A capacitor consists of square conducting plates \(25 \mathrm{cm}\) on a side and \(5.0 \mathrm{mm}\) apart, carrying charges \(\pm 1.1 \mu \mathrm{C}\). Find (a) the electric field, (b) the potential difference between the plates, and (c) the stored energy.

The electric field within a capacitor is a crucial concept as it determines the force felt by charges within the capacitor's vicinity. To calculate the electric field, we use the formula \(E = \frac{Q}{A*\varepsilon}\), where \(Q\) is the charge on one of the plates, \(A\) is the plate's area, and \(\varepsilon\) represents the permittivity of free space.

In the case of the given problem, we have a charge of \(\pm 1.1 \mu C\) and square plates that are \(25 cm\) on a side. This calculation will provide us with a value in Newtons per Coulomb (N/C), which defines the intensity of the electric field at any point between the plates. This field is uniform and perpendicular to the plates when they are closely spaced and parallel, which is the common scenario in capacitors.

The potential difference, often called voltage, between two points is defined as the work done per unit charge to move a small charge between those two points. For the capacitor in question, the potential difference between the plates can be found with the formula \(V = E*d\) where \(E\) is the electric field calculated in the previous step and \(d\) is the separation distance between the plates.

In our example, once we've calculated the electric field, we multiply it by the distance between the plates, which is \(5.0 mm\) or \(5.0 \times 10^{-3} m\), to find the potential difference in volts. This value represents how much energy is required to move a charge from one plate to the other within the electric field.

Capacitors store energy in the form of an electric field created between the plates, and this stored energy can be quantified. The formula to calculate this energy is \(U = \frac{1}{2}*C*V^2\), where \(U\) is the energy in Joules, \(C\) is the capacitance, and \(V\) is the potential difference between the plates.

After determining the capacitance and the potential difference, these values must be inserted into the energy formula. The stored energy can then be visualized as the energy needed to build up the electric field between the plates from no charge to its current state or, alternatively, as the energy released when the capacitor discharges.

Capacitance is a measure of a capacitor’s ability to store charge for a given potential difference between its plates. The capacitance formula \(C = \frac{\varepsilon*A}{d}\) describes how geometry and material properties determine this ability.

Breaking it down, \(A\) represents the surface area of one of the plates, \(\varepsilon\) is the permittivity constant for the medium between plates (free space in this case), and \(d\) is the separation between the plates. Larger plate areas or smaller distances between plates lead to a higher capacitance, allowing more charge to be stored at a lower potential difference. Understanding this relationship is essential in selecting and designing capacitors for various electronic components and circuits.