PHYS 208 Lecture 10

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Energy stored in capacitor

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

Dielectrics

The insulating stuff between the two conductors in the capacitor

The atoms inside dielectric material polarize and orient themselves in the e-field formed between the conductors, producing their own field in the opposite direction. This "inner" field weakens the "outer" field, allowing more charge to be stored in the conductors.

Dielectric Constant

Capacitance with dielectric = dielectric constant × Capacitance without dielectric.

the constant is the dielectric constant:

$K=\frac{C_{with}}{C_{without}}$

ε₀ is the permittivity of free space (i.e. no dielectric), and this constant increases by a factor of $K$ when a dielecric is inserted:

$ϵ=K{ϵ}_0$

Energy per unit volume

Energy divided by volume: [J/m³]

$u=\frac{1}{2}ϵE^2$
For free space, use ε₀ for ε

Thus $U=uV$, where V is the volume between the conductors in a capacitor.

Chapter 25: Current and Resistance

Before, we were just studying static charges in equilibrium. Now these are moving targets. Enter DC circuits, stage left.

Before, there could not be any electric field within a conductor, but since charges are moving now, there must be an electric field along the path of the current in the conductor.

Electrons move against field, but bang around erratically instead of in a straight line (like plinko).

Velocity depends on acceleration ($v=at$), but we use the average velocity or drift velocity as $v_{drift}=\frac{\mathrm{\Delta }x}{\mathrm{\Delta }t}$

Current

Measured in Amperes/Amps, or Coulombs per second: [A = C/s]

$I=\frac{\mathrm{d}Q}{\mathrm{d}t}=n|q|v_{drift}A$

• $n$ is the number of charge carriers per unit volume (a big number)
  • For example, the number of charge carriers for Cu is 8.4 × 10²⁸ charges/m³.
  • Interesting fact: the electrons in copper at 10 A move 9.47 × 10^-4 m/s
• Current does not depend on the sign of the charge $q$, so we take the absolute value

Current flow is conventionally said to flow from positive to negative. In other words, the direction that positive charges would move in the circuit. In reality, both types of charges move in opposite directions simultaneously

We can do whatever we want in a circuit, but the total current is conserved (otherwise we'd have a buildup or loss of charges)

Current Density

$\overrightarrow{J}=\frac{I}{A}=n|q|{\overrightarrow{v}}_{drift}$

Note that current (I) is not a vector, and current density (J) points in the direction of drift velocity.

Based on this definition, current is the flux of current density:

$I=\oint \overrightarrow{J}\cdot \mathrm{d}\overrightarrow{A}$

Example

How much charge passes by a point in a circuit in 60 seconds when carrying a current of 1 A?

$\mathrm{\Delta }Q=I\mathrm{\Delta }t=60\mathrm{C}$

Resistivity and Conductivity

Resistivity

Represented by Greek letter rho (ρ) with units [Ω-m]

An alternate form of Ohm's law (more on this later):

$E=\rho J⟶\rho =\frac{E}{J}$

Conductors have very low resistivity, insulators are very high, and semiconductors are somewhere between.

Conductivity

Represented by the Greek letter sigma (σ)

Reciprocal of resistivity

$\sigma =1/\rho$

$J=\sigma E$

Ohm's Law

$V=IR$

"Ohmic materials" have a linear relationship between current and voltage, and this linear slope is 1/R

Thus

$R=\frac{\mathrm{\Delta }V}{I}$

Example

$\begin{array}{rl}R& =\frac{|\mathrm{\Delta }V|}{I}\\ & =\frac{|-\int \overrightarrow{E}\cdot \mathrm{d}\ell |}{\oint \overrightarrow{J}\cdot \mathrm{d}\overrightarrow{A}}\\ & =\dots \\ & =\frac{\rho L}{A}\end{array}$