When a lens of refractive index `n_(1)`, then the lens looks to be dissapeared only, if

Options

A. `n_(1)=n_(2)//2`
B. `n_(1)=3n_(2)//2`
C. `n_(1)=n_(2)`
D. `n_(1)=5n_(2)//2`

Explanation

To determine when a lens of refractive index \( n_1 \) appears to disappear, we can analyze the situation using the lens maker's formula and the concept of optical power.

### Step-by-Step Solution:

1. **Understanding the Concept of Disappearance**:
- A lens appears to disappear when it does not refract light, which means its optical power is zero.

2. **Using the Lens Maker's Formula**:
- The lens maker's formula relates the focal length \( f \) of a lens to its refractive indices and radii of curvature:
\[ \frac{1}{f} = (n_1 - n_2) \left( \frac{1}{r_1} - \frac{1}{r_2} \right) \]
- Here, \( n_1 \) is the refractive index of the lens, and \( n_2 \) is the refractive index of the surrounding medium.

3. **Setting the Power to Zero**:
- For the lens to appear to disappear, its power must be zero, which implies:
\[ f \to \infty \quad \text{(or equivalently, } \frac{1}{f} = 0\text{)} \]
- Therefore, we set the right-hand side of the lens maker's formula to zero:
\[ (n_1 - n_2) \left( \frac{1}{r_1} - \frac{1}{r_2} \right) = 0 \]

4. **Analyzing the Equation**:
- The equation can be satisfied if either:
- \( n_1 - n_2 = 0 \) (which means \( n_1 = n_2 \))
- or \( \frac{1}{r_1} - \frac{1}{r_2} = 0 \) (which means \( r_1 = r_2 \), indicating a plano-convex or plano-concave lens).

5. **Conclusion**:
- Since we are interested in the case where the lens disappears due to the refractive index, we focus on the first condition:
\[ n_1 = n_2 \]
- Therefore, the lens will appear to disappear when the refractive index of the lens \( n_1 \) is equal to the refractive index of the surrounding medium \( n_2 \).

6. **Final Answer**:
- The correct option is:
\[ \text{Option 3: } n_1 = n_2 \]