Tuesday week 10

Electric potential

PE(electric) per unit charge.

Lets first consider gravity and a two dimensional surface.  The hill represent the gravitational potential.  This is quantified by the height at each point.  Let us consider the lowest point and select this point to be called h=0.  We can now label each point in our space with a value of h.   If we were to place a ball at this point it would have gravitational PE of mgh.

It is critical that you understand that potential and potential energy are different but related concepts.  For gravity the fact that g is treated as a constant for problems on the earthÕs surface means that h is a measure of the gravitational potential (i.e. proportional to the gravitational potential).

If you were running a restaurant and sales of various items (dinners, wine, desserts and appetizers) were proportional to the number of people that visited the restaurant. Then the success or failure would hinge on the number of people.  The actual dollar amounts could be calculated and compared with costs.  The owner however might phrase the success or failure in terms of numbers of people.  I need at least 300 people per night to succeed.

For record keeping I could record all the values of h.  It would then be a simple matter to multiply each value of h by g to get the actual potential.  Most of the information related to the gravitational potential is provided by just giving the heights.

Given the value of h for all points on our surface we can construct the terrain.  In this case we need to combine the force of gravity and the normal force of the hill so as to define the Ņclimbing forceÓ.  Thus the terrain will determine how difficult it is to move between two point in terms of the force necessary to climb.  On a level surface you do not need to climb at all. Two interesting aspects of the terrain are direction a ball would role if placed at a point in our space.  An obvious answer is downhill (opposite the climbing force required).  The direction of this Ņclimbing force will be the direction where the height is changing the greatest.  Perhaps surprisingly, the direction perpendicular to this direction will have no change in height and there is no component of the force along this direction.  This of course is the way many maps are labeled. Lines are drawn on the terrain where one can walk without changing height.  These contours are usually drawn at regular intervals h=0,1,2,3É ft.

The goal of this picture is to help relate the potential and field.  For every point on path along a terrain you can imagine a force that can be expended in the direct uphill direction and the size of this force is based on how your height changes in this direction (steepness).  If you choose a direction other than directly up the hill the less you climb and the less force you will need to exert. However you will need to walk a greater distance to get to the same height.  Energy conservation, force, work, PE, field and potentials are tangible parts of this problem.

This picture might give you some insight into electric fields and electrical potentials.  You may realize that characterizing your path by height is sufficient.  The details of the climb are contained in this information.  It is important to know that you will need to climb as you go up the mountain and therefore there are forces at play but they are clearly contained just knowing the height changes.  Most of the relevant features of a circuit are given by measuring and/or calculating voltage.  Yes the charges are being pushed around by the electric field and it is large in some regions and small in other regions but we never need to calculate the field or the forces we just deal with voltage.

If there is an electric field then the electrical potential can be defined as the potential energy per unit charge.

Because the force law for gravity is straightforward

while the force law for electric fields can be complicated, relating all the aspects of a field is simpler for the gravitational case (at least on the surface of the earth where g is treated as a constant).

FOR ELECTRIC CIRCUITS MOST ANALYSIS CAN BE PERFORMED USING ELECTRIC POTENTIAL V WITHOUT WORRYING ABOUT THE FIELD OR THE ELECTRIC FORCE.

THE COMPONENTS CREATE PATHS THROUGH SPACE SIMILAR TO A TRAIL ON A MOUNTAIN.  ONE CAN DETERMINE THE CHANGE IN HEIGHT ALONG A MOUNTAIN PATH OR THE CHANGE IN VOLTAGE ALONG CIRCUIT PATH.  THE ANALYSIS OF AN ELECTRIC CIRCUIT RELIES ON THIS CHANGE IN VOLTAGE AND THE PROPERTIES OF THE TRAIL (RESISTANCE) TO FIND THE CURRENT.

 BASIC IDEAS Fields (E,B,G) & how they are created, Forces (electric, magnetic, gravitational) on charges and masses in these fields, energy, power,  general case where charges and mass  move in fields, special case where a flow can characterizes the motion (current¸ charge/sec=Amp), work (done on a mass or charge moving through a field), potential energy, potential (V¸voltage).   The general case simplifies for electric circuits where paths are defined and a flow characterizes the steady state motion. Voltage can be found along these paths. Energy, work, power, can be determined. A charge [or mass] that experiences a change of V [height] gains or losses energy  The equations above are written two ways.  The key to our analysis will be finding changes in potential.  The change in a quantity  However in many cases we can simplify the problem by choosing    or in some cases the formulation requires the student to recognize that the change is what is meant. So in traversing a circuit a charge will gain and loose energy.  It gains energy as it passes through the PS and looses energy in the resistors.  This can be seen simply by looking at how the voltage changes along this path. Power can be determined in a system of objects moving along paths.  For any segment  (again here the change in voltage is relevant). So if a group of backpackers transport food up to a mountain restaurant the energy to get the food to the top can be calculated and knowing how fast the backpackers are moving the power or energy/unit time employed to lift the food can be ascertained.  If the backpackers move slowly then they still must invest the same energy but not at the same rate as when they move quickly (higher power as speed increases). Since the people are moving one can assign a flow to the transport of food.  Students can imagine a line of  people transporting or a conveyor belt that moves the food.  In each case a flow can be imagined as the amount of stuff that passes a point on the trail in a given time [lbs of food/ hour ¸ flow].  Here the power will depend only on the flow and the height that the food is lifted. Fields (forces) can be found if the potential is known.  For gravity we discussed the fact that at every point on a hill there is a direction of steepest descent. Perhaps this reassures the student that both a direction and magnitude can be determined so that there is a relationship that defines field in terms of the potential and vice versa.  In any case one can find the field from the potential and the potential from the field but the method will not be covered in detail in this class. It will be important that the student recognize that a ball can be  lifted at constant speed by an external force in a gravitational field.  The work done by this force will then be transformed into PE and the potential of the ball will be changed as it is lifted.  Thus external energy can be added to the ball to change its potential (and of course PE). So as the potential changes one could formulate the problem in terms of forces pushing things around. However, although this is a valid way to think about the problem most analysis involves V, I, R and not E.

Electric circuits

Several models will be introduced to help explain electrical concepts such as voltage current, resistance and capacitance. We will discuss three models. Some ideas are presented below but these ideas will need to be developed.

 Model Description Stair Model People walk (current) through corridors (wires) are pulled upstairs (power supplies) and are propelled by gravity down staircases (resistors). Ski Slopes Skiers ski (current) down hills (pathways: resistors and wires). A lift (power supply) pulls them to the top of the hill (voltage). Water Model Pump (power) lifts water to a specific height (voltage). Water flows through pipes (resistors).

In order to understand electricity we need to develop an intuition (understanding) about the electrical quantities.  With mathematics we can precisely define what we mean by these quantities. For example a resistor is an object that allows electric current to flow through it according to Ohm's law.  Ohm's law states that V=IR.  Knowing the math I understand what a resistor is. An alternative to using mathematics is to make measurements with many resistors in many situations so that you know what will happen based on experience. Then you combine these measurements with a model that has the correct behavior. This is what we are trying to do. We gain insight by establishing models.  Models usually have limitations.  If one wants to compare electrical circuits with skiers one needs to restrict skiers from climbing hills under their own power.  Also charges in an electric circuit tend to move at an average speed at all points in the circuit whereas skiers typically speed down some sections and stop and rest at points.  To actually correctly model circuits are skiers need to continually fall or stop every few feet then get up and keep going so that they have a slow average speed with no overall accumulation of kinetic energy.

The table summarizes the quantities of importance and their relationship to our models.

 Electric Quantity Analog Idea Voltage Volts V Drives current through a circuit. Pushes + and - charges. Height; provides the push or pressure.  Water tower. Charge Coulombs two types + and -, like repel. Unlike attract. Water, skiers, people Current Amps I Flow of electrical charge. Flowing water, moving stuff. Resistance Ohms W Bottle neck,  V=IR. Tubes: large resistors are thin tubes, small resistors are wide tubes. Wires Pathways for current flow with no resistance. Very large diameter tubes. Capacitance farad C Ability to hold or store charge, q=C/V. Bucket. LED Current flows in one direction only and they have a turn-on voltage. One way valve with a spring holding it closed. Voltages are same across parallel components. Water pressure in pipes at same level is the same. Voltages add along a series of components. Standing on a chair, yopur height is the sum of you + chair. Current entering a series sequence of components is the current that leaves. Put water in one end it comes out the other. Current splits at circuit junctions. Total = sum of the parts. Water divides in pipes.

 For circuits A circuit consists of pathways. Charge moves along the pathways and voltage changes. Voltage add along paths in series Voltage is the same across paths in parallel Current divides at Ts in the path such that the total in equals the total out Current along paths in series is the same Each resistor needs a share of the voltage to drive the current through it according to OhmÕs law

The analysis so far has dealt with DC circuits.  Here you turn on the circuit and wait a nanosecond for things to adjust and then analyze the steady state properties of the circuit.  Now we would like to consider an almost identical situation but allow the parameters of the circuit to change slowly in time.  To consider these characteristics one plots the current/voltage as a function of time.

Plot can be either the voltage on y-axis versus time on the x-axis or it could represent the current vs time.

Voltages and currents in our homes follow these sinusoidal variations at a frequency of 60 Hz.

Could represent a clock to regulate a process such as a computer algorithms stepping through a process.

Data analog and digital.

Data about our world will be represented as an electrical value (current or voltage).  The data can change in time.  If we examine our two plots the first could represent an analog piece of information.  It could represent a microphones response to a tuning fork.  The sound doesnÕt start as an electric signal but the microphone a transducer converts it.  Electrically we see a voltage that is changing in time with the same features that the pressure on our ear drum feels from the vibrating tuning fork.  The second signal might be a digital signal representing the logical (Yes/No) nature of some data.  It could represent the answer to the question is it daytime (the time axis would not about 2 hours per division so that the cycle would correctly represent 24 hours).

Motivation----------------------------

Lets imagine that we want to move a robot what is the most straightforward approach?

Wheels and a motor.

What does the motor do?

Rotates at some speed that turns the wheel.

How do we regulate the speed?

Gears & power to the motor.  [car you can shift and you can press on the gas]

Hydraulic systems have same facility.  Change force

Here is where the mechanics that we considered would be applied. Just like in the problem with the pulleys [block and tackle] you can get different forces but not an overall increase in the energy.  Also the power or energy delivered per unit time will be important.

Source ¸ How much energy does it contain and how fast can it deliver it.

Most robots use electric motors and many sensors are electric.

Sensor to detect an obstacle ¸ switch  [try to be simple.  If you donÕt understand whether a sensor is simple or complicated it shows that you need a clearer understanding!  Is it simple because you know where to get the stuff. Simple because you have seen it work somewhere. Simple because the implementations requires only basic concepts and components]

switch ¸ basic

camera ¸ complex but available and software is available but how do you detect distance with a camera.?

ultrasonic ranger ¸ available but complex in principle but the complexity may be removed if you find the correct sensor that does all the work.

A sensor (also called detector) is a device that measures a physical quantity and converts it into a signal which can be read by an observer or by an instrument

 typical temperature Position via on/off switch there not there Range - ultrasonic How far, speed, acceleration motion Multiple methods gps Location, speed, altitude grade gyroscopes Orientation Voltage, power Monitor robots energy and delivery force pressure Hydraulic systems vision Multi faceted (facial rec., location, threat [cliff], object id) color IR Imaging or point conductance Distinguish metal nonmetal acoustic Volume, distinct sounds, voice recognition, commands magnetic Earths field as a  reference Beacons/sensing Send out a signal that is used for navigation or other other radiation chemical weather Humidity, wind, cloud cover (radar) flow Fluids electric Capacitance, resistance, inductance, É.

Some of the material was not covered so it is repeated but there are some additional comments.

Thursday Nov 3, 2011, week 10

CIRCUITS SIMPLIFY THE APPLICATION OF THE GENERAL PRINCIPLES OF E&M.  CIRCUITS BEHAVIOR IS LIMITED BY THE COMPONENTS.

FOR ELECTRIC CIRCUITS MOST ANALYSIS CAN BE PERFORMED USING ELECTRIC POTENTIAL V WITHOUT WORRYING ABOUT THE FIELD OR THE ELECTRIC FORCE.

THE COMPONENTS CREATE PATHS THROUGH SPACE SIMILAR TO A TRAIL ON A MOUNTAIN.  ONE CAN DETERMINE THE CHANGE IN HEIGHT ALONG A MOUNTAIN PATH OR THE CHANGE IN VOLTAGE ALONG CIRCUIT PATH.  THE ANALYSIS OF AN ELECTRIC CIRCUIT RELIES ON THIS CHANGE IN VOLTAGE AND THE PROPERTIES OF THE TRAIL (RESISTANCE) TO FIND THE CURRENT.

 DC-PS Battery or DC-power delivers a fixed voltage and whatever current is required. R Resistors require a voltage difference to push current through them. The amount is based on OhmÕs law Along a path that spans the PS all elements get a share in the voltage.  There is a voltage budget and each element gets a fraction of the total. wires A wire is a resistor with resistance approaching zero.  The ideal wire has zero resistance and so the current flows through without a change in the voltage across the wire. [It is problematic to have special rules for wires.  Our sense of the workings of a system are complicated by the number of separate rules that need to be followed. By realizing that a wire is a resistor with very small resistance we do not add a new rule rather name certain components with special properties (small R). ]

[There is an ambiguity in notation.  When finding V it is always a voltage difference. The  is understood and so is not always explicitly stated.]

The equation above tells us that  the amount of charge  that  moves by a point in  will gain an amount of energy  so that the energy delivered per unit time is  .

So here the power is related to charge moving across some potential difference.  If there are three resistors in series then you can calculate the power that is lost as the current passes through each resistor or the power lost from all three. You determine the section of the circuit 1,2,3 resistors. Find the voltage drop across the chosen path and multiply by the current. You can calculate how much power is provided by the PS by noting the change in voltage across the poser supply and the current through it.

Electric circuits

Several models will be introduced to help explain electrical concepts such as voltage current, resistance and capacitance. We will discuss three models. Some ideas are presented below but these ideas will need to be developed.

 Model Description Stair Model People walk (current) through corridors (wires) are pulled upstairs (power supplies) and are propelled by gravity down staircases (resistors). Ski Slopes Skiers ski (current) down hills (pathways: resistors and wires). A lift (power supply) pulls them to the top of the hill (voltage). Water Model Pump (power) lifts water to a specific height (voltage). Water flows through pipes (resistors).

In order to understand electricity we need to develop an intuition (understanding) about the electrical quantities.  With mathematics we can precisely define what we mean by these quantities. For example a resistor is an object that allows electric current to flow through it according to Ohm's law.  Ohm's law states that V=IR.  Knowing the math I understand what a resistor is. An alternative to using mathematics is to make measurements with many resistors in many situations so that you know what will happen based on experience. Then you combine these measurements with a model that has the correct behavior. This is what we are trying to do. We gain insight by establishing models.  Models usually have limitations.  If one wants to compare electrical circuits with skiers one needs to restrict skiers from climbing hills under their own power.  Also charges in an electric circuit tend to move at an average speed at all points in the circuit whereas skiers typically speed down some sections and stop and rest at points.  To actually correctly model circuits are skiers need to continually fall or stop every few feet then get up and keep going so that they have a slow average speed with no overall accumulation of kinetic energy.

The table summarizes the quantities of importance and their relationship to our models.

 Electric Quantity Analog Idea Voltage Volts V Drives current through a circuit. Pushes + and - charges. Height; provides the push or pressure.  Water tower. Charge Coulombs two types + and -, like repel. Unlike attract. Water, skiers, people Current Amps I Flow of electrical charge. Flowing water, moving stuff. Resistance Ohms W Bottle neck,  V=IR. Tubes: large resistors are thin tubes, small resistors are wide tubes. Wires Pathways for current flow with no resistance. Very large diameter tubes. Capacitance farad C Ability to hold or store charge, q=C/V. Bucket. LED Current flows in one direction only and they have a turn-on voltage. One way valve with a spring holding it closed. Voltages are same across parallel components. Water pressure in pipes at same level is the same. Voltages add along a series of components. Standing on a chair, yopur height is the sum of you + chair. Current entering a series sequence of components is the current that leaves. Put water in one end it comes out the other. Current splits at circuit junctions. Total = sum of the parts. Water divides in pipes.

 For circuits A circuit consists of pathways. Charge moves along the pathways and voltage changes. Voltage add along paths in series Voltage is the same across paths in parallel Current divides at Ts in the path such that the total in equals the total out Current along paths in series is the same Each resistor needs a share of the voltage to drive the current through it according to OhmÕs law

The analysis so far has dealt with DC circuits.  Here you turn on the circuit and wait a nanosecond for things to adjust and then analyze the steady state properties of the circuit.  Now we would like to consider an almost identical situation but allow the parameters of the circuit to change slowly in time.  To consider these characteristics one plots the current/voltage as a function of time.

Plot can be either the voltage on y-axis versus time on the x-axis or it could represent the current vs time.

Voltages and currents in our homes follow these sinusoidal variations at a frequency of 60 Hz.

Could represent a clock to regulate a process such as a computer algorithms stepping through a process.

Data analog and digital.

Data about our world will be represented as an electrical value (current or voltage).  The data can change in time.  If we examine our two plots the first could represent an analog piece of information.  It could represent a microphones response to a tuning fork.  The sound doesnÕt start as an electric signal but the microphone a transducer converts it.  Electrically we see a voltage that is changing in time with the same features that the pressure on our ear drum feels from the vibrating tuning fork.  The second signal might be a digital signal representing the logical (Yes/No) nature of some data.  It could represent the answer to the question is it daytime (the time axis would not about 2 hours per division so that the cycle would correctly represent 24 hours).

Motivation----------------------------

Lets imagine that we want to move a robot what is the most straightforward approach?

Wheels and a motor.

What does the motor do?

Rotates at some speed that turns the wheel.

How do we regulate the speed?

Gears & power to the motor.  [car you can shift and you can press on the gas]

Hydraulic systems have same facility.  Change force

Here is where the mechanics that we considered would be applied. Just like in the problem with the pulleys [block and tackle] you can get different forces but not an overall increase in the energy.  Also the power or energy delivered per unit time will be important.

Source ¸ How much energy does it contain and how fast can it deliver it.

Most robots use electric motors and many sensors are electric.

Sensor to detect an obstacle ¸ switch  [try to be simple.  If you donÕt understand whether a sensor is simple or complicated it shows that you need a clearer understanding!  Is it simple because you know where to get the stuff. Simple because you have seen it work somewhere. Simple because the implementations requires only basic concepts and components]

switch ¸ basic

camera ¸ complex but available and software is available but how do you detect distance with a camera.?

ultrasonic ranger ¸ available but complex in principle but the complexity may be removed if you find the correct sensor that does all the work.

A sensor (also called detector) is a device that measures a physical quantity and converts it into a signal which can be read by an observer or by an instrument

 typical temperature Position via on/off switch there not there Range - ultrasonic How far, speed, acceleration motion Multiple methods gps Location, speed, altitude grade gyroscopes orientation Voltage, power Monitor robots energy and delivery force pressure Hydraulic systems vision Multi faceted (facial rec., location, threat [cliff], object id) color IR Imaging or point conductance Distinguish metal nonmetal acoustic Volume, distinct sounds, voice recognition, commands magnetic Earths field as a  reference Beacons/sensing Send out a signal that is used for navigation or other other radiation chemical weather Humidity, wind, cloud cover (radar) flow fluids electric Capacitance, resistance, inductance, É.