st04 – electronics – particle level: forces and fields 1 electronics – particle level: forces...

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ST04 – Electronics – particle level: forces and fields

Electronics – particle level: forces and fields

Lecturer:Smilen Dimitrov

Sensors Technology – MED4

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Introduction

• The model that we introduced for ST

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Introduction

• Goal – continue with focus on electronics• Discuss concept of electrostatic field and potential• Discuss graphical solving of electrostatic field from point charges• Concept of conservative force fields• Concept of electric voltage as difference of electric potential

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Electrostatics

• Time is not taken into account directly – forces are static, “frozen” in time - hence electrostatics

• Time can be resolved through usage of second Newton law F=ma (through which we can derive coordinates and time)

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Electrostatic force (Coulomb force)

• The force that occurs between electrically charged particles.

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Action at a distance and fields

• The electrostatic force is a non-contact force, and it apparently acts over a distance.

• Problem, since action-at-a-distance in physics means that there is no known mechanism that would account for the force acting

• Field theory – Faraday - the medium for transferring a force between two objects on a distance is a field – a mechanism existing in the empty space

• The field - as a specific disturbance in [empty] space, which can be perceived as stress in the space

• We can only detect a field if we place a charged particle in it and measure a force acting on it

• Field is a mechanism for electrostatic force – not action-at-a-distance anymore

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Electrostatic field of a single point charge

• Look for the el-stat. force between a source charge Q and a (small) test charge q, at all points in space

• Then eliminate the test charge from the equation

• You get a vector assigned to each point in space, only due to source charge Q – electrostatic field vector E

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Electrostatic field of two point charges

• We can look for the electrostatic field from two point charges purely graphically

Again we have an electrostatic field vector assigned to each point in space.

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Electric dipole

• The previous example was about two point charges of same sign• A structure which is very common is one with two point charges of

opposite sign at a given distance from each other – electric dipole

For instance, the water molecule can be seen to behave like a dipole – in an electric field, dipoles turn to align with the field.

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Work and energy

• We have described the electric state of space around a source charge with a vector (electrostatic field) – we would like to use a scalar quantity

• Possible through introduction of electrostatic potential

• Potential is only defined for conservative force fields

• These are defined in terms of work and energy

• Work – anytime we conclude that a displacement of an object occurs in direction of the applied force; force times distance, measured in Joules

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ST04 – Electronics – particle level: forces and fieldsConservative nature of the electrostatic field and potential energy

• For systems which involve conservative force fields, there is an “equilibrium” state – meaning a state of the system if it is left alone, after an infinite amount of time (for two like charges – infinite distance apart; two unlike charges – zero distance apart)

• Any difference from this state, means that work was done to change the state

• Electrostatic and gravity fields are conservative

• Conservative force fields will always tend to return to the equilibrium state

• This means that they will always try to ‘return’ the work (invested in changing the original equilibrium state) – that is, they can ‘store’ energy

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• Objects placed in a conservative field thus gain energy due to a position in the field – potential energy

• For example – a positive test charge q at distance d from a source charge Q has some potential energy due to this position (which indicates that work was invested to bring the test charge from equilibrium at infinity, to the distance

d) • The source charge field ‘tries’ to return the invested work, by

exerting electrostatic force on the test charge in order to restore equilibrium

• To look for potential energy U, we find the work done us, in moving a test charge from a to b against the conservative force – force times distance (as an integral)

• Negative sign (field returns work)

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• Finally – the potential energy U a test charge q obtains, by being placed in the field of source charge Q at distance r is

• If we eliminate the test charge q from the equation, we obtain the electrostatic potential V

• V is now only due to source charge• To find the potential energy a test charge q gains in the field of

source charge Q, we just have to know the potential V (at distance r) and the test charge q

• Force is a derivative of potential energy

r

Qqk

r

QqrU

1

4)(

0

r

QkV

qrVrU )()(

dr

dUrF )(

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• Important notes about conservative force fields

– The work done in taking a charge around a closed loop in an electric field generated by fixed charges is zero.

– The work done in taking a charge between two points in an electric field generated by fixed charges is independent of the path taken between the points

• This means that an electrostatic field can never sustain directed current on its own - as it cannot move a charge placed in it in a closed loop

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• Important notes about conservative force fields– It should be possible to associate a potential energy (i.e., an energy a body

possesses by virtue of its position) with any conservative force-field. – Any force-field for which we can define a potential energy must necessarily

be conservative. For instance, the existence of gravitational potential energy is proof that gravitational fields are conservative.

– The concept of potential energy is meaningless in a non-conservative force-field (since the potential energy at a given point cannot be uniquely defined).

– Potential energy is only defined to within an arbitrary additive constant. In other words, the point in space at which we set the potential energy to zero can be chosen at will. This implies that only differences in potential energies between different points in space have any physical significance.

– The difference in potential energy between two points represents the net energy transferred to the associated force-field when a body moves between these two points. In other words, potential energy is not, strictly speaking, a property of the body--instead, it is a property of the force-field within which the body moves.

– The sum of the kinetic energy and the potential energy remains constant as the body moves around in the force-field.

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Electrostatic potential

• We have already come to the concept – potential energy with the test charge q eliminated

• Proper: start by definition of the change in potential energy ΔU associated with moving a particle from position r1 to r2 against a given conservative force

• Then, define the change in the electric potential ΔV in terms of the change in the electric potential energy of a given charge, per unit charge

• electrostatic potential is now a scalar field (function on all the coordinates of space) from which we can easily find the (change in the) potential energy of a given charge

• Measured in Joules / Coulomb, more commonly Volts [V]

1

0

1

0

)()(r

r

r

r

rdrEqrdrFU

1

0

)(lim0

r

rq

rdrEq

UV

VqU

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Electrostatic potential

• Generally two ways to represent potential in graphical terms:

• Change of brightness (potential) across a spatial coordinate – gradient

• Electrostatic field is mathematically gradient of electrostatic potential

• Means – we can extract direction of the field E (vector) by looking at the change of potential V (scalar) across r (direction – along r, orientation by the sign)

1. The value of the potential in a point is mapped to a brightness or color of the point.

2. All points that have the same potential are connected on a line known as an equipotential line.

Vr

VE

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Review: Force, Field, Potential Energy and Potential

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ST04 – Electronics – particle level: forces and fieldsPoster: negative test charge (q) in the field of negative source charge (Q)

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ST04 – Electronics – particle level: forces and fieldsPoster: negative test charge (q) in the field of positive source charge (Q)

• _

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Fields and potential visualisation

Scalar fields representation

Vector fields representation

1. Contour Maps – Equipotential Lines

2. Color-Coding 3. Relief Maps

1. Particle Sources and Sinks In Fluid Flows

2. The “Vector Field” Representation of A Vector Field

3. The “Field Line” Representation Of A Vector Field

4. “Grass Seeds” and “Iron Filings” Representations

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Fields and potential visualisation

Important things to remember:

1. The direction of force vectors / field vectors / field lines depend on what we consider as a test charge, but most common directions are for negative test charge – negative source charge radiates outwards (away from itself), positive source charge radiates inwards (toward itself).

2. Electrostatic field lines are always tangent to the electric field vector direction at any point.

3. The equipotential lines (or surfaces in 3D) represent points that have the same electrostatic potential.

4. The equipotential lines and field lines must be normal (perpendicular) to each other at all times.

5. Relief maps (a 3D plot of the potential) can be related to “hills” and “valleys” in the gravitational field – thus simulating the common conception we have of potential: for a body with mass, it is hard to go uphill and easy to go downhill due to gravitation – though on a plain, there will be no influence to movement just because of gravity (although there will be a gravity pull, or weight). Of course, here we talk about electrostatic charge instead of mass, and electrostatic field instead of gravitational field.

These are several applets where different electrostatic fields can be visualised, in several of the abovementioned ways.

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Electrostatic potential of distribution of charge

• The task of finding field is complex for many charges (arranged in arbitrary collection)

• Can be made a bit easier by looking at the problem graphically through electrostatic potential

• The color brightness of a given point in space as an indication of the electrostatic potential

• we can notice that it changes depending on the number of charges that extend their electrostatic influence in that point in space

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Electric voltage – difference of electrostatic potential

• Work done by the external force is equal to the change in the electrostatic potential energy of the particle.

• The difference in electrostatic potential between two points is the work required to move one unit charge (the work for a different

amount of charge can simply be obtained by multiplication)• Electric voltage - difference of electric

potential (potential difference) between two points in a field.

2112 VVU

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• Work done by the external force is equal to the change in the electrostatic potential energy of the particle.

• The difference in electrostatic potential between two points is the work required to move one unit charge (the work for a different

amount of charge can simply be obtained by multiplication)• Electric voltage - difference of electric

potential (potential difference) between two points in a field.

2112 VVU

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• Applet

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Wave aspect of electrostatic field

• As the charges move, they ’carry’ their fields with them• How fast the field can change in space is limited by the speed of

light c• The change of the electric field in space is wave-like; but wave

phenomena need restoring forces. Here, must involve magnetic phenomena (not discussed in this course)

• Thus, dynamically we talk about electro-magnetic force (Maxwell)

Can be visualised as a wave-like change of potential:

st04 – electronics – particle level: forces and fields 1 electronics – particle level: forces...

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