AcademyCapacitors and Dielectrics
Academy
Molecular View of Induced Charge
Level 1 - Physics topic page in Capacitors and Dielectrics.
Principle
Dielectric molecules polarize so bound charge partly opposes the field from free charge.
Notation
\(\vec p\)
electric dipole moment
\(\mathrm{C\,m}\)
\(q\)
charge magnitude in a simple dipole
\(\mathrm{C}\)
\(\ell\)
charge separation in a simple dipole
\(\mathrm{m}\)
\(\vec\tau\)
torque on a dipole
\(\mathrm{N\,m}\)
\(\vec P\)
polarization dipole moment per volume
\(\mathrm{C\,m^{-2}}\)
\(\sigma_b\)
bound surface charge density
\(\mathrm{C\,m^{-2}}\)
Method
Derivation 1: Dipoles align with the field
A molecular dipole has equal and opposite charges separated by a small distance. A uniform field gives opposite forces on the two ends, producing a torque.
Dipole moment
\[p=q\ell\]
Dipole torque
\[\tau=pE\sin\theta\]
Potential energy
\[U=-\vec p\cdot\vec E\]
Derivation 2: Polarization creates bound surface charge
When many dipoles align, their interior charges mostly cancel. Uncancelled bound charge remains where the dielectric ends.
Polarization
\[\vec P=\frac{\text{dipole moment}}{\text{volume}}\]
Surface bound charge
\[\sigma_b=\vec P\cdot\hat n\]
Derivation 3: Bound charge opposes the free-charge field
For a slab between capacitor plates, the bound charge on the dielectric faces produces a field opposite the field from the free plate charge.
Free-charge field
\[\vec E_0\text{ points from }+\text{ plate to }-\text{ plate}\]
Bound-charge field
\[\vec E_b\text{ points opposite }\vec E_0\]
Reduced net field
\[\vec E=\vec E_0+\vec E_b\]
Rules
These are the molecular polarization relations.
Dipole moment
\[p=q\ell\]
Dipole torque
\[\tau=pE\sin\theta\]
Dipole energy
\[U=-\vec p\cdot\vec E\]
Bound surface
\[\sigma_b=\vec P\cdot\hat n\]
Examples
Question
A dipole with
\[p=2.0\times10^{-29}\,\mathrm{C\,m}\]
is at \[30^\circ\]
to a \[5.0\times10^5\,\mathrm{V\,m^{-1}}\]
field. Estimate the torque.Answer
\[\tau=pE\sin\theta=(2.0\times10^{-29})(5.0\times10^5)\sin30^\circ=5.0\times10^{-24}\,\mathrm{N\,m}\]
Checks
- Bound charge is not free to move through a circuit.
- Polarization reduces the field in a dielectric capacitor; it does not reverse it in ordinary cases.
- A conductor cancels electrostatic field internally by free-charge motion, not by small molecular polarization.
- Dipole energy is lowest when \(\vec p\) aligns with \(\vec E\).