91Ó°ÊÓ

Electrostatics is the study of electric charges at rest. It deals with the forces, fields, and potentials arising from stationary charges. A fundamental principle in electrostatics is Coulomb's law, which describes the force between two point charges. Another important concept is the electric field, which represents the force per unit charge exerted on a small positive test charge.
In the case of two parallel plate capacitors, like the ones given in the exercise, electrostatics helps us to understand how charges interact with each other and generate an electric field. The electric field between the plates is uniform, meaning its strength is the same at every point between the plates. This uniformity simplifies the calculation of other properties, such as the charge on the plates.
Understanding electrostatics is crucial for solving problems involving forces between charges or the behavior of charges in electric fields. It also provides the basis for studying more complex scenarios involving electric potential and energy.

The electric field strength, denoted by \(\vec{E}\), is a vector quantity that indicates the force per unit charge exerted by an electric field on a test charge. The strength of the electric field between two parallel plates is determined by the surface charge density and the separation distance between the plates.
In the provided exercise, the electric field strength is given as \(55 \, \mathrm{N}/\mathrm{C}\). This value allows us to calculate the charge on the plates using the relationship between the electric field and the surface charge density \( \sigma \). The surface charge density \( \sigma \) is defined as the charge per unit area and is given by:
\( \sigma = \epsilon_0 \vec{E} \)
Where \( \epsilon_0 \) is the permittivity of free space, approximately equal to \( 8.85 \times 10^{-12} \, \mathrm{C}^2/\mathrm{N} \cdot \mathrm{m}^2\).
Knowing the electric field strength helps us understand how much force a test charge would experience in the field, which thereby allows us to find the surface charge density and, consequently, the actual charge on the plates.

To find the charge on each plate, we use the relationship between the electric field strength, surface charge density, and permittivity of free space. The following steps can be taken to calculate the charge:
1. **Determine the surface charge density**: Using the given electric field strength, we calculate the surface charge density \( \sigma \) with the formula:
\( \sigma = \epsilon_0 \vec{E} \).
Plugging in the values, we get:
\( \sigma = (8.85 \times 10^{-12} \mathrm{C}^2/\mathrm{N} \cdot \mathrm{m}^2) \times (55 \mathrm{N}/\mathrm{C}) \approx 4.8675 \times 10^{-10} \mathrm{C}/\mathrm{m}^2 \).
2. **Find the total charge**: The total charge on each plate can be found by multiplying the surface charge density by the area of the plate
\( Q = \sigma \times A \).
Given the area \(A = 1.0 \mathrm{~m}^2\), we find
\( Q = (4.8675 \times 10^{-10} \mathrm{C}/\mathrm{m}^2) \times (1.0 \mathrm{~m}^2) = 4.8675 \times 10^{-10} \mathrm{C}\).
These calculated steps show the link between the electric field strength and the amount of charge on each plate, enabling us to solve the exercise efficiently.