A parallel-plate capacitor has square plates 12 cm on a side separated by 0.10 mm of plastic with a dielectric constant of K = 3.8. The plates are connected to a battery, causing them to become oppositely charged. Since the oppositely charged plates attract each other, they exert a pressure on the dielectric. If this pressure is 40.0 Pa, what is the battery voltage?

Verified step by step guidance

1. Calculate the area of the capacitor plates using the formula for the area of a square, \( A = s^2 \), where \( s \) is the side length of the square plates.
2. Use the formula for the capacitance of a parallel-plate capacitor with a dielectric, \( C = \kappa \epsilon_0 \frac{A}{d} \), where \( \kappa \) is the dielectric constant, \( \epsilon_0 \) is the permittivity of free space (approximately \( 8.85 \times 10^{-12} \) F/m), \( A \) is the area of the plates, and \( d \) is the separation between the plates.
3. Calculate the electric field \( E \) across the dielectric using the relation between the pressure \( P \), dielectric constant \( \kappa \), and electric field, given by \( P = \frac{\kappa \epsilon_0 E^2}{2} \). Solve for \( E \) by rearranging the formula to \( E = \sqrt{\frac{2P}{\kappa \epsilon_0}} \).
4. Determine the voltage across the capacitor using the relationship between electric field and voltage, \( V = Ed \), where \( E \) is the electric field and \( d \) is the separation between the plates.
5. Substitute the values obtained from the previous steps into the voltage formula to find the battery voltage.

Key Concepts

Capacitance

Capacitance is the ability of a capacitor to store charge per unit voltage. It is defined by the formula C = εA/d, where C is capacitance, ε is the permittivity of the dielectric material, A is the area of the plates, and d is the separation between them. The presence of a dielectric material increases capacitance by a factor equal to the dielectric constant (K), which enhances the capacitor's ability to store electrical energy.

Dielectric Constant

The dielectric constant (K) is a measure of a material's ability to reduce the electric field within it compared to a vacuum. It affects the capacitance of a capacitor; higher values of K indicate better insulating properties and greater charge storage capacity. In this scenario, the dielectric constant of the plastic used between the plates is 3.8, which influences the overall capacitance and the voltage across the capacitor.

Pressure in Dielectrics

When a capacitor is charged, the electric field between the plates exerts a force on the dielectric material, leading to a pressure that can be calculated using the formula P = εE²/2, where P is the pressure, ε is the permittivity, and E is the electric field strength. This pressure is crucial for understanding the mechanical effects of the electric field on the dielectric, and it can be related to the voltage across the capacitor through the electric field and the geometry of the capacitor.