AcademyCapacitors and Dielectrics
Academy
Dielectrics
Level 1 - Physics topic page in Capacitors and Dielectrics.
Principle
A dielectric increases capacitance by reducing the field needed for a given free charge.
Notation
\(\kappa\)
dielectric constant
1
\(C_0\)
capacitance without dielectric
\(\mathrm{F}\)
\(C\)
capacitance with dielectric
\(\mathrm{F}\)
\(\epsilon\)
permittivity of the dielectric
\(\mathrm{F\,m^{-1}}\)
\(E_0\)
field without dielectric for the same free charge
\(\mathrm{V\,m^{-1}}\)
\(E\)
field inside the dielectric
\(\mathrm{V\,m^{-1}}\)
Method
Derivation 1: Field reduction at fixed free charge
For a dielectric filling the gap, polarization produces bound charge whose field partly opposes the field from the free plate charge.
Field reduction
\[E=\frac{E_0}{\kappa}\]
Voltage reduction
\[V=Ed=\frac{E_0d}{\kappa}=\frac{V_0}{\kappa}\]
Capacitance increase
\[C=\frac{Q}{V}=\kappa\frac{Q}{V_0}=\kappa C_0\]
Derivation 2: Permittivity form
The dielectric constant is usually folded into the material permittivity.
Material permittivity
\[\epsilon=\kappa\epsilon_0\]
Plate capacitance
\[C=\epsilon\frac{A}{d}=\kappa\epsilon_0\frac{A}{d}\]
Derivation 3: Fixed voltage versus fixed charge
The mechanical and energy consequences depend on whether the capacitor stays connected to a battery.
Fixed voltage
\[Q=CV\quad\Rightarrow\quad Q\text{ increases by }\kappa\]
Fixed charge
\[V=\frac{Q}{C}\quad\Rightarrow\quad V\text{ decreases by }\kappa\]
Rules
These relations assume the dielectric completely fills the field region.
Dielectric capacitance
\[C=\kappa C_0\]
Permittivity
\[\epsilon=\kappa\epsilon_0\]
Filled plates
\[C=\kappa\epsilon_0\frac{A}{d}\]
Fixed charge field
\[E=\frac{E_0}{\kappa}\]
Fixed charge voltage
\[V=\frac{V_0}{\kappa}\]
Examples
Question
A
\[40\,\mathrm{pF}\]
air capacitor is filled with dielectric \[\kappa=3.0\]
Find the new capacitance.Answer
\[C=\kappa C_0=(3.0)(40\,\mathrm{pF})=120\,\mathrm{pF}\]
Checks
- \(\kappa\\ge 1\) for ordinary dielectrics, so a filled capacitor has larger capacitance.
- Fixed \(Q\) and fixed \(V\) are different physical constraints.
- A dielectric reduces the internal field for a given free charge.
- The simple \(C=\\kappa C_0\) rule requires the dielectric to fill the same field region.