PHYS 208 Lecture 11

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Resistance

measured in Ohms [Ω]

$V=IR\,\!$ $R={\frac {V}{I}}={\frac {\rho \ell }{A}}$

Current

Current is conserved as it moves through resistors

Current Density:

$J={\frac {I}{A}}$

Hose Analogy

Flow of water through a hose depends on:

• pressure of water main (voltage)
• kinks and friction along the hose (resistance)
• diameter of hose (conductivity)

Example

concentric sphere resistor: current flows from inner sphere (radius $r_{1}$), through ohmic material, to outer sphere (radius $r_{2}$) (battery terminals connected to each cylinder)

${\vec {J}}(r)={\frac {I}{4\pi r^{2}}}\,{\hat {r}}$ (radially outward)

${\begin{aligned}R&={\frac {\left|-\int {\rho {\vec {J}}(r)\cdot \mathrm {d} {\vec {\ell }}}\right|}{I}}\\&={\frac {\rho }{4\pi }}\left|{\frac {1}{r_{2}}}-{\frac {1}{r_{1}}}\right|\\&={\frac {\rho }{4\pi }}\left({\frac {1}{r_{1}}}-{\frac {1}{r_{2}}}\right)\end{aligned}}$

Batteries

Experiments with current, resistivity, etc was impossible until the invention of the battery:

Modern Battery configuration:

• Carbon electrode (positive terminal) through center (not touching bottom)
• Electrolyte paste for storage of ions
• zinc casing for negative electrode

Batteries have an internal resistance (usually less than 1Ω). That's why batteries have a slightly different voltage than what is advertised on the side.

The source of the voltage within the battery is referred to as Electromotive Force (emf) (represented by ℰ). In order for a battery to work, the external resistance (load) must be greater than the internal resistance. When this condition is not met, the battery cannot supply enough current to keep the system running.

Batteries can be measured in ampere-hours: the amount of current that can be supplied for a given amount of time.

Electric Power

Amount of work done per time; measured in Watts [W = J/s]

$P={\frac {\mathrm {d} W}{\mathrm {d} t}}=IV={\frac {V^{2}}{R}}$

Example

2000 watt heater designed to work at 120 V of current:

1. what is the resistance? $R={\frac {120^{2}}{2000}}=7.2\Omega$
2. what is the current? $I={\frac {120}{7.2}}=16.67\mathrm {A}$

Chapter 26: DC Circuits

Whoa! what just happened‽

Tools for measuring DC currents:

• amp-meter: connected in series with a circuit so current to be measured flows through it
• volt-meter: connected in parallel with a circuit that you want to measure the voltage acress