Question

Capacitors tend to be more expensive as their capacitance and maximum voltage \(V_{\max }\) increase. The voltage \(V_{\max }\) is limited by the field strength at which the dielectric breaks down, \(E_{B D}\). Which of these dielectrics will give the largest \(C V_{\max }\) product for equal plate areas? \((a)\) Air: \(\epsilon_{r}=1\), \(E_{B D}=3 \mathrm{MV} / \mathrm{m} .(b)\) Barium titanate: \(\epsilon_{r}=1200, E_{B D}=3 \mathrm{MV} / \mathrm{m} .(c)\) Silicon dioxide: \(\epsilon_{r}=3.78, E_{B D}=16 \mathrm{MV} / \mathrm{m} .(d)\) Polyethylene: \(\epsilon_{r}=2.26, E_{B D}=\) \(4.7 \mathrm{MV} / \mathrm{m} .\)

Short answer

The dielectric that will give the largest CVmax product for equal plate areas is Barium titanate with a relative permittivity of 1200 and a breakdown field strength of 3 MV/m.

Step-by-step solution

Calculate \(\epsilon_r E_{BD}\) product for each dielectric

For each dielectric, we will calculate the \(\epsilon_r E_{BD}\) product: 1. Air: \(\epsilon_r E_{BD} = (1)(3 \times 10^6 \,\text{V/m}) = 3 \times 10^6 \,\text{V/m}\) 2. Barium titanate: \(\epsilon_r E_{BD} = (1200)(3 \times 10^6 \,\text{V/m}) = 3.6 \times 10^9 \,\text{V/m}\) 3. Silicon dioxide: \(\epsilon_r E_{BD} = (3.78)(16 \times 10^6 \,\text{V/m}) = 6.048 \times 10^7 \,\text{V/m}\) 4. Polyethylene: \(\epsilon_r E_{BD} = (2.26)(4.7 \times 10^6 \,\text{V/m}) = 1.0622 \times 10^7 \,\text{V/m}\)

Identify the largest \(\epsilon_r E_{BD}\) product

Comparing the \(\epsilon_r E_{BD}\) products calculated in step 1, we can see that the largest product is for Barium titanate with a product of \(3.6 \times 10^9 \,\text{V/m}\).

Conclusion

The dielectric that will give the largest \(C V_{\max }\) product for equal plate areas is Barium titanate with a relative permittivity of 1200 and a breakdown field strength of 3 MV/m.

Key concepts

Capacitance

Capacitance is a fundamental electrical parameter that measures a capacitor's ability to store electric charge relative to the applied voltage. It is denoted by the symbol C and the unit of capacitance is the farad (F). A capacitor's capacitance is determined by the surface area of its plates, the distance between the plates, and the dielectric material used between the plates. The equation for capacitance is often given as:
\[ C = \frac{\epsilon_r \epsilon_0 A}{d} \],
where A is the plate area, d is the separation between the plates, \(\epsilon_0\) is the vacuum permittivity constant, and \(\epsilon_r\) is the relative permittivity of the dielectric. In the context of dielectric breakdown, the capacitance value is an important consideration for designing capacitors that can withstand high voltages without failing.

Maximum Voltage

Maximum voltage, often symbolized as \(V_{\max}\), is the highest voltage a capacitor can handle before it undergoes dielectric breakdown, leading to a permanent loss of its insulating properties. This value is crucial for ensuring the reliability and safety of electronic devices. When the applied voltage exceeds \(V_{\max}\), the electric field strength may reach a point where it forces electrons across the insulating material, causing the dielectric to break down. In electrical engineering, understanding and preventing dielectric breakdown is integral to designing capacitors that are robust enough for their intended applications, without incurring unnecessary costs by overdesigning for excessively high voltage tolerances.

Field Strength

The electric field strength, E, within a capacitor influences how electrons are distributed between the plates and is defined as the force per unit charge experienced by a stationary point charge. For practical purposes, dielectric breakdown occurs when the electric field strength reaches a critical value, known as the breakdown field strength, \(E_{BD}\), which is different for each dielectric material. The equation for the electric field E in a capacitor is:
\[ E = \frac{V}{d} \],
where V is the voltage applied across the capacitor, and d is the distance between the plates. This concept is invaluable for predicting the performance limits of capacitors under high-voltage conditions and is a fundamental aspect of electrical engineering education.

Relative Permittivity

Relative permittivity, denoted as \(\epsilon_r\), is a dimensionless quantity that represents the ratio of the permittivity of a material to the permittivity of a vacuum, signifying how much the dielectric material can reduce the electric field within the capacitor. Materials with high relative permittivity, also called dielectric constant, can store more charge for the same voltage across the capacitor. This is why \(\epsilon_r\) is a critical factor in determining the efficiency of capacitors, especially when seeking a large \(CV_{\max}\) product. The product of \(\epsilon_r\) and \(E_{BD}\) can serve as a good comparative measure of how different dielectric materials perform under the constraint of dielectric breakdown when designing capacitors.

Electrical Engineering Education

Developing a strong foundation in the principles of electrical engineering is essential for students to excel in the field. Understanding concepts like capacitance, maximum voltage, field strength, relative permittivity, and dielectric breakdown forms the core knowledge necessary for designing and analyzing electronic components. Electrical engineering education blends theoretical knowledge with practical implementation, helping students navigate complex design challenges and innovate solutions. In the case of capacitors, grasping how these concepts interplay ensures that students can design capacitors with optimal size, cost, and reliability parameters. Effective education in this realm emphasizes case studies and problem-solving exercises, such as choosing the best dielectric material for a capacitor based on its \(CV_{\max}\) product, to equip future engineers with the critical thinking and technical skills they need.