25.9:
Gauss’s Law in Dielectrics
Consider a charged parallel plate capacitor separated by a vacuum. Now, applying Gauss's law to a Gaussian surface, the electric flux is proportional to the charge enclosed by the surface.
When the vacuum is replaced with a dielectric, free charges develop on the surface of the capacitor plates, and bound charges are induced on the dielectric surface, decreasing the net electric field.
So, the magnitude of the electric flux through the Gaussian surface inside this capacitor with the dielectric is proportional to the net charge inside the surface.
The dielectric polarization causes a decrease in the electric field by a factor of the dielectric constant; so, the net charge equals the ratio of free charge to the dielectric constant.
This net charge, when substituted into Gauss's law yields Gauss's law in dielectrics, where the electric field is replaced with the product of the dielectric constant and electric field.
The product of the vacuum permittivity, dielectric constant, and electric field is called the electric displacement. So, Gauss's law can be rewritten in terms of electric displacement.
25.9:
Gauss’s Law in Dielectrics
Consider a polar dielectric placed in an external field. In such a dielectric, opposite charges on adjacent dipoles neutralize each other, such that the net charge within the dielectric is zero. When a polar dielectric is inserted in between the capacitor plates, an electric field is generated due to the presence of net charges near the edge of the dielectric and the metal plates interface. Since the external electrical field merely aligns the dipoles, the dielectric as a whole is neutral. An equal magnitude of bound charges with opposite polarity are induced on the surface of the dielectric. These induced charges produce an additional electric field which opposes the previous external field.
A similar effect is produced when the molecules of a dielectric are nonpolar. A nonpolar molecule acquires an induced electric-dipole moment because the external field causes a separation between its positive and negative charges. The induced dipoles of the nonpolar molecules align with the external electric field in the same way as the permanent dipoles of the polar molecules. Hence, the electrical field within the dielectric weakens, regardless of whether its molecules are polar or nonpolar.
A capacitor filled with dielectric consists of an electric field due to the free charges on the capacitor plates and an electric field due to the induced charges on the dielectric surfaces. Their vector sum gives the net electric field within the dielectric between the capacitor plates. This net field is produced by an effective charge that is equal to the difference between the net free and the surface charges. Since the effect of the dielectric is to weaken the original field by a factor of the dielectric constant, the electric field term in the Gauss Law is replaced by the product of the electric field and dielectric constant, giving the modified Gauss Law in dielectrics.
Suggested Reading
- Young, H. D., and Freedman, R.A. (2012). University Physics with Modern Physics. San Francisco, CA: Pearson. pp. 807.
- OpenStax. (2019). University Physics Vol. 2. [Web version]. pp. 369-370 Retrieved from https://openstax.org/books/college-physics/pages/19-5-capacitors-and-dielectrics