PHSI191 Module 4 - Electricity & DC Circuits (Notes) PDF

Preview

Summary

PHSI191 Module 4Electricity & DC CircuitsLecture 1Electric ForceThe Electric ForceThere are four fundamental forces in the universe; the strong nuclear force, weak nuclear force, gravity, and electric force. Nuclear forces are short ranged and are important because they hold nuclei and nucleons ...

Description

PHSI191 Module 4 Electricity & DC Circuits

Lecture 1 Electric Force The Electric Force There are four fundamental forces in the universe; the strong nuclear force, weak nuclear force, gravity, and electric force. Nuclear forces are short ranged and are important because they hold nuclei and nucleons together. Gravity is important because it keeps you on earth and it keeps the earth orbiting the sun. The electric force is important because it is responsible for everything else. It: - Binds nuclei and electrons together to form atoms. - Binds atoms together to form molecules. - Binds molecules together to form bulk matter.

Static Electricity Electrical forces act on things that have chase. There are two types of charge, positive and negative. Two like charges will repel one another, while opposite charges with attract one another. The unit for charge is the coloumb which has the symbol C. Charge is quantised, meaning that the smallest nonzero charge that an objet can have is qe and the charge on any object will be an integer of qe. The value of qe is 1.6 x 10-19 C. Ordinary matter is made of protons, neutrons, and electrons. Protons have a charge of +1.6 x 10-19 C while electrons have a charge of -1.6 x 10-19 C (neutrons are neutral). An object will therefore have a negative charge is it has more electrons than protons, and will have a positive charge if it has more protons than electrons. Charge cannot be created or destroyed, only redistributed. For normal, low energy processes, this is because numbers of protons and electrons don’t change. But charge conservation is much more fundamental than that. For any process, including much higher energy processes (like those happening in nuclear reactions and particle accelerators), charge is conserved. We can charge objects through friction. Rubbing two insulators together can transfer electrons from one to the other. This means that one object becomes positively charged and the other becomes negatively charged.

Conductors & Insulators Conductors are materials where charge can move freely: - Metals - contain electrons that are not bound to any one atom and are free to move. - Salty water - salts disassociate into positive and negative ions when dissolved, and these ions can move freely. Insulators are materials where charge cannot move freely: - Plastics - electrons are bound tightly. - Oil - molecules are free to move but are all neutral and there are no ions. - Glass. - Undissolved salt.

Polarisation Let's suppose we bring a charged rod up to a metallic conductor. The charge movement is very slight within the conductor. This separation of charges in a neutral object is called polarisation. This can also occur in insulators two ways: - The presence of a charged object induces a polarisation in

each atom or molecule.

- By aligning molecules that have a built in charge separation.

Coloumb’s Law Coloumb’s law tells you the force between point charges.

Forces act along the line joining the two charges. The two forces are equal and opposite (an action-reaction pair). The force is repulsive if the two charges have the same sign, and attractive if the two charges have opposite signs. The size of the force is given by:

F =k

| q1 | | q2 | r2

Where k is 9.00 x 109 Nm2 / C2. So from this equation, we know that the force depends on the inverse square of the separation. Suppose we have a charged object Q that is in the presence of two other charges, q1 and q2. To work out the force on Q: - Calculate the force due to q1. - Calculate the force due to q2. - The the vector sum of these two forces.

Lecture 2 Electric Fields & Potential Electric Fields The forces described by Coloumb’s law are due to a two step process.

The reason for the force (F) on q2 is because q1 produces an electric field everywhere in space. q2 experiences a force due to this electric field. Suppose we have a bunch of charges and we want to know about the force on one of them. We have a collection of source charges (Q1, Q2…) and a test charge (q1). We are interested in the force on the test charge. The source charges are the charges that produce a force on the test charge. We can work out the force on the test charge using coloumb’s law. Then suppose that we keep all source charges the same but replace the test charge with a 2μC charge. The force on the test charge is going to change.

This tells us that the force on a charge at point P is equal to the charge, multiplied by some vector. This vector is called the electric field at the point P, and is given the symbol E:

F = qE The magnitude of the electric field at a point a distance away from a source charge of a specific strength is:

|E | =

k|Q| r2

If the source charge is positive, the electric field points away from the source charge. If the source charge is negative, the electric field points towards the source. If there is more than one source charge, then the superposition principle says you calculate the electric fields due to each of the source charges individually, then sum the result.

Electric Potential Let's say we have fixed charge and a moving charge. The electric field from the fixed charge will provide a force on the moving charge that points to the right. The charge will slow down, stop, and then accelerate to the right (as it is being repelled).

Energy is conserved, so as the moving charge approaches the fixed charge, kinetic energy decreases and potential energy increases. As the moving charge starts to accelerate away, kinetic energy increases and potential energy decreases. This potential energy is called electrical potential energy. The electric potential is a scalar field, and V is defined by:

ΔV =

ΔU q

Where ΔU is change in potential energy, and V is voltage, which has the units of volts (V). Just like all other potential energies, we are only ever interested in the difference in the electric potential between two points. Voltage essentially means the difference in potential energy (potential difference).

Lecture 3 Capacitance & Current Capacitance Supposed we have two conductors that are initially uncharged. Then, we transfer an amount of Q from one to the other. Overall the conductors are still neutral, but there is now an electric field between them. Therefore, a potential difference is created. The potential difference is proportional to the charge transferred. The capacitance of conductors is defined as:

|Q| C = |V | The units of capacitance is farads (F). Unless the geometry is specially chosen, capacitances tend to be very small. In order to achieve big capacitances the capacitors used in electronics are generally large area plates that are closely spaced and parallel. The capacitance of a parallel plate capacitor is given by:

C =ε

A d

Where A is the surface area, d is the distance between the plates, and ε is the permittivity of the insulator. Sometimes permittivities are expressed as relative permittivities:

ε = εr × ε0 Where ε0 is the permittivity of a vacuum, equal to 8.85 x 10-12 Fm-1. Relative permittivities for most materials are in the range 1 to 100.

Cell Membranes Cells have very small values for d, and relatively large values for A. This means that there will be a large capacitance. The capacitance of the nerve cell membranes slows down your reaction time. It determines how fast you think.

Energy In Capacitors Once you start to charge a capacitor, further charging involves doing work against the electric field. The energy stored in a capacitor is given by:

U=

1 CV 2 2

The energy does not equal the charge times the voltage because the voltage changes as the amount of charge changes. For each small packet of change in potential energy will be:

ΔU = ΔQ × V

Capacitors In Circuits The total capacitance of a collection of capacitors in parallel is the sum of the total capacitances:

Ctotal = C1 + C2 + C3 This is because each capacitor has the same potential difference between the plates. For a collection of capacitors in series, the total capacitance is given by:

1 Ctotal

=

1 1 1 + + C1 C2 C3

In this case, the charge on each capacitor is the same, and the potential difference across all the capacitors is the sum of the individual potential differences.

Adding capacitors in serious reduces the capacitance in series therefore reduces the capacitance. For capacitors in series, the capacitance is smaller than any of the individual capacitances.

Electric Current Most of the common uses of electricity involve a flow of charge. This flow is called electric current. The electric current is defined as the amount of charge per unit time that is crossing some area:

I=

ΔQ ΔT

The unit for current is the ampere (A). Positive charges moving one direction produce the same current as negative charges moving in the opposite direction. The conventional current points in the direction that positive charge carries would need to flow. Conduction in metals is due to the movement of negative electrons. This means that the current is in the opposite direction to the movement of the electrons.

Lecture 4 Ohms Law & Electric Circuits Resistance & Ohms Law For the vast majority of materials:

V∝I The potential difference across an object and the current flowing through it are related by:

V = IR R is resistance of the object, measured in ohms (Ω). Good conductors, like metal wires, have very small resistance. The resistance of an object depends on its geometry. For an object with a uniform composition and cross section, resistance is given by:

R=ρ

l A

ρ is the resistivity of the material, l is the length of the object, and A is the area of the object.

Simple Circuits In most applications, charge is recycled in a circuit. Here is a simple circuit:

The ideal battery keeps a constant potential difference across its terminals. This potential difference is called the electromotive force (EMF) of the battery (ε). Electrons are pushed through the resistor due to this EMF.

Kirchhoff’s Laws There are two basic rules that are used when analysing circuits, called Kirchhoff’s laws: 1. The sum of potential changes ground a loop are zero (due to conservation of energy). 2. The current flowing into a junction is the same as the current flowing out (due to conservation of charge).

Because the electric forces are conservative:

ΔVAB + ΔVBC + ΔVCD + ΔVDA = 0 Note the minus sign in ΔVBC, because in the direction of the current, the potential drops as you go across a resistor.

I1 = I2 + I3 Energy In The Circuit The electrons travel through the resistor in the same direction as the electric force upon them. The electric force does work on the electrons, transferring energy to them. The electrons are continually losing this energy as they bash into metal atoms. As a result, electrical energy is turned into heat. When electrons go through the battery, the force and the movement are in opposite directions. Negative work is done on the electrons, but they don’t slow down. Energy is provided by the battery. As a result, chemical energy is turned into electrical energy.

Resistors In Series & Parallel Resistors in series are those connected end upon end:

The effective resistance is the sum of the individual resistances:

RS = R1 + R2 + R3 So adding resistors in series results in a bigger resistance. This is consistent with our formula for resistors with a uniform composition and cross section, as longer length means more resistance. Resistors in parallel all share the same two terminals. For resistors in parallel the effective resistance is:

1 1 1 1 RP = R1 + R2 + R3 Adding resistors in parallel results in a smaller resistance. This is consistent with our formula for resistors with uniform composition and cross section, as a bigger cross sectional area means less resistance. Note that the series and parallel rules for resistors are opposite to the rules for capacitors.

Lecture 5 Power, Electric Shocks & Capacitors In Circuits Energy In Electric Circuits Supposed we have a circuit element with a potential drop across it of V and a current flowing through it of I. Every Δt, IΔt of charge flows through the resistor. Each coulomb of charge dissipates V of energy.

P=

e n erg y e n erg y ch arge × = VI = time ch arge time

If the potential decreases in the direction of current flow, the power is dissipated. For the potential to increase in the direction of the current flow, power input is required. We can also almost resistance in the equation for power:

P=

V2 = I 2R R

Electrical Safety Electrical current in the body produces many varied effect. It is sometimes even beneficial. Most fatalities are due to heart fibrillation. The major factors determining how dangerous an electric shock is are: - The amount of current (not necessarily the voltage). - The path taken by the current. - The duration of the shock. - The frequency (direct current is more dangerous that alternating current). The effect of a one second shock across the trunk of an average male is shown by this table: Current (mA)

Effect

1

You can feel it.

5

Maximum harmless current.

10-20

Involuntary muscular contraction.

100-300

Ventricular fibrillation (often fatal).

>300

Burns.

A standard 12V car battery is capable of producing a current over 200A. Is it dangerous? To answer this we need a model of the human body:

RTOTAL = RSKIN + RIN TER IOR = RSK IN The current that flows if you are exposed to 12V between your hands is equal to:

I=

12 V = = 120 × 10−6 A R 100 × 103

The current that flows if you are exposed to 230V between your hands is equal to:

I=

V 230 = = 2 × 10−3 A 3 R 100 × 10

So in terms of a car battery electric shock, you will feel it, but it won’t kill you if you can let it go.

Capacitors As Circuit Elements All circuits considered so far have been steady state. When the circuit is energised this steady state is reached almost instantly. In contrast, biological systems produce electric pulses which take a really long time to turn on and turn off. The reason for this slow turn on and turn off behaviour is because of the capacitance of the membrane. We need to understand how capacitors behave in circuits. Charging A Capacitor In the case where initially the capacitor is uncharged, the switch is closed at t = 0.

Discharging A Capacitor In the case where initially the capacitor is charged, the switch is closed at t = 0.

The timescale of decay or increase is given by:

τ = RC

This is called the time constant. For the discharging of a capacitor:

V = V0e

−t τ

For the charging of a capacitor: t

V = V0(1 − e − τ ) Where V0 is the across the capacitor....