To determine the area [tex]\( A \)[/tex] of the plates of a capacitor, we can use the formula for the capacitance of a parallel plate capacitor. The formula is:
[tex]\[ C = \varepsilon_0 \frac{A}{d} \][/tex]
Given values:
- Capacitance, [tex]\( C = 2.18 \times 10^{-8} \text{ F} \)[/tex]
- Distance between the plates, [tex]\( d = 3.89 \times 10^{-8} \text{ m} \)[/tex]
- Permittivity of free space, [tex]\( \varepsilon_0 = 8.85 \times 10^{-12} \text{ F/m} \)[/tex]
Rearrange the formula to solve for the area [tex]\( A \)[/tex]:
[tex]\[ A = \frac{C \cdot d}{\varepsilon_0} \][/tex]
Now, substitute the given values into the equation:
[tex]\[ A = \frac{2.18 \times 10^{-8} \text{ F} \times 3.89 \times 10^{-8} \text{ m}}{8.85 \times 10^{-12} \text{ F/m}} \][/tex]
Perform the multiplication in the numerator:
[tex]\[ 2.18 \times 10^{-8} \times 3.89 \times 10^{-8} = 8.4802 \times 10^{-16} \][/tex]
Now, divide by the denominator:
[tex]\[ A = \frac{8.4802 \times 10^{-16}}{8.85 \times 10^{-12}} \][/tex]
Perform the division:
[tex]\[ A \approx 9.58214689265537 \times 10^{-5} \text{ m}^2 \][/tex]
To express the area in the form [tex]\((\text{Coefficient} \times 10^{\text{Exponent}})\)[/tex]:
Given that the area [tex]\( A \approx 9.58214689265537 \times 10^{-5} \text{ m}^2 \)[/tex], we need to break this down:
[tex]\[ 9.58214689265537 \times 10^{-5} \text{ m}^2 \][/tex]
The coefficient of this scientific notation is approximately 0.958214689265537, and the exponent is -4 when adjusted to one significant digit before the decimal point.
Hence, the area of the plates is:
[tex]\[ A \approx 0.958 \times 10^{-4} \text{ m}^2 \][/tex]
Therefore, the area of the plates [tex]\( A \)[/tex] is approximately
[tex]\[ 0.958 \times 10^{-4} \text{ m}^2 \][/tex]
