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?

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Convert all given quantities to SI units. The side length of the square plates is 12 cm, which is 0.12 m. The separation between the plates is 0.10 mm, which is 0.0001 m. The dielectric constant is K = 3.8, and the pressure is 40.0 Pa.

Recall the relationship between the pressure exerted on the dielectric and the electric field in the capacitor: \( P = \frac{1}{2} \varepsilon_0 K E^2 \), where \( P \) is the pressure, \( \varepsilon_0 \) is the permittivity of free space (\( 8.85 \times 10^{-12} \ \text{F/m} \)), \( K \) is the dielectric constant, and \( E \) is the electric field.

Rearrange the formula to solve for the electric field \( E \): \( E = \sqrt{\frac{2P}{\varepsilon_0 K}} \). Substitute the known values for \( P \), \( \varepsilon_0 \), and \( K \) into this equation to calculate \( E \).

The relationship between the electric field \( E \) and the voltage \( V \) across the plates is given by \( E = \frac{V}{d} \), where \( d \) is the separation between the plates. Rearrange this formula to solve for \( V \): \( V = E \cdot d \). Substitute the value of \( E \) from the previous step and the given value of \( d \) to calculate \( V \).

Combine all the steps to express the battery voltage \( V \) in terms of the given quantities: \( V = d \cdot \sqrt{\frac{2P}{\varepsilon_0 K}} \). This formula can now be used to compute the battery voltage if needed.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 the capacitance by a factor equal to the dielectric constant (K).

Dielectric Constant

The dielectric constant (K) is a measure of a material's ability to store electrical energy in an electric field. It is a dimensionless number that indicates how much the electric field is reduced within the material compared to a vacuum. A higher dielectric constant means the material can store more charge, which affects the overall capacitance of the capacitor.

Pressure in Dielectrics

When a capacitor is charged, the electric field between the plates exerts forces on the dielectric material, leading to a pressure. This pressure can be calculated using the formula P = εE^2/2, where P is the pressure, ε is the permittivity of the dielectric, and E is the electric field strength. Understanding this relationship is crucial for determining the voltage across the capacitor when the pressure is known.