A cylindrical capacitor has an inner conductor of variable...

A cylindrical capacitor has an inner conductor of variable radius $a>0$ and an outer conductor of fixed radius $b$. If the breakdown electric field is given by $E_b$ (a fixed value), determine the relation between $a$ and $b$ such that the capacitor is able to store the most energy. What if we wish to maximize the potential difference between the two conductors?

Key Concepts

Potential Difference Maximization

Maximizing the potential difference between the conductors, under the same breakdown field limit, follows a similar optimization principle. By setting up the relation between the voltage and the geometric parameters, and then finding the derivative with respect to the variable element (here the inner radius), one can determine the specific geometry that allows the largest voltage drop before breakdown occurs. This concept emphasizes the trade-offs between geometry, breakdown constraints, and the achievable electrical performance.

Optimization under Electrical Constraints

This concept covers the method of optimizing device performance (such as energy storage) subject to constraints like the maximum allowed electric field. By expressing energy (or potential difference) in terms of the capacitor's geometry and then applying calculus to find the extrema, one determines the optimal relation between design parameters. This process is fundamental in electromagnetic design and illustrates how physical limits influence optimum configurations.

Cylindrical Capacitor Geometry

This concept involves understanding the structure of a cylindrical capacitor, consisting of two coaxial cylinders with different radii. The geometry determines how the electric field is distributed between the conductors and directly influences the expressions for capacitance and potential difference. In problems like this, the relation between the inner radius a and the fixed outer radius b is pivotal in determining the device's electrical performance.

Electric Breakdown

The breakdown electric field is a critical parameter that defines the maximum electric field the dielectric (or the surrounding medium) can sustain before it becomes conductive, leading to failure. In capacitor design, ensuring that the maximum field (usually occurring at the surface of the inner conductor) does not exceed this threshold is essential to prevent breakdown, thereby setting a constraint on the allowed voltage and affecting the design optimization.

Capacitance and Energy Storage

Capacitance quantifies a capacitor's ability to store electrical charge per unit potential difference, and for a cylindrical capacitor, it depends logarithmically on the ratio of the outer to inner radii. The energy stored in a capacitor is related to its capacitance and the voltage applied across it. Optimizing the energy storage involves finding the optimal values of the geometric parameters while considering the breakdown limits imposed by the maximum electric field.

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