You are asked to construct a capacitor having a capacitance near 1 nF and a breakdown potential

step 1 In this question we are asked to find out the capacitance of a cylindrical capacitor and they ha

You are asked to construct a capacitor having a capacitance...

You are asked to construct a capacitor having a capacitance near $1 \mathrm{nF}$ and a breakdown potential in excess of $10000 \mathrm{~V}$. You think of using the sides of a tall Pyrex drinking glass as a dielectric, lining the inside and outside curved surfaces with aluminum foil to act as the plates. The glass is $10 \mathrm{~cm}$ tall with an inner radius of $3.6 \mathrm{~cm}$ and an outer radius of $3.8 \mathrm{~cm}$. What are the (a) capacitance and (b) breakdown potential of this capacitor?

Key Concepts

Cylindrical Capacitor Geometry

This concept involves understanding the geometry of a capacitor consisting of two coaxial cylindrical conductors separated by a dielectric material. The capacitance of such a setup is determined by the radii of the inner and outer cylinders and the height (or length) of the capacitor. The cylindrical configuration leads to a logarithmic dependence on the ratio of the outer to inner radii, which is a crucial factor in the design and analysis of such capacitors.

Capacitance Calculation Using Dielectric Properties

Capacitance in any capacitor depends not only on its geometry but also on the dielectric material inserted between the conductors. The dielectric constant (or relative permittivity) of the material, along with the vacuum permittivity, factors into the capacitance calculation. In the case of a cylindrical capacitor, the formula incorporates both the physical dimensions and the dielectric's ability to reduce the effective electric field, thereby increasing the capacitance.

Dielectric Breakdown Voltage

This key concept refers to the maximum voltage that a dielectric material can withstand before its insulating properties fail and it becomes conductive. The breakdown voltage is directly related to the dielectric strength of the material, which is typically given as a maximum electric field (volts per meter) that the material can support. In capacitor design, ensuring that the operating voltage is well below this threshold is essential for reliable performance and to prevent catastrophic failure.

Material Dielectric Strength and its Role in Capacitor Design

The dielectric strength is an intrinsic property of the material used in the capacitor. It defines the maximum electric field the material can handle without undergoing breakdown. When designing a capacitor, especially with unconventional materials like a specific type of glass, understanding and utilizing the material’s dielectric strength ensures that the capacitor’s breakdown potential meets the required specifications. This understanding also allows the designer to determine the maximum voltage the capacitor can safely handle based on the thickness of the dielectric.

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