To determine the charge required to create a 1000 N/C electric field between two parallel plates with an area of [tex]\(0.00100 \, \text{m}^2\)[/tex], we can follow these steps:
1. Identify Key Constants and Variables:
- The permittivity of free space, [tex]\( \epsilon_0 \)[/tex], is [tex]\( 8.854187817 \times 10^{-12} \, \text{F/m} \)[/tex].
- The area of the plates, [tex]\( A \)[/tex], is [tex]\( 0.00100 \, \text{m}^2 \)[/tex].
- The electric field, [tex]\( E \)[/tex], is [tex]\( 1000 \, \text{N/C} \)[/tex].
2. Relate the Electric Field to Surface Charge Density:
The electric field [tex]\( E \)[/tex] between parallel plates is related to the surface charge density [tex]\( \sigma \)[/tex] by:
[tex]\[
E = \frac{\sigma}{\epsilon_0}
\][/tex]
where [tex]\( \sigma = \frac{Q}{A} \)[/tex] is the surface charge density and [tex]\( Q \)[/tex] is the charge on the plates.
3. Rearrange the Formula to Solve for Charge [tex]\( Q \)[/tex]:
[tex]\[
\sigma = E \cdot \epsilon_0
\][/tex]
[tex]\[
Q = \sigma \cdot A
\][/tex]
Substituting [tex]\( \sigma \)[/tex] into the equation gives:
[tex]\[
Q = (E \cdot \epsilon_0) \cdot A
\][/tex]
4. Substitute the Values:
[tex]\[
Q = (1000 \, \text{N/C} \times 8.854187817 \times 10^{-12} \, \text{F/m}) \times 0.00100 \, \text{m}^2
\][/tex]
5. Perform the Calculation:
[tex]\[
Q = 8.854187817 \times 10^{-12} \times 1000 \times 0.00100
\][/tex]
[tex]\[
Q = 8.854187817 \times 10^{-15} \, \text{C}
\][/tex]
6. Express the Result in the Desired Format:
The result is [tex]\( 8.854187817 \times 10^{-15} \, \text{C} \)[/tex].
Thus, the required charge [tex]\( Q \)[/tex] to create a 1000 N/C electric field between the parallel plates with an area of [tex]\( 0.00100 \, \text{m}^2 \)[/tex] is:
[tex]\[
\boxed{8.854187817} \cdot 10^{\boxed{-15}} \, \text{C}
\][/tex]
