Let's solve this problem step-by-step to determine the area of the capacitor plates using the given values and the formula provided:
1. Given Information:
- Capacitance, [tex]\( C = 2.18 \cdot 10^{-8} \, \text{F} \)[/tex]
- Separation distance between the plates, [tex]\( d = 3.89 \cdot 10^{-8} \, \text{m} \)[/tex]
- Vacuum permittivity, [tex]\( \varepsilon_0 = 8.85 \cdot 10^{-12} \, \text{F/m} \)[/tex]
2. Formula to Use:
The relationship between capacitance [tex]\( C \)[/tex], plate area [tex]\( A \)[/tex], plate separation distance [tex]\( d \)[/tex], and vacuum permittivity [tex]\( \varepsilon_0 \)[/tex] is given by:
[tex]\[
C = \frac{\varepsilon_0 \cdot A}{d}
\][/tex]
3. Rearrange the Formula to Solve for [tex]\( A \)[/tex]:
To find the area [tex]\( A \)[/tex], we rearrange the equation:
[tex]\[
A = \frac{C \cdot d}{\varepsilon_0}
\][/tex]
4. Substitute the Given Values:
[tex]\[
A = \frac{(2.18 \cdot 10^{-8} \, \text{F}) \cdot (3.89 \cdot 10^{-8} \, \text{m})}{8.85 \cdot 10^{-12} \, \text{F/m}}
\][/tex]
5. Calculate the Area [tex]\( A \)[/tex]:
Performing the multiplication and division step-by-step:
[tex]\[
A = \frac{(2.18 \cdot 3.89) \cdot 10^{-8} \cdot 10^{-8}}{8.85 \cdot 10^{-12}}
\][/tex]
Simplify the exponents:
[tex]\[
A = \frac{8.4802 \cdot 10^{-16}}{8.85 \cdot 10^{-12}}
\][/tex]
Dividing the coefficients and exponents:
[tex]\[
A = 0.958214689265537 \cdot 10^{-4}
\][/tex]
Converting to a standard scientific notation:
[tex]\[
A = 9.58214689265537 \cdot 10^{-5} \, \text{m}^2
\][/tex]
6. Identify the Coefficient and Exponent:
The coefficient is [tex]\( 9.58214689265537 \)[/tex]
The exponent is [tex]\( -5 \)[/tex]
Therefore, the area [tex]\( A \)[/tex] of the capacitor plates, in scientific notation, is:
[tex]\[
A \approx 9.582 \cdot 10^{-5} \, \text{m}^2
\][/tex]
So, the coefficient is [tex]\(\boxed{9.582}\)[/tex] and the exponent is [tex]\(\boxed{-5}\)[/tex].
