Phys 1401 - Chapter 15

Flux and Gauss's Law

Flux is related to something passing through a window or surface.
Momentum flux involves an object with mass and velocity moving through a surface.
Gauss's law uses the term "surface" instead of "window."
The direction of the window or surface is always perpendicular to it.
The Greek letter {\Phi} represents flux.
The flux of the electric field is the magnitude of the electric field multiplied by the area of the surface and the cosine of the angle {\theta} between the electric field and the surface direction.

Gauss's law involves closed two-dimensional surfaces.
Examples of closed surfaces include cubes, spheres, and cylinders.
Gauss's law states that the flux of the electric field through a closed surface equals all the charge within the enclosure divided by a constant ({\epsilon_0}).
Source and Perturbation: In physics, there's always a source (e.g., charge) and a perturbation (e.g., electric field).
Gauss's law describes the relationship between the flux of the ripples and the sources themselves.
The electric flux is equal to the enclosed charge divided by {\epsilon_0}.
Analogy: Source and effect; cause and effect.
Mathematical Representation: {\Phi = \frac{Q{enclosed}}{\epsilon0}}

Consider a charge at the center of a sphere.
The electric field for a single charge is E = \frac{kQ}{r^2}.
The area of the surface is 4 \pi r^2.
According to Gauss's law, the flux is the electric field times the area, which is equal to Q / \epsilon_0.
E(4 \pi r^2) = \frac{Q}{\epsilon_0}
E = \frac{Q}{4 \pi r^2 \epsilon_0} = \frac{kQ}{r^2}
The source is like the cause, and the perturbation is like the effect.
Newton's second law: F=ma, where force is the cause, and acceleration is the effect.

The Millikan oil drop experiment involves tiny droplets of oil between two oppositely charged parallel metal plates.
The electric field between the plates is {\sigma / \epsilon_0}, where {\sigma} is the charge density.
The electric field is constant and points downward.
Oil droplets pick up electrons, becoming charged.
Forces acting on the charge: electric force (qE) and gravity (mg).

The electric force on a negative charge (electron) is upward, and the gravitational force is downward.
Coordinate system: upward is positive.
The electric force is Fe = qE, and the gravitational force is Fg = mg.
Some droplets pick up one or more excess electrons.
The charge on one electron is 1.6 \times 10^{-19} coulombs.
The charge on the plates is adjusted so that the electric force on the excess electrons exactly balances the weight of the droplet.

If the electric field is 4.9 \times 10^4 N/C, the electric force on one electron is F_e = (1.6 \times 10^{-19} C)(4.9 \times 10^4 N/C) = 7.85 \times 10^{-15} N.
To balance the weight of the droplet, Fe = Fg or qE = mg.
Density is mass divided by volume (\rho = \frac{m}{V}), so mass is density times volume (m = \rho V).
The volume of a sphere is V = \frac{4}{3} \pi r^3.
Therefore, qE = \rho \frac{4}{3} \pi r^3 g. Solve for the radius (r) of the oil droplet.
Radius: r = \sqrt[3]{\frac{3qE}{4 \pi \rho g}}.
Conducting plates have charges along the surface, with no electric field inside.
Bringing two plates closer causes positive charges on one to attract negative charges on the other.
There's a constant electric field between the plates: E = \frac{\sigma}{\epsilon_0}.
When the plates are baked down, electric field is constant and equal to charge density over epsilon zero.
The electric field always points from the positive plate to the negative plate.

Work done on a charge by the field is force times displacement.
Change in potential energy is the negative of the work done.
\Delta U = -qE \Delta x
The change in kinetic energy plus the change in potential energy must always be zero.
Kinetic energy: T = \frac{1}{2}mv^2.
Change in kinetic energy: \Delta T = \frac{1}{2} m vf^2 - \frac{1}{2} m vi^2 .
I Energy conservation: \Delta T + \Delta U = 0 .
Voltage ($\Delta V$) is the change in potential energy per charge: {\Delta V = \frac{\Delta U}{q} = -E \Delta x}.
Voltage is a property of the electric field, not the charge.
12-volt battery provides 12 joules of energy per Coulomb of charge.
Capacitors are associated with the loss of potential energy per Coulomb, while batteries provide it.
The source of energy for a capacitor is the electrical potential energy due to difference in charge between the plates.
Capacitors stores energy. Battery provides energy.
Batteries stores energy by performing chemical reactions that free up electrons to move. Capacitor stores energy by generating an electric field.

Capacitance is the relationship between the charge (Q) and the voltage drop (V).
{\Delta V = Ed = \frac{Q}{A \epsilon_0}d}
Q = CV, where C is the capacitance ({C = \frac{A \epsilon_0}{d}}).