21. Gauss’s Law “The Prince of Mathematics” Carl Friedrich Gauss (1777 – 1855) Wikemedia Commons

Topics Electric Field Lines Electric Flux Gauss’s Law Using Gauss’s Law Gauss’s Law and Conductors

Electric Field Lines

Electric Field Lines A field line shows the direction of the electric force on a positive point charge

Electric Field Lines By using a convention for the number of lines per unit charge, one can use field lines to indicate the strength of an electric field.

Electric Flux

Electric Flux Flux is the “flow” of any quantity through a surface. For example, it could be sunlight through a window, or water through a hole. In particular, it can be electric field through a surface

The electric flux Df through a small surface element DA is defined by where The unit vector is normal to the surface element

The total electric flux f through a surface is the sum of the individual fluxes Df This is an example of a surface integral

Electric Flux – A Closed Surface Let’s compute the flux through a spherical surface about a point charge: + We see that the flux is proportional to the enclosed charge

Gauss’s Law

Gauss’s Law The electric flux through any closed surface is proportional to the net charge enclosed by the surface: which is usually written as + - + + - - + + + where e0 = (4pk)-1 = 8.85 x 10-12 C2/Nm2 is called the permittivity constant

Gauss’s Law Gauss’s law is always true for any closed surface. However, it is most useful when the charge distribution and the enclosing surface have a high degree of symmetry. + - + + - - + + + a

Using Gauss’s Law

A Uniformly Charged Sphere The enclosed charge is Q. Gauss’s law is Because of the spherical symmetry of the charge distribution, we can infer that the magnitude of the electric field is constant on any spherical surface enclosing the charge

A Uniformly Charged Sphere The symmetry makes is easy to evaluate the surface integral The electric field of a spherically symmetric charge distribution is like that of a point charge

A Uniformly Charged Sphere We can apply Gauss’s law within the sphere by drawing a Gaussian surface of radius r. The charge enclosed within this surface is therefore,

A Uniformly Charged Sphere Within the sphere the field varies linearly with radius Outside, the field looks like that of a point charge

A Hollow Spherical Shell The shell contains a net charge Q distributed uniformly over its surface. Because of the spherical symmetry, the field outside the shell is like that of a point charge. But the field inside is zero! Why? Because the field from A cancels that from B.

Recap By convention, the electric field points away from a positive charge and towards a negative charge. Electric field lines can be used to visualize an electric field. By convention, the number of field lines is proportional to the charge.

Recap Electric field is additive: the field at any point is the vector sum of the electric fields of all charges. Gauss’s law: the net electric flux through any closed surface is proportional to the net enclosed charge:

An Infinite Line of Charge By symmetry, the electric field is radial. Therefore, a suitable Gaussian surface is a cylinder of length L, radius r placed symmetrically about the line charge. The enclosed charge is q = l L, where l is the charge per unit length

An Infinite Line of Charge From Gauss’s law, we deduce that the electric field of a long (strictly infinite) line charge is

A Sheet of Charge For an infinite sheet of charge the field is perpendicular to the sheet. The flux through a cylindrical Gaussian surface is EA + EA. The enclosed charge is q = sA, where s is the charge per unit area. Therefore, the field is E+ E- A

A Charged Disk The electric field at a point P along the axis of a disk is closely related to the field of a sheet of charge: P dq r x q

Gauss’s Law & Conductors

Conductors An applied electric field causes the free positive and negative charges to separate until the field they create exactly cancels the applied field, at which point the charge migration stops. The conductor is then in electrostatic equilibrium.

Charged Conductors Since like charges repel, all excess charge must reside on the surface of a conductor. This is also consistent with the fact that, in equilibrium, the electric field within a conductor is zero.

A Hollow Conductor A conductor carries a net charge of 1 mC and has a 2 mC charge in the internal cavity. The charges must distribute themselves as shown in order to be consistent with Gauss’s law.

A Hollow Spherical Conductor +q E = 0 Consider a neutral spherical conductor in equilibrium with a cavity containing a net charge +q. The charge on the inner surface of the cavity is –q. Why? The charge on the outer surface of the conductor must therefore be +q. Why? And this charge is uniformly distributed. Why?

Field at a Conductor Surface The flux through a cylindrical Gaussian surface is just EA since the field inside the conductor is zero, in equilibrium. The enclosed charge is q = sA, therefore, the field at the surface of a charged conductor is E+ E- = 0 (inside conductor)

Applications

Electric Shielding The tendency of conductors to exclude electric fields from their bulk has many applications. For example: Co-axial cables Lightning Safety Sensitive Compartmented Information Facility (SCIF)

Co-axial Cables Co-axial cables connect, for example, iPods to ear-phones. If the electric fields are too strong, the dielectric can suffer dielectric breakdown Wikemedia Commons

Co-axial Cables and Dielectrics Some molecules, like H2O, have permanent dipole moments. Others can be distorted by an electric field, and become dipolar; that is, acquire induced dipole moments. These materials are called dielectrics

Lightning Safety http://www.lightningsafety.noaa.gov/lightning_map.htm

SCIFs Wright-Patterson Air Force Base in Dayton, Ohio, is one of the major command posts of the U.S. Air Force (USAF). It contains a giant Faraday cage that houses a Sensitive Compartmented Information Facility (SCIF) + -

SCIFs Any externally generated electric field causes electrons in the Faraday cage to migrate in the direction opposite the field. + -

SCIFs The induced field exactly cancels the externally generated fields. Consequently, any electronic equipment inside is immune from an electromagnetic attack. + -

Summary Electric Flux Gauss’s Law The electric flux through a closed surface is determined by the enclosed charge

Summary Conductors The electric field within a conductor, in electrostatic equilibrium, is zero because the charge rushes to, and distributes itself on, the surface of the conductor