Question

Polystyrene has dielectric constant 2.6 and dielectric strength 2.0 $\times$ 10${}^7$ V/m. A piece of polystyrene is used as a dielectric in a parallel-plate capacitor, filling the volume between the plates. (a) When the electric field between the plates is 80% of the dielectric strength, what is the energy density of the stored energy? (b) When the capacitor is connected to a battery with voltage 500.0 V, the electric field between the plates is 80% of the dielectric strength. What is the area of each plate if the capacitor stores 0.200 mJ of energy under these conditions?

Short answer

(a) Energy density is approximately 2.95 J/m³. (b) The area of each plate is approximately 0.0032 m².

Step-by-step solution

Compute Electric Field

The electric field in the capacitor is given as 80% of the dielectric strength of polystyrene. Calculate this electric field, $E$, as $E=0.8\times 2.0\times {10}^7\text{V/m}=1.6\times {10}^7\text{V/m}$.

Energy Density Formula

The energy density $u$ of the stored energy in a dielectric material can be calculated using the formula: $$u=\frac{1}{2}\epsilon E^2$$ where $\epsilon ={\epsilon }_0\times \text{dielectric constant of polystyrene}$. ${\epsilon }_0$ is the permittivity of free space, $8.85\times {10}^{-12}\text{F/m}$. Substituting the values we have $\epsilon =8.85\times {10}^{-12}\times 2.6$.

Calculate Energy Density

Substituting $\epsilon$ and $E$ into the energy density formula: $$u=\frac{1}{2}\times (8.85\times {10}^{-12}\times 2.6)\times (1.6\times {10}^7{)}^2$$. Compute $u$ to find the energy density.

Relate Energy to Capacitor Specifications

We know the formula for energy stored in a capacitor: $$U=\frac{1}{2}C{V}^2=0.200\text{mJ}=0.200\times {10}^{-3}\text{J}$$. Since $V=500\text{V}$, we substitute $U$ and $V$ to find $C$, the capacitance.

Capacitance Calculation

Rearrange the energy formula to solve for $C$: $$C=\frac{U}{\frac{1}{2}{V}^2}=\frac{0.200\times {10}^{-3}}{0.5\times (500{)}^2}$$. Calculate $C$.

Relate Capacitance to Physical Dimensions

Capacitance for a parallel-plate capacitor is given by $C=\frac{\epsilon A}{d}$ where $d$ is the plate separation (can calculate from $V=Ed$), and $A$ is the area of the plates. Substitute $C$ from step 5 and solve for $A$.

Calculate Plate Area

Using $d=\frac{500}{1.6\times {10}^7}$ and $C=0.0000016F$, substitute these into the equation for capacitance $C=\frac{(8.85\times {10}^{-12}\times 2.6)A}{d}$. Solve to find $A$, the area of each plate.

Key concepts

Dielectric Constant

The dielectric constant, also known as relative permittivity, is a measure of how much a dielectric material can increase the capacitance of a capacitor compared to when there is a vacuum between the plates. For example, the dielectric constant of polystyrene is 2.6. This means that when polystyrene is used as the dielectric material in a capacitor, the capacitance becomes 2.6 times higher than it would be without any dielectric material.

Dielectric materials are insulators, and they do not conduct electricity but can support electrostatic fields. This ability enhances the capacitor's potential to store charge, an essential aspect of various electrical applications.

• The dielectric constant is a dimensionless number and has no units.
• It varies depending on the material type and influences the energy-storing capacity of a capacitor.

Understanding the dielectric constant of different materials is important when designing capacitors for particular applications, as it influences both the effectiveness and efficiency of the capacitor.

Dielectric Strength

Dielectric strength is a critical property of dielectric materials, indicating the maximum electric field the material can handle before breaking down and conducting electricity. It is measured in volts per meter (V/m). In the case of polystyrene, the dielectric strength is 2.0 × 10⁷ V/m.

When a dielectric material reaches or exceeds its dielectric strength, it can no longer function as an insulator and may allow the enclosed charges to move freely across the material, causing a short circuit or damage to the capacitor.

• Dielectric strength ensures that under normal operating conditions, the material prevents breakdown and maintains its insulating properties.
• It is vital to select a material with a sufficient dielectric strength for applications with high voltages to prevent failure.

When designing or using capacitors, always ensure that the operational electric field stays within a safe margin below the dielectric strength to ensure longevity and reliability.

Energy Density

Energy density is the amount of energy stored per unit volume within a capacitor. It is an important measure indicating how effectively a capacitor can store energy in the field it creates. In formula terms, the energy density $u$ is given by $$u=\frac{1}{2}\epsilon E^2$$where $\epsilon$ is the permittivity of the dielectric material, and $E$ is the electric field strength.

For polystyrene with a dielectric constant of 2.6 and an electric field of 1.6 × 10⁷ V/m, we substitute these values to calculate the energy density. This expresses how the internal structure of a capacitor is capable of storing energy given the specific properties of the material used.

• Higher permittivity and dielectric constants allow for greater energy density.
• Energy density is a critical factor in applications where size and efficiency are constraints.

Understanding energy density helps in designing capacitors that are both compact and efficient, catering to advanced technology solutions.

Parallel-Plate Capacitor

The parallel-plate capacitor is one of the most simple and commonly used capacitor designs. It consists of two conductive plates separated by a dielectric material. The key feature of this capacitor is that the capacitance $C$ is directly related to the area of the plates $A$, the separation distance $d$, and the dielectric constant ${\epsilon }_r$ of the material, with the formula:
$$C=\frac{{\epsilon }_0{\epsilon }_{r}A}{d}$$
where ${\epsilon }_0$ is the permittivity of free space.

The parallel-plate capacitor serves a wide array of applications due to its straightforward design and reliable functionality. Typically, it is used whenever a precise and stable capacitance value is needed, as the physical parameters influencing the capacitance can be accurately controlled.

• The area of the plates and the nature of the dielectric material directly affect the overall capacitance.
• The simplicity of the design makes it easy to manufacture and integrate into various circuits.

Designing with parallel-plate capacitors requires understanding how changes to the physical structure or materials impact the overall behaviour and efficiency of the capacitor.