Capacitance is the ability of a Capacitor to store charge.
Capacitor
A Capacitor is an electrical component that can store energy in an electric field. They consist at least of two electrical conductors separated by a dielectric (insulator), although many variations can be found.
A potential voltage difference between the two fields will create a static electric field develops across the dielectric. Positive charge will collect on one of the plates and negative charge on the other one.
Charge
the charge stored equals to the Capacitance of the object and the Voltage across the object.
Energy
To derive an expression for how much energy is stored in a capacitor we consider the infinitesimal small voltage due to the infinitesimal little charge \(y\)
Energy is defined as:
The Energy for every small amount of voltage can therefore be expressed as:
At this point we can take our Capacitance expression and plug it in.
Then we can add all these infinitesimal small Energies up to get the total amount of energy:
Capacitance
The Capacitance between two plates can be derived as followed
Electric Field
The Electric Field between two plates:
Voltage
The Voltage between two plates:
Capacitance
The Capacitance between two plates:
Examples
In these examples we are trying to find:
- \(E\), the Electric Field
- \(\Delta V = \int_A^B E(r) dr\), the difference in Voltage
- \(C = \frac{q}{V}\), the Capacitance
Cylinder
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Electric Field
The \(\cos\) term will fall away since the Field is perpendicular to the area.
The Area will just add up to the total Area since again all the small Area’s are at the same angle to the Electric Field.
Therefore the Electric Field will be:
Voltage Difference
Voltage is defined as:
Therefore we can define the change in Voltage as
We can plug in our result from earlier to get
We can factor out all the constants to get
Capacitance
Capacitance is defined as
Plug and play
Solving Magic
Sphere
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Electric Field
Therefore the Electric Field will be:
Voltage Difference
Same steps as for the cylinder can be applied:
The integral can simply be solved by usage of the power rule on \(r^{-2}\).
Capacitance
Capacitance is defined as
Plug and play
Sources
- My Notes