 # Current, Resistance and Power

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Current, Resistance and Power

Battery + - Negative Electrode Positive Electrode electrolyte

Battery + - Negative Electrode Positive Electrode electrolyte

Current Simple flow of charge I (Note Convention) - MKS unit - Ampere
See Active Figure 27.09 - I (Note Convention) MKS unit - Ampere Current

Current Charge carriers + A
n - concentration of charges per unit volume vd – drift velocity A – cross sectional area of conducting wire q – charges carried by each particle

The volume contains the passing charges can be found with vd, A and Dt
After some time Dt, the particles will pass beyond a particular point on the wire + + + A + + + The volume contains the passing charges can be found with vd, A and Dt

Volume of passing charges
+ + + A + + +

+ + + + + +

Current Density and Ohm’s Law
Current per unit Area

Current Density and Ohm’s Law
s - conductivity

A more familiar form

Resistivity where Volt/amp= W - MKS Unit Resistance Ohm’s Law:
Macroscopic form where Volt/amp= W - MKS Unit Resistance

Note Dependencies If you double the area (ie. Adding an addition wire) the effective resistance halves If you add the wire to the length the effective resistance doubles The resistivity is an intrinsic property of the material the resistor is made of. If you change material keeping physical geometry the same, the resistance changes

Ohmic (or linear) device
Slope = 1/R I Non-Ohmic (or nonlinear) device V

Microscopic View of Conductor
copper Electron Charge Cross sectional area acceleration Time between collisions Independent of Electric Field: Ohmic But can depend on conditions which effect t, such as temperature See Active Figure 27.09

Resistivity vs. Temperature
r(T) – characteristic of material T r r T semi-conductors insulators metallic

– thermal coefficient of resistivity
r0 – resistivity at T0 – thermal coefficient of resistivity

Superconductors Many materials will below a specific characteristic temperature, Tc, have a pronounced decrease in resistivity.

Power Battery – “works” to push current through circuit
Powersource = VI V – Potential Source I – Current sent from source through circuit I V

Thermal energy dissipated through resistors
Voltage drop across resistor Rate of Thermal Energy dissipation through Resistor

Example Problem: Suppose we wanted to design a small heater for your to work before your car warmed up. We want 500Watts using the 12V of your car battery. How much Nichrome wire with a crossectional area of 0.1 cm2 do we need?

- actual potential difference between electrodes of battery (EMF)
Battery (Source) e - actual potential difference between electrodes of battery (EMF) r – internal resistance of battery

r Battery (Source) e I R By attaching the battery to a circuit including a load resistor R, the current drawn through the battery will effect the actual potential difference in the battery

Kirchoff’s Voltage Loop Theorem
The algebraic sum of the changes in electric potential encountered in a complete traversal of the circuit must be zero. A circuit is closed path through which current (electrons) may be forced to move through circuit elements (resistors).

V = e - Ir Battery voltage terminal to terminal r e I R Jumping from the negative to the positive end of the battery, the potential increases by e, but after going through the resistor, the potential drops by IR

To find the current… r e I R Kirchoff’s Voltage Loop

Resistors in Series and Parallel: Equivalent Resistance

Resistors in Series R1 e R3 I R2 I

Resistors in Series e I Rseries

Kirchoff’s Junction Theorem
At any junction (point where current can split) the algebraic sum of the currents into and out of the wires of the junction must add to zero. By convention the current into a junction is positive and the current out of a junction is negative.

Resistors in Parallel I e R1 R2 R3 I

Resistors in Parallel e I R3 R2 R1 I2 I1 I3

Resistors in Parallel e I R3 R2 R1 I2 I1 I3

Resistors in Parallel e I R3 R2 R1 I2 I1 I3

Resistors in Parallel e I R3 R2 R1 I2 I1 I3

Resistors in Parallel I e I Rparallel I

Resistors in Parallel

Break circuit down into series and parallel resistors
Solve for the currents going through each of the resistors by circuit reduction (equivalent resistance) 42 V 2 W 12 W 4 W 1 W Break circuit down into series and parallel resistors

Currents in the various branches
2 W 12 W 4 W 1 W I1 I2 I

42 V 2 W 12 W 4 W 1 W I1 I2 I Find equivalent resistance for the Series Resistors

Find Equivalent Parallel Resistance
12 W I I2 42 V 3 W 6 W I1 Find Equivalent Parallel Resistance

Find Equivalent Series Resistance
12 W I 42 V 2 W Find Equivalent Series Resistance

Find Equivalent Series Resistance
14 W I Find Equivalent Series Resistance

42 V 14 W I

12 W I I2 1 W 2 W 42 V I1 2 W 4 W

42 V 2 W 12 W 4 W 1 W I1 I2 I

Kirchoff’s Analysis Solve simultaneously for the unknown currents