Gauss's Law and Electric Fields

• An important tool for simplifying problems in physics is utilizing the symmetry properties of systems
• Cylindrical bodies look the same after rotation around their axis
• Charged metal spheres look the same when turned about any axis through their center

Gauss's Law

• A new principle used alongside symmetry to simplify electric-field calculations
• Using Gauss's law, the field of a straight-line or plane-sheet charge distribution, previously derived using strenuous integrations in Section 21.5, can be obtained in a few steps
• It is a fundamental statement of the relationship between electric charges and electric fields
• It helps with understanding how electric charge distributes itself over conducting bodies.
• Gauss's law involves surrounding any general charge distribution with an imaginary surface
• Examining the electric field at various points on this imaginary surface determines the relationship between the field and the total charge enclosed within the surface.
• Provides insights into the character of electric fields, proving to be a tremendously useful relationship.

Charge and Electric Flux

• Chapter 21 contemplated about electric field production at a point P from a given charge distribution
• The answer involved representing the distribution as an assembly of point charges, each contributing an electric field E, as calculated by a previous equation
• The total field at P is the vector sum of the fields caused by all the point charges
• Another relationship exists between charge distributions and electric fields.
• If the electric-field pattern is known in a given region, then one can determine of the charge distribution in that region.
• Electric charge may or may not be contained in a box fashioned out of a material that has no effect on any electric fields.
• The box is an imaginary surface that may or may not enclose some charge.
• The box is a closed surface because it completely encloses a volume.
• One can determine the amount of electric charge inside the box
• A charge distribution produces an electric field, and an electric field exerts a force on a test charge
• A test charge q₀ is moved around the vicinity of the box.
• Measuring the force F experienced by the test charge at different positions makes a three-dimensional map of the electric field E = F/q₀ outside the box.
• In the case that the map turns out to be the same as that of the electric field produced by a positive point charge, from the details of the map, the exact value of the point charge inside the box is discovered.
• Measuring E is needed only on the surface of the box to determine the contents of the box
• The electric field patterns for single or double positive charges inside the box are different, yet in each case the electric field points out of the box
• For single or double negative charges inside the box, the details of E are different for the two cases, but the electric field points into each box.

Electric Flux and Enclosed Charge

• An analogy exists between electric-field vectors and the velocity vectors of a fluid in motion
• There’s an outward electric flux when electric-field vectors point out of the surface
• The E vectors point into the surface, hence the electric flux is inward.
• A simple relationship is suggested: Positive charge inside the box is connected to an outward electric flux through the box's surface, and negative charge inside goes with an inward electric flux.

Zero Net Charge Cases

• Box is empty and E = 0 everywhere and has no electric flux into or out of it
• One positive and one negative point charge of equal magnitude are enclosed within the box, so the net charge inside the box is zero
• There is an electric field, but it "flows into" the box on half of its surface and "flows out of" the box on the other half, and hence there is no net electric flux into or out of it
• Charge is present outside empty box, which has been placed with one end parallel to a uniformly charged infinite sheet, producing a uniform electric field perpendicular to the sheet
• On one end of the box, E points into the box
• On the opposite end, E points out of the box; and on the sides, E is parallel to the surface and so points neither into nor out of the box
• The inward electric flux on one part of the box exactly compensates for the outward electric flux on the other part.
• No net electric flux through the surface of the box, and no net charge is enclosed in the box in all cases.
• There is a connection between the sign (positive, negative, or zero) of the net charge enclosed by a closed surface and the sense (outward, inward, or none) of the net electric flux through the surface
• Keeping in mind the fluid-flow analogy, the net outward electric flux is also twice as great
• The net electric flux through the surface of the box is directly proportional to the magnitude of the net charge enclosed by the box
• The above conclusion is irrespective of the box size
• Despite the magnitude of the electric field of a point charge decreasing with distance, with a larger box, each face has four times the area
• Provided electric flux is defined as taking each face of the box product of the average perpendicular component of E with the area, then add up the faces.
• The net electric flux independently correlates with the net charge inside the box instead of the size.

Charge and rectangular relationship to electric flux

• Net outward or inward electric flux corresponds to the sign of the enclosed charge in rectangular and charged sheets
• Charges outside the surface contributes zero net value to electric flux
• Net electric flux that goes in or out is directly proportional to the value of the charge inside