Charged Capacitor

CLAIMED BY: GA HYUN OH

The Main Idea

A capacitor is when two uniformly, but oppositely (-Q and +Q), charged metal plates are held very close to each other with a separation of s.
This page is dedicated to understanding and calculating the electric field of a capacitor through definition, mathematical models, computational models, and example problems.

A Mathematical Model

The mathematical model to the electric field of a charged capacitor (near the center of the capacitor) is [math]\displaystyle{ E \approx {\frac{Q/A}{{\epsilon}_0}} }[/math], where Q is the magnitude of the plate charges and A is the area of each plates. The direction is perpendicular to the plates.
The fringe field (field located near the center of the disks but right outside of the plates) is [math]\displaystyle{ E_{fringe} \approx {\frac{Q/A}{2{\epsilon}_0}} (\frac{s}{R}) }[/math]

Derivation

Take the origin at the surface of the left plane, with the z-axis running to the right. We assume that each disk has a uniformly charge density (s [math]\displaystyle{ \ll }[/math] R).

Then, the contribution of the negative capacitor is [math]\displaystyle{ E_- \approx {\frac{Q/A}{2{\epsilon}_0}} [1-\frac{z}{R}] }[/math] (to the left) and the positive capacitor is [math]\displaystyle{ E_+ \approx {\frac{Q/A}{2{\epsilon}_0}} [1-\frac{s-z}{R}] }[/math] (to the left).
If we add up the contributions, [math]\displaystyle{ E_{total} \approx {\frac{Q/A}{2{\epsilon}_0}} [1-\frac{z}{R}] + {\frac{Q/A}{2{\epsilon}_0}} [1-\frac{s-z}{R}] \approx {\frac{Q/A}{{\epsilon}_0}} [1-\frac{s/2}{R}] }[/math]. Since s [math]\displaystyle{ \ll }[/math] R, [math]\displaystyle{ E \approx {\frac{Q/A}{{\epsilon}_0}} }[/math].

Gauss's Law

Say [math]\displaystyle{ \sigma }[/math] is the surface charge density, and the area on each side of the cylinder is A.
Since Gauss's Law is [math]\displaystyle{ \Phi = \lmoustache \vec{E} \cdot d\vec{A} }[/math], [math]\displaystyle{ \Phi_{left} }[/math] = 0, [math]\displaystyle{ \Phi_{circular sides} }[/math] = 0, and [math]\displaystyle{ \Phi_{right} = \frac{Q_{enclosed}}{\varepsilon_0} }[/math]; while [math]\displaystyle{ \vec{E}_{left} = 0 }[/math], [math]\displaystyle{ \vec{E}_{circular sides} \perp \vec{A} }[/math].
Thus, EA = [math]\displaystyle{ \Phi = \lmoustache \vec{E} \cdot d\vec{A} }[/math] = [math]\displaystyle{ \Phi_{right} = \frac{Q_{enclosed}}{\varepsilon_0} }[/math] = [math]\displaystyle{ \frac{\sigma A}{\varepsilon_0} }[/math]. Therefore, E = frac{\sigma}{\varepsilon_0}</math>

A Computational Model

Uniformly charged capacitors can be further explored through PhET Interactive Simulations. Here, the user can explore how a capacitor works by changing the size of capacitors and add different objects such as dielectrics to observe how they affect capacitance.

Charged capacitors can also be visualized through this vpython code created on the website Teach hands-on with GlowScript

Examples

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