A spherical capacitor with a 1.0 mm gap between the spheres...

Key Concepts

Capacitance

Capacitance is a measure of a system's ability to store electric charge for a given potential difference between conductors. It is defined as the ratio of the electric charge on one conductor to the potential difference between the conductors. This concept is central to understanding energy storage in capacitors and is used across various electronic and electrical applications.

Spherical Capacitors

Spherical capacitors consist of two concentric conducting spheres separated by an insulating region, often vacuum or a dielectric material. The capacitance of such a system is computed using a specific formula that accounts for the radii of the inner and outer spheres. This concept highlights how geometry directly influences the ability to store charge in systems with spherical symmetry.

Geometric Relationships in Spherical Structures

Understanding the geometric relationships inherent in spherical objects is essential when analyzing systems like spherical capacitors. This involves relating the radius and diameter of spheres, as well as the spacing between them. These geometric considerations are crucial in deriving and applying formulas that determine the physical dimensions required to achieve a desired electrical property, such as a specific capacitance.

Permittivity of Free Space

The permittivity of free space, denoted as ??, is a fundamental physical constant that appears in equations describing electric fields and capacitance in vacuum. It quantifies the ability of a vacuum to permit electric field lines and is a key parameter in the calculations of capacitance for systems like the spherical capacitor. Its value is essential for accurately determining capacitance when no additional dielectric materials are present.

Transcript

00:01 Okay, so we know the capacitance for a spherical capacitor can be equal to 4 pi epsilon 0 times r1 r2 over r2 minus r1.

00:09 Okay? so if you do some arrangement here, we have r1, r2 is equal to c times r2 minus r1 over 4x00 so we know c, which is a capacitance is given as a hundred picofera, which is 1 times 10 to p .10 phara.

00:23 And r2 minus r1 here is the gap between the outer radius and the inner radius, which is about 1 millimeter, which is 1 times 10 to 0 .93 meter.

00:36 And epsilon 0 here is equal to 8 .85 times 10 to 012 koulon square per newton times meter square.

00:44 So therefore we have r1, r2, is simply equal to 8 .99 times 10 to 0 .94 meters square.

00:52 And then we know r1 plus r2 can be equal to square root r2 minus r1 square plus 4 r1r2.

00:58 We know r2 minus r1 is 1 times 10 to power negative 3 meter.

01:01 And r1r2 is equal to 8 % to 02.

01:04 2 .99 times 10 to 0 .94 meter square.

01:06 So therefore we have r1 plus r2 is equal to 6 times 10 to 0 meter, which means that r2 plus r1 is also equal to 6 times 102 0.

01:20 Remember r1 is equal to 1 times 102 0 0 0 meter.

01:25 So therefore we have, okay, let's say this one is one equation and this one is the second equation.

01:34 And we can just say, simply use 1 plus 2.

01:38 And this will give us 2r2 is equal to 6 times 102 0 .92 meter plus 1 times 102 power negative 3 meter.

01:55 Ok? because the r1 cancel out.

01:58 And this will give us 2r2 is equal to 0 .061 meter.

02:05 This will give us r2.

02:15 Let's just keep 2r2, okay? because 2r2 is actually equal to d2, which is the diameter based on the outer radius, which is the outer diameter, which is equal to 0 .061 meter...

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