PHYS 1008 Electric Circuits

www.ibiblio.org/ pub/Linux/apps/circuits/

This is a "need to know" for the first week's labs. By the end of this you should be able to do the following

Use Ohm's law.
Find the resistance of a material
Interpret circuit diagrams
Solve simple circuits

Resistivity

Some materials allow charges to move around: these are conductors.

Current: the rate at which charge moves through a wire (C s^-1), but this is so important that it gets its own name

1 Ampere = 1 Amp = 1 C s^-1.

It is very useful to think of current as the flow of a liquid. We can think of wire consisting of a large number of charges, which all flow with the same speed.

If there are

n positive charges/unit volume

charge q on each

velocity v

in a wire with X-sect area A

Total charge in this piece of wire is
δQ = nqLA

Then the charge that moves past a given point in time δt = L/v is given by
```
I = δQ = nqLA = nqAv
    δt    L/v 
```
Note (very important!) you cannot distinguish a positive charge moving in one direction with a negative charge moving in the other: qv is a constant.

or even a mixture

Unfortunately Franklin guessed wrong: he assumed that current consists of +ve charges, whereas it (in most metals) consists of electrons.

Resistance

What makes a current flow? Only thing that can move charges is an electric field.The wire acts as a "pipe" for the electrons to move in. If there is a field inside a wire, this means there must be a potential difference (or "voltage") along the wire.

Ohm's law relates the Current, Resistance and Voltage

V = I R

Actually, this is always true, by definition: what makes it useful is that for most metals, the resistance R is a constant. Metals conduct equally well in all directions: the resistance is just 1/slope of curve (why?). A diode is a device designed to conduct current in one direction only.

Units: if I is measured in amps, V in volts then R is in ohms Ω

Expect resistance to depend on shape of conductor, R

∝ length of conductor
∝1/area

so
```
R = ρ L
      A 
```
where ρ is the resistivity of the material (units are ohm m).

Circuits

Next step is to hook together different elements to make a circuit. Batteries provide as source of potential, or electromotive force (EMF). Ɛ (say curly E!). It isn't actually a force..

..The simplest possible circuit is just a source of EMF and a resistor: e.g. a battery and a light

However, circuits like this are always represented by schematics: standard notation:
So the above circuit becomes
Also, by convention we assume that connecting wires have no resistance, so we can draw them in any way we wish.

Kirchoff's Laws

Almost every circuit can be solved by Kirchoff's First and Second laws

Kirchoff's Point Law: the algebraic sum of currents into a junction is 0. (algebraic sum means current in is positive, current out is negative)

(What goes in must come out!)

I₁ - I₂ -I₃ = 0

Kirchoff's Loop Law: the sum of P.D.'s round any circuit is zero (What goes up must come down!

The fluid analogy: if we think of the wire as a pipe, and charge as a liquid

The current is the rate of flow of the liquid
The potential difference is the pressure that keeps the liquid moving
The resistance is the "narrowness" of the pipe

This makes a number of points almost obvious.

e.g. Kirchoff's 1st Law says that the algebraic sum of currents into a junction is 0

I₁ - I₂ -I₃ = 0

(algebraic sum means current in is positive, current out is negative) but this is almost obvious if we think of the charge as a liquid: what flows in must flow out

To use this, must draw a loop round the circuit, and proceed round the circuit until you get back where you started. Count

+Ɛ when you go through a battery from negative to positive,
-IR when you go through a resistor in the direction of the current.

(What happens if you guess the direction of a current wrong? Nothing: you just get a negative result)
Kirchoff's loop law is "obvious":

A positive charge passing through the battery would have a potential increase of Ɛ, followed by a potential drop of IR. Hence
Ɛ-IR = 0 or I = Ɛ/R (which is obvious!).

Measurement of Current and Voltage

All devices actually measure a very small current: galvanometer consists of coil suspended in magnetic field: torque corresponds to current, measuring (say) 1 mA. How do we measure large currents or large voltages with such a device?

e.g.: an ammeter measuring a current of 20 A has to be constructed from a galvanometer, with a max current of 10 mA and an internal resistance of 100Ω. What should the "shunt resistance" R be?
e.g. a voltmeter is to measure a voltage of 12 V has to be constructed from the same galvo. What should the series resistance be?
****Yes, I know that we don't have to do this explicitly these days: however what a multi-meter does is to switch resistances electronically.

Now lets start at the beginning

Note (very important!) you cannot distinguish a positive charge moving in one direction with a negative charge moving in the other: qv is a constant.
or even a mixture