# st part forcefield 04

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ST04 – Electronics – particle level: forces and fields

Electronics – particle level: forces and fields

Lecturer:
Smilen Dimitrov

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ST04 – Electronics – particle level: forces and fields
Introduction

•   The model that we introduced for ST

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ST04 – Electronics – particle level: forces and fields
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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ST04 – Electronics – particle level: forces and fields
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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ST04 – Electronics – particle level: forces and fields
Electrostatic force (Coulomb force)

•   The force that occurs between electrically charged particles.

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ST04 – Electronics – particle level: forces and fields
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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ST04 – Electronics – particle level: forces and fields
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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ST04 – Electronics – particle level: forces and fields
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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ST04 – Electronics – particle level: forces and fields
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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ST04 – Electronics – particle level: forces and fields
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 fields
Conservative 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
•   Conservative force fields
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

•   Electrostatic and gravity fields are conservative

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

•   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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ST04 – Electronics – particle level: forces and fields
Conservative nature of the electrostatic field and potential energy

•   Finally – the potential energy U a test charge q obtains, by being placed in
the field of source charge Q at distance r is
Qq 1     Qq
U (r )        k
4 0 r    r
•   If we eliminate the test charge q from the equation, we obtain the
electrostatic potential V
Q
V k
r
•   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
U (r )  V (r )  q

•   Force is a derivative of potential energy                    dU
F (r )  
dr
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ST04 – Electronics – particle level: forces and fields
Conservative nature of the electrostatic field and potential energy

•   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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ST04 – Electronics – particle level: forces and fields
Conservative nature of the electrostatic field and potential energy

•   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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ST04 – Electronics – particle level: forces and fields
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                        r1
       
r1
        
U    F (r )  dr  q  E (r )  dr
r0                 r0

•   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
U
r1
        
V  lim         E (r )  dr
q 0 q
r0
•   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 U  qV
•   Measured in Joules / Coulomb, more commonly Volts [V]

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ST04 – Electronics – particle level: forces and fields
Electrostatic potential

•   Generally two ways to represent potential in graphical terms:
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.

•   Change of brightness (potential) across a spatial coordinate – gradient
•   Electrostatic field is mathematically gradient of electrostatic potential
   V
E     V
r

•   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)

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ST04 – Electronics – particle level: forces and fields
Review: Force, Field, Potential Energy and Potential

•   _

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

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

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ST04 – Electronics – particle level: forces and fields
Fields and potential visualisation

Scalar fields representation
1. Contour Maps –                       2. Color-Coding                     3. Relief Maps
Equipotential Lines

Vector fields representation
2. The “Vector Field”        3. The “Field Line”     4. “Grass Seeds” and
1. Particle Sources and   Representation of A          Representation Of A     “Iron Filings”
Sinks In Fluid Flows      Vector Field                 Vector Field            Representations

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ST04 – Electronics – particle level: forces and fields
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
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
field.

These are several applets where different electrostatic fields can be visualised, in several
of the abovementioned ways.
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ST04 – Electronics – particle level: forces and fields
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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ST04 – Electronics – particle level: forces and fields
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. U  V  V
12        1       2

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ST04 – Electronics – particle level: forces and fields
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. U  V  V
12        1       2

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ST04 – Electronics – particle level: forces and fields
Electric voltage – difference of electrostatic potential

•   Applet

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ST04 – Electronics – particle level: forces and fields
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:

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