Two parallel plates have an area of [tex]0.188 \, m^2[/tex]. How much charge must you place on them to create a [tex]37000 \, N/C[/tex] field between the plates?

[tex][?] \cdot 10^? \, C[/tex]



Answer :

To determine the charge required to create a specified electric field between two parallel plates, we can use the relationship between the electric field (E), the surface charge density (σ), and the permittivity of free space (ε₀).

First, recall the formula that relates the electric field to the surface charge density:

[tex]\[ E = \frac{\sigma}{\epsilon_0} \][/tex]

where:
- [tex]\( E \)[/tex] is the electric field.
- [tex]\( \sigma \)[/tex] is the surface charge density.
- [tex]\( \epsilon_0 \)[/tex] is the permittivity of free space, which has a value of approximately [tex]\( 8.854 \times 10^{-12} \, \text{F/m} \)[/tex].

The surface charge density (σ) is related to the charge (Q) and the area (A) of the plates by the formula:

[tex]\[ \sigma = \frac{Q}{A} \][/tex]

Combine the two formulas to express charge (Q) in terms of the given quantities:

[tex]\[ E = \frac{Q}{A \cdot \epsilon_0} \][/tex]

Rearrange this equation to solve for Q:

[tex]\[ Q = E \cdot A \cdot \epsilon_0 \][/tex]

Given the values:
- [tex]\( A = 0.188 \, \text{m}^2 \)[/tex]
- [tex]\( E = 37000 \, \text{N/C} \)[/tex]
- [tex]\( \epsilon_0 = 8.854 \times 10^{-12} \, \text{F/m} \)[/tex]

Substitute these values into the formula:

[tex]\[ Q = 37000 \, \text{N/C} \times 0.188 \, \text{m}^2 \times 8.854 \times 10^{-12} \, \text{F/m} \][/tex]

After performing the calculation, the charge [tex]\( Q \)[/tex] is found to be approximately:

[tex]\[ Q \approx 6.16 \times 10^{-8} \, \text{C} \][/tex]

Therefore, the charge required to create the given electric field is:

[tex]\[ Q = 6.16 \times 10^{-8} \, \text{C} \][/tex]

To express this in the format given:

[tex]\[ 6.1589730455052 \times 10^{-8} \, \text{C} \][/tex]

or approximately:

[tex]\[ [615.90] \times 10^{-10} \, \text{C} \][/tex]

So the final answer in the format [tex]\([?] \times 10^? \, \text{C}\)[/tex] is approximately:

[tex]\[ 6.16 \times 10^{-8} \, \text{C} \][/tex]
or in the provided required format:

[tex]\[ 615.90 \times 10^{-10} \, \text{C} \][/tex]