When a charge of amount Q is given to an isolated metal plate X of surface area A, its surface charge density becomes `sigma_(1)`. When an isolated identical plate Y is brought close to X the surface charge density on X becomes `sigma_(2)`. When Y is earthed the surface charge density on X becomes `sigma_(3)`. Choose the incorrect option.

Options

A. `sigma_(1)=Q/A`
B. `sigma_(1)=Q/(2A)`
C. `sigma_(1)=sigma_(2)`
D. `sigma_(3)=Q/A`

Explanation

To solve the problem regarding the surface charge densities on two isolated metal plates X and Y, we will analyze each scenario step by step.

### Step 1: Initial Charge Distribution on Plate X
When a charge \( Q \) is given to an isolated metal plate X of surface area \( A \), the surface charge density \( \sigma_1 \) on plate X can be calculated using the formula:
\[ \sigma_1 = \frac{Q}{A} \]

### Step 2: Bringing Plate Y Close to Plate X
When an identical isolated plate Y is brought close to plate X, it influences the electric field around plate X. The presence of plate Y causes the surface charge density on plate X to change to \( \sigma_2 \). The new surface charge density \( \sigma_2 \) will depend on the interaction between the two plates, but we can say that:
\[ \sigma_2 < \sigma_1 \]
This is because the electric field due to plate Y will partially cancel the electric field due to plate X.

### Step 3: Earthing Plate Y
When plate Y is earthed, it allows charge to flow between the earth and plate Y. This process will further affect the surface charge density on plate X. The surface charge density on plate X becomes \( \sigma_3 \). Since earthing allows charge to neutralize the effect of plate Y, we can conclude that:
\[ \sigma_3 < \sigma_2 \]
This is because earthing plate Y will reduce the electric field produced by plate Y, leading to a decrease in the surface charge density on plate X.

### Step 4: Analyzing the Options
Now, we need to analyze the options provided in the question to identify the incorrect statement. The relationships we have established are:
- \( \sigma_1 > \sigma_2 > \sigma_3 \)

If any option contradicts this relationship, it will be the incorrect option.

### Conclusion
Based on the analysis, we can conclude that the incorrect option will be the one that states a relationship that contradicts \( \sigma_1 > \sigma_2 > \sigma_3 \).