Discussion Notes and Class Agenda
Physics 240
February 6, 2006
Problem Assignments
Problem Groups
24.44 1, 3
24.71 5, 7
24.40 2, 4
24.34 6, 8
Discussion: Dielectrics and Polarization Induced Surface Charge.
We have seen that many objects can store charge, and that we can relate the charge stored to the potential applied
The proportionality constant C is called the capacitance of the object, and depends only on the geometry of the object or device which stores the charge. The capacitance has units of ε0×(length). This is useful for roughly estimating the capacitance of objects.
We have also seen that if we introduce a dielectric material between the electrodes of a capacitor we can define a dielectric constant
where C0 is the original capacitance of the capacitor before the dielectric material was introduced.
The effect of the dielectric material is to reduce the electric field between the electrodes of the capacitor, thereby reducing the potential across the electrodes for a given amount of stored charge. This makes the capacitor more efficient and the capacitance is increased by a factor of Κ for the same electrode geometry. The physics of how the electric field is reduced can be understood in terms of an induced charge density on the surfaces of the dielectric when it is placed in the electric field between the electrodes of the capacitor. The electric field separates the charges in the dielectric slightly, resulting in a surface charge density on the dielectric surfaces. The induced surface charge will be negative near the positively charged plate and positive near the negatively charged plate of a parallel plate capacitor as shown:
If we apply Gauss Law near the plate, we see that the electric field will be reduced by the presence of the induced charge on the surface of the dielectric, since the Gaussian surface will now enclose less total charge:
Applying Gauss’ Law we find that
and the electric field is clearly reduced between the plates due to the presence of σi . The introduction of the dielectric reduces the potential across the plates by 1/Κ, so the E field between the plates is also reduced by this amount since V=Ed where d is the distance between the plates:
We can write the E fields before and after the introduction of the dielectric as
Substituting for E and E0 into the last equation yields
For large values of Κ, the induced charge density is nearly as large as the free charge density σ, the electric field is greatly reduced, and the capacitance is greatly increased over the dielectric-free value.
The point to take away here is that we can model the complicated effects of polarization in terms of simple induced surface charge densities and Gauss’ Law lets us quantify how the polarization reduces the electric field between the plates and therefore increases the capacitance of the system.
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