Sure, let's solve the problem step by step.
### Given:
1. Separation distance between the plates: \( d = 8.08 \times 10^{-5} \, \text{m} \)
2. Charge on the plates: \( Q = 2.24 \times 10^{-9} \, \text{C} \)
3. Potential difference across the plates: \( V = 855 \, \text{V} \)
4. Permittivity of free space: \( \epsilon_0 = 8.854187817 \times 10^{-12} \, \text{F/m} \)
### Steps to solve for the area \( A \) of the plates:
1. Calculate the electric field (E) between the plates:
The electric field can be calculated from the potential difference and the separation distance using the formula:
[tex]\[
E = \frac{V}{d}
\][/tex]
where
[tex]\[
V = 855 \, \text{V} \quad \text{and} \quad d = 8.08 \times 10^{-5} \, \text{m}
\][/tex]
2. Calculate the electric field (E):
[tex]\[
E = \frac{855}{8.08 \times 10^{-5}} \, \text{V/m}
\][/tex]
3. Relate the charge (Q) to the electric field (E) and the area (A):
Using the formula for the capacitive relationship involving the electric field and plate area,
[tex]\[
Q = \epsilon_0 \cdot A \cdot E
\][/tex]
we can solve for \( A \):
[tex]\[
A = \frac{Q}{\epsilon_0 \cdot E}
\][/tex]
where
[tex]\[
Q = 2.24 \times 10^{-9} \, \text{C} \quad \text{and} \quad \epsilon_0 = 8.854187817 \times 10^{-12} \, \text{F/m}
\][/tex]
4. Using the electric field (E) from the previous step:
5. Plug the values into the equation for \( A \):
[tex]\[
A = \frac{2.24 \times 10^{-9}}{8.854187817 \times 10^{-12} \cdot \left(\frac{855}{8.08 \times 10^{-5}}\right)}
\][/tex]
6. Simplify the expression to find the area.
### Final Calculation Result:
The area of the plates is approximately
[tex]\[
2.3908070856726704 \times 10^{-5} \, \text{m}^2.
\][/tex]
Thus, the answer is:
[tex]\[
2.3908070856726704
\][/tex]
So the area of the plates is [tex]\( \boxed{2.3908070856726704} \cdot 10^{-5} \, \text{m}^2 \)[/tex].
