Certainly! Let's go through the solution step-by-step.
1. Convert the dimensions to meters:
- The electrode dimensions are given in centimeters. We need to convert these to meters since the SI unit for calculations is meters.
- Length: [tex]\(0.1156 \, \text{cm} = 0.1156 \times 10^{-2} \, \text{m} = 0.001156 \, \text{m}\)[/tex]
- Width: [tex]\(0.1156 \, \text{cm} = 0.1156 \times 10^{-2} \, \text{m} = 0.001156 \, \text{m}\)[/tex]
2. Convert the separation to meters:
- The separation between the electrodes is given in millimeters. We also need this in meters.
- Separation: [tex]\(0.5982 \, \text{mm} = 0.5982 \times 10^{-3} \, \text{m} = 0.0005982 \, \text{m}\)[/tex]
3. Calculate the area of the electrodes:
- The area [tex]\(A\)[/tex] of one electrode can be found using the formula:
[tex]\[
A = \text{length} \times \text{width}
\][/tex]
- Substituting the values:
[tex]\[
A = 0.001156 \, \text{m} \times 0.001156 \, \text{m} = 1.336336 \times 10^{-6} \, \text{m}^2
\][/tex]
4. Calculate the capacitance:
- To find the capacitance [tex]\(C\)[/tex] of the parallel plate capacitor, use the formula:
[tex]\[
C = \frac{\epsilon_0 \cdot A}{d}
\][/tex]
where [tex]\(\epsilon_0\)[/tex] is the permittivity of free space [tex]\((8.854 \times 10^{-12} \, \text{F/m})\)[/tex], [tex]\(A\)[/tex] is the area of the plates, and [tex]\(d\)[/tex] is the separation between the plates.
- Substituting the values:
[tex]\[
C = \frac{8.854 \times 10^{-12} \, \text{F/m} \times 1.336336 \times 10^{-6} \, \text{m}^2}{0.0005982 \, \text{m}} = 1.978 \times 10^{-14} \, \text{F}
\][/tex]
5. Calculate the charge:
- The charge [tex]\(Q\)[/tex] on the plates can be found using the relationship:
[tex]\[
Q = C \times V
\][/tex]
where [tex]\(V\)[/tex] is the voltage applied.
- Substituting the values:
[tex]\[
Q = 1.978 \times 10^{-14} \, \text{F} \times 73 \, \text{V} = 1.444 \times 10^{-12} \, \text{C}
\][/tex]
6. Convert the charge to nanoCoulombs:
- Finally, to convert the charge from Coulombs to nanoCoulombs:
[tex]\[
1 \, \text{C} = 1 \times 10^9 \, \text{nC}
\][/tex]
- Substituting the value:
[tex]\[
Q = 1.444 \times 10^{-12} \, \text{C} \times 10^9 = 1.444 \, \text{nC}
\][/tex]
So, the charge on each plate is approximately [tex]\(1.444 \, \text{nC}\)[/tex].
