Recap of Chapter 21-24

Electric Charge and Electric Field

• There are two kinds of electric charge. Positive and negative.
• Charge is to be treated algebraically. It's unit is Coulombs (C).
• Electric charge is conserved
• Charge of an electron $-e=-\mathrm {e^{-}} =-1.6\times 10^{-19}{\textrm {C}}$

Coulomb's law:

• The magnitude of the force one point charge exerts on another is proportional to the product of their charges, and inversely proportional to the square of the distance between them.

$F=k{\frac {Q_{1}Q_{2}}{r^{2}}}={\frac {1}{4\pi \epsilon _{0}}}{\frac {Q_{1}Q_{2}}{r^{2}}}$

Electric Field

• The electric field, ${\vec {E}}$, due to one one or more charges, is defined as the force per unit charge that would act on a positive test charge $q$ placed at that point:

${\vec {E}}={\frac {\vec {F}}{q}}$

• The magnitude $\left|{\vec {E}}\right|=E$ is

$E=k{\frac {Q}{r^{2}}}$

• The total electric field at a point of space is equal to the vector sum of the individual fields (principle of superposition)
• Electric field lines start on positive charge, and terminate on negative charge. Their direction indicates the direction of force a positive test charge would feel (i.e. at each point of space, it is the vector sum of all ${\vec {E}}$). The intensity of the ${\vec {E}}$ is proportional to number of electric field lines drawn per unit space.
• The static electric field inside a conductor is zero, since the free electrons in the conductor would keep on moving freely until the field is zeroed.

Electric Dipole

• An electric dipole is a combination of two equal but opposite charges $+Q$ and $-Q$ separated by a distance $l$.
• The dipole moment is

$p=Ql$

• A dipole placed in a uniform electric field experiences a torque.
• The electric field produced by the dipole decreases as the third power of the distance $r$ from the dipole ($E\propto 1/r^{3}$ when $r\gg l$)

Gauss's Law

Electric Flux

• The electric flux $\Phi _{E}$ passing through an area $A$ is

 $\Phi _{E}=\int {\vec {E}}\cdot d{\vec {A}}$ 

where ${\vec {A}}$ is a vector with a magnitude equal to the the infinitesimal area in question, and direction along the unit normal (the vector that points outward from the enclosed surface).

• The flux through a surface is proportional to the number of field lines passing through it.

Gauss's law

• Gauss's law states that the net flux passing through any closed surface is equal to the net charge $Q_{\textrm {encl}}$ enclosed by the surface divided by $\epsilon _{0}$

$\oint {\vec {E}}\cdot d{\vec {A}}={\frac {Q_{\textrm {encl}}}{\epsilon _{0}}}$

Electric Potential

• Electric potential is defined as electric potential energy per unit charge.
• electric potential difference between any two points in space is defined as the difference in potential energy of a test charge q placed at those two points, divided by the charge q:

$V_{ba}={\frac {U_{b}-U_{a}}{q}}$

• the unit of potential is Volts ( 1 V = 1 J/C)
• the change in potential energy for a charged particle when it moves through a potential difference is

 $\Delta U=qV_{ba}$

• The potential difference between two points, a and b, can be obtained from ${\vec {E}}$

 $V_{ba}=V_{b}-V_{a}=-\int _{a}^{b}{\vec {E}}.d{\vec {l}}$

• An equipotential line or surface is all at the same potential, and thus it is always perpendicular to the electric field at all points by definition.

• The electric potential due to a single point charge Q, relative to zero potential at infinity, is given by

$V={\frac {1}{4\pi \epsilon _{0}}}{\frac {Q}{r}}$

• The potential due to any charge distribution can be obtained by summing (or integrating) over the potentials for all the charges.
• The potential due to an electric dipole drops off as $1/r^{2}$
• The dipole moment is $p=Q.l$ where is the distance between the two equal but opposite charges of magnitude Q.
• The relationship between potential difference and electric field works both ways (i.e. electric field can be obtained from potential)

 $E_{x}=-{\frac {\partial }{\partial x}}V$, $E_{y}=-{\frac {\partial }{\partial y}}V$, $E_{z}=-{\frac {\partial }{\partial z}}V$,${\vec {E}}=E_{x}{\hat {x}}+E_{y}{\hat {y}}+E_{z}{\hat {z}}$

Capacitance, Dielectrics, Electric Energy Storage

• A capacitor is a device used to store charge and electric energy).
• The ratio of the charge held in the capacitor Q to the potential V capacitor is supplied with is called the capacitance, C:

 $C={\frac {Q}{V}}$

• The capacitance of a parallel-plate capacitor is proportional to the area A of each plate and inversely proportional to their separation d:

$C=\epsilon _{0}{\frac {A}{d}}$

• Capacitors in parallel

 $C_{eq}=C_{1}+C_{2}+\dots$

• Capacitors in Series

 ${\frac {1}{C_{eq}}}={\frac {1}{C_{1}}}+{\frac {1}{C_{2}}}+\dots$

• A charged capacitor stores electric energy

given by

 $U={\frac {1}{2}}QV={\frac {1}{2}}CV^{2}={\frac {1}{2}}{\frac {Q^{2}}{C}}$

• In any electric field in free space the energy density u (energy per unit volume) is

 $u={\frac {1}{2}}\epsilon _{0}E^{2}$

• the capacitance is proportional to a property of dielectrics called the dielectric constant, K

(nearly equal to 1 for air). For a parallel-plate capacitor

 $C=K\epsilon _{0}{\frac {A}{d}}=\epsilon {\frac {A}{d}}$
  

where $\epsilon$ is called the permittivity of the dielectric material.

• When a dielectric is present, the energy density is

 $u={\frac {1}{2}}\epsilon E^{2}$