Ch.6 Current and Resistance

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Overview

current is the flow of charge through a conductor, indicated by

• nearby compass needle is deflected
• wire heats up

Current $I$ is measured in amperes

Model of conduction:

• connecting wire to battery causes nonuniform surface charge distribution
• surface charges create electric field within wire
• field pushes electrons through wire
• current describes the motion of positive charge

Kirchhoff's Junction law: current is the same if no junctions; if there is, sum going in equals sum going out

collisions of electrons with other atoms causes resistance
resistivity: electrical property of a material
resistance: property of circuit based on material, size, and shape

Ohm's law: $I=\Delta V/R$

Electron Current

current: the motion of positive charge
drift speed$v_d$: net motion of charges through a wire, typically small (around $10^{-4}\text{ m/s}$)
define electron current$i_e$ (unit $s^{-1}$) as the number of electrons per second going through a cross section of wire
the number of electrons $N_e$ passing through time $\Delta t$ is $N_e=i_e\Delta t$
If electron number densitiy is $n_e$, we have
$i_e=n_eAv_d$
This means $i_e$ can be incerased by increasing the density, the area they flow through, or the drift speed
To neutralize a capacitor, some electrons from the completely filled wire push some electrons onto the positive plate and the negative plate pushes some electrons onto the wire.

Creating a Current

In electrostatic equilibrium, a conductor has a field of $\vec{E}=0$ inside. But with a current, it is not in equilibrium, so a field pushes the electrons, with some of the energy converting to thermal energy.

Connecting the plates creates a strong negative end, a strong positive end, and neutral center -> nonuniform charge distribution
surface charges do not move; current is inside the wire

If there is an electric field (pointing in negative$x$ direction), then the acceleration of electrons between collisions with other atoms is
$a_x=\frac{F}{m}=\frac{eE}{m}$ Then
$v_x=(v_0)_x+\frac{eE}{m}\Delta t$ Where $\Delta t$ is the time until next collision, where the velocity gets reset to $\vec{v}_0$. Thus, the average velocity, $v_d$, is
$v_d=\overline{v_x}=\overline{(v_0)_x}+\frac{eE}{m}\tau=\frac{e\tau}{m}E$ where $\tau$ is the mean time between collisions, a property of materials
Substituting, we see that
$i_e=\frac{n_eAe\tau}{m}E$

current: time rate of charge transfer; essentially flow of positive charge, despite positive charges not moving
$I=\frac{dq}{dt}=nAv_de=nAve$
electron current: flow of electrons, as is actually happening
1 ampere$\text{A}\equiv\text{C}/\text{S}$ (coulomb per second)

Since $I=ei_e$ (current equals charge times electron flow rate), we also have that
$i_e=n_eAv_d$
current density is
$J=\frac{I}{A}=n_eev_d$
the current per square meter of cross section, units $\text{J}/\text{m}^2$

Due to conservation of charge, current cannot change as it passes through a light bulb; the current must be the same at all points in an individual current-carrying wire.
A direct consequence is Kirchhoff's Junction law
$\sum I_\text{in}=\sum I_\text{out}$

Charge and Resistance

From the current density formula, substituting $v_d$, we have
$J=n_eev_d=n_ee\left(\frac{e\tau E}{m}\right)=\frac{n_ee^2\tau }{m}E$
All quantities depend only on the material. Increasing electron density $n_e$ or time between collisions $\tau$ increases the current density, making the material a better conductor.
The conductivity$\sigma$ is
$\sigma=\frac{n_ee^2\tau}{m}$
This definition makes $J=\sigma E$. Conductivity is dependent on the structure of the material, impurities, and temperature (higher temperature means more collisions, so smaller $\tau$)
The inverse is resistivity$\rho$
$\rho=\frac{1}{\sigma}=\frac{m}{n_ee^2\tau}$
Conductivity has units $\text{A C}/\text{N}\text{ m}^2=\Omega^{-1}\text{ m}^{-1}$

At extremely low temperatures, some materials lose resistivity entirely, called superconductivity

Ohm's Law

In a constant diameter wire of length $L$ and potential difference $\Delta V$, from $E=-dV/ds$, the field magnitude is
$E=\Delta V/L$
Because $J=\sigma E$ and $I=JA$,
$I=JA=\sigma EA=\frac{A}{\rho L}\Delta V$
So, current is proportional to potential difference. Let $R=\rho L/A$ and we have the resistance of a specific conductor with specific length.
SI Unit ohm, $1\text{ ohm}\equiv 1\text{ V/A}$

Ohm's Law: establishing a potential difference across a conductor with resistance $R$ creates a current (note this is only valid for constant $A$)
$I=\frac{\Delta V}{R}$

Recall the battery charge-escalator model:

1. A battery creates a potential difference $\Delta V_\text{bat}$. An ideal battery has $\Delta V_\text{bat}=\epsilon$
2. The battery creates a potential difference $\Delta V_\text{wire}=\Delta V_\text{bat}$ in the wire
3. The potential difference in the wire creates an electric field $E=\Delta V_\text{wire}/L$
4. The field establishes current $I=JA=\sigma EA$
5. The current magnitude is determined by the conductor's resistance and the battery $I=\Delta V_\text{wire}/R$

Materials which Ohm's law applies to (constant $R$) are ohmic
Batteries ($\Delta V=\epsilon$ determined by chemical reactions) and capacitors (different relationship between $I$ and $\Delta V$) are nonohmic

Ohmic materials:

1. wires- metals with extremely small resistivities $\rho$ and resistances ($R\ll 1\text{ }\Omega$); ideal wires have $R=0\text{ }\Omega$ even if there is current
2. resistors- poor conductors with resistances usually between $10^1$ and $10^6$$\Omega$; most have a specified value, e.g. $500$$\Omega$
3. insulators- glass, plastic, air, etc with large resistances (typicall $R\gg10^9$$\Omega$); ideal insulators have $R=\infty\Omega$, giving no current even if there is a potential difference