\[ \mu_2 \left(1 - \frac{0.78}{3.075}\right) = 1 - \frac{0.78}{3.075} \]

Now, calculate the numerical values:

\[ 1 - \frac{0.78}{3.075} = 1 - 0.254 = 0.746 \]

Now, we can express \(\mu_2\):

\[ \mu_2 \left(1 - 0.254\right) = 0.746 \]

Calculating the left side:

\[ \mu_2 (0.746) = 0.746 \]

Finally, solving for \(\mu_2\):

\[ \mu_2 = \frac{0.746}{0.746} = 1.0 \]

### Step 5: Conclusion
Thus, the refractive index of the human eye is approximately:

\[ \mu_2 \approx 1.0 + 0.254 \approx 1.254 \]

### Final Answer
The refractive index of the eye is approximately **1.254**.