# MOTION contributors include Maxwell whom developed electromagnetic theory

MOTION OF CHARGES IN NON-PARALLEL ELECTRIC FIELD CONFIGURATION
By: Tlotlo Montshwari Oepeng
201502772
Supervisor: Dr M. Tshipa

1.1 INTRODUCTION
The electromagnetic force between charged particles is one of the fundamental forces of nature. It was developed due to contributions from many scientists. The region around a charged object where it exerts a force is called its electric field. Another charged object placed in this field will have a force exerted on it. Coulomb’s rule is used to calculate this force.

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Coulomb, a French physicist, made a detailed study of electrical attractions and repulsions between various charged bodies and concluded that electrical forces follow the same type of law as gravitation. Coulomb found a similar principle linking the relationship of magnetic forces. He believed electricity and magnetism, however, to be two separate ‘fluids’. It was left to Hans Christian Oersted, Andre-Marie Ampere and Michael Faraday to enunciate the phenomenon of electromagnetism. In 1831 – following the demonstration by Hans Christian Oersted that passing an electric current through a wire produced a magnetic field around the wire, Michael Faraday had shown that when a wire moves within the field of a magnet, it causes an electric current to flow along the wire, electromagnetic induction. contributors include Maxwell whom developed electromagnetic theory of light propagation 19th century CITATION Ros00 l 1033 m Geo12 1, 2 .

The basic electric quantity is a charge, an isolated charge is surrounded by electric field that exerts a force on all other charges. This Q1 at r from Q2 experiences a coulombs force
F=KeQ1Q2 r2 r(1.1)
A region is said to be characterized by an electric field if particle q moving with velocity v experiences a force Fe, independent of v.
Fe= qE(1.2)
Equation (1.2) plays an analogous role to Schrödinger’s equation and allows to specify position of a charge at all future time in an electric field by using Newton’s second law CITATION Ser82 l 1033 3. The aim of the paper is to establish a physical model that would account the electrical forces that act on the charged particle driven through an electric field at different plate orientations. This model can be used to optimize the design of devices which use this principle such as a novel tribo-aero-electrostatic separator and in medical acoustics. Several types of electrode structures are employed in microfluidic systems as well as in integrated optical systems for in-line analytical biochemistry and dielectrophoretic particle control or for guided wave control respectively CITATION LiJ13 l 1033 m epo06 m Zez18 m Asi14 4, 5, 6, 7. Electric field analysis of out-of plane electrodes is also required to study local electric field distribution in 2D for in-vivo electrochemotherapy and gene electro transfer and in hybrid microelectronic circuits CITATION Bue09 l 1033 8.

But, electric field analysis for inclined plane electrodes or non-parallel electrode-pair, placed on arbitrary geometrical surface configuration, is not straight forward. Nowadays, thanks to the performance that offered by computers, different iterative algorithms are used to solve the equations that govern the behaviour of plates of such geometry, these algorithms are designed in order to obtain the superficial charge distribution, the electric field values and the capacitance. The disadvantage of these algorithms is that they are designed for a few specific cases, so that if the geometry changes, the algorithm should be reconstructed completely. CITATION Asi14 l 1033 7
Fig SEQ Figure * ARABIC 1: Electric Field distribution on inclined plateCITATION Jos09 l 1033 92 Physical model
The electric field of plate electrode inclined at angle is very complex, techniques for electric field analysis for non-parallel electrode pair include using Schwarz-Christoffel mapping (SCM) and conformal transformations and although accurate lead to two second order simultaneous differential equations which are almost impossible to solve since such require 4 equations. The modelled electric field ignores fringing and assumes all geometries have straight electric fields. This will be compared with a known electric field. CITATION CSh87 l 1033 102.1 Model of inclined plates
Assumptions must be made to analyses motion of a charge so that the equations become less cumbersome and most importantly solvable. The most prominent of assumption is that the electric fields are straight.

41910099060436245043180di0di-869952685415Figure SEQ Figure * ARABIC 2: Electric plates inclined at angle ? separated by distance doFigure SEQ Figure * ARABIC 2: Electric plates inclined at angle ? separated by distance docenter-876935E+ + + + + + + + + + + + + + + + + + + + + + + + + + +
– – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – –
Y
x
?
E+ + + + + + + + + + + + + + + + + + + + + + + + + + +
– – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – –
Y
x
?

-209550176530d00d0
Figure 2 shows the non-parallel plate configuration of the electric field; the electric plates are at a potential ?V separated by distance do. The negative plate is inclined at angle ?.

The electric potential ?V between the plates is
?V=E?dy?V=E d0+diWhere di=xtan? therefore the electric field can becomes
E=?Vd0+xtan?j(2.1)
Which is electric field strength between the plates.

Figure SEQ Figure * ARABIC 3: The electric field strength between plates for d0=15cm at incline angles ?=0°,10°,20°.

Figure 3 shows the repartition of the electric filed strength E along the plates. Because the electrodes are inclined at an angle ?, the electric field near the origin is stronger than else ware. The decay of the electric field strength along the electrodes is steeper for ?= 20° than for ?= 10° and constant at no inclination.
2.2 Launching the charge
A charge of +q is launched into the electric field of the plate configuration at angle ? to the horizontal with initial velocity v0
vosin?left217170vovo
219075219075??
v0cos?Figure 4: positive charge launched at angle ? to the horizontal with voThe charge experiences electric forces defined
F=qE The acceleration due to the forces from newtons 2nd law of motion
Fe=may jHence;
ay=qEmAlong the horizontal there is no acceleration hence
x=vocos?t t=xvocos?(?) (2.2)
Along the vertical however
ay=qm?Vd0+xtan(?)j (2.3)
2.3 velocity function
Along the vertical the velocity of the charge is found by integrating the acceleration function.
y=qm?Vd0+xtan(?)By replacing the value of x by equation (2.2) and integrating from 0 to t.

y=µ0tdtd0+vocos?tan? t ;µ=q?Vm- constant J/kg
y t=µtan?vocos?lntan?vocos?dot+1+vosin? (2.4)
The vertical velocity component in terms of position can be found replacing t with the value x.

vy x=µtan?vocos?ln tan?dox+1+vosin? (2.5)
What determines a charge’s velocity component along the non-force acting axis is the ? constant. For a proton its ? is so large; µ=2.87 GJ/Kg, that compare to initial conditions to the overall motion initials are almost negligible. We assume a theoretical charge of m=1kg charge Q=+0.1C under a potential distance ?V=30V it’s constant ? would be µ=3 J/kg, making effects of initial trajectory significant.

A special condition is when ?=0. This is found by taking the limit as ??0By applying L’Hospital’s rule.

lim??0y= µtdo+vosin?(3.5)
vyx=µxv0docos?+vosin? (3.6)
This equation shows that the velocity along the vertical increases linearly with x when ?=0°

Figure SEQ Figure * ARABIC 5: The velocity along the y axis for incline plates at angles ?=0°,20°,30°. separated by d0=15cm at potential difference ?V=30.Figure 5 depicts the dependence of the of vertical velocity on the horizontal distance for a charge moving between plates with different of angle of inclination imposed.
3.2 The position function.
The position can be found be found by integrating the velocity function.

y=dydtdty=0tµtan?(?)cos?volntan?vocos?dot+1+vosin?dt+yi
y=?tan?v0cos?0tlntan?vocos?dot+1dt +0tvosin?dt+yiLet ;
y1=?tan?v0cos?0tlntan?vocos?dot+1dt To solve this use integration by parts the substitution is; u=lntan?vocos?dot+1 , dv=dtAnd it simplifies to:
y1=tlntan?(?)vocos?(?)dot+1 -0ttan?vocos?dottan?vocos?dot+1dty1=tlntan?vocos?dot+1-t+dotan?vocos?lntan?vocos?dot+1 y1=t+dotan?vocos?lntan?vocos?dot+1-tHence;
yt=?tan?(?)v0cos?(?)t+dotan?(?)vocos?(?)lntan?(?)vocos?(?)dot+1-t+vosin?(?)t+yiWith same set of initial conditions but this time choosing yi=0 the plot of position function. Using again yx=?tan?(?)v0cos?(?)xv0cos?(?)+dotan(?)vocos?(?)lntan(?)dox+1-xv0cos?(?)+xtan(?)Initial position at origin.

The very special condition when ?=0 can also be found by taking the limit as ? approaches zero using again L’ Hospitals rule but an easier alternative is to take equation 3.6 and integrate it
yx-yi=?v0docos?(?)0xxdxyx=?2v0docos?(?)x2+yiWhich shows regardless of the initial conditions then charge must follow a parabolic path.

Figure SEQ Figure * ARABIC 6: Force on charge due to electric field of non-parallel plate configuration separated by do=15cm at potential difference ?V=30V. The angles of inclination are ?=0°,10°,20°,30°

Figure SEQ Figure * ARABIC 7:Position of particles inclined plates at angles ?=0, 10, 20, 30 separeted by do=15cm at electric potential ?V=30At angles of inclination ?>0, the charge flows an almost straight path ascending while it moves parabolic path when plates are parallel, ?=0.

3.3 The Charge trajectories in 3D
The radial vector can be written as;
rt=vocos?t i+?tan?v0cos?t+dotan?vocos?lntan?vocos?(?)dot+1-t+ vosin?(?)t j
The parametric curves followed by the charges are as follows
Conditions; vo=10ms-1, ?=10°,?=30°
Figure SEQ Figure * ARABIC 8: three dimensional view of trajectory of particle in inclined plates angle ?=10, separeted by do=15cm at electric potential ?V=30.

When plates are parallel , ?=0.

Figure SEQ Figure * ARABIC 9: three dimensional view of trajectory of particle in parallel plates, angle ?=0 separeted by do=15cm at electric potential ?V=30Figure SEQ Figure * ARABIC 10:side view

4. Discussion
Although the modelled electric field does not represent a precise or exact representation of electric field configuration the assumption to ignore fringing and the horizontal component of electric field are valid because the acceleration this component is very weak to produce a significant change in velocity compared to vertical change in velocity such that it can be ignored. The modelled field was also in agreement with a more precise modelling of a such a configuration from a journal CITATION LiJ13 l 1033 4. It also makes calculations much simpler. the exponential decay of electric field strength as inclination gets larger is the cause of practices velocity decreasing as it moves further along the configuration because the slope f velocity time represents the acceleration hence it. for the parallel plate configuration, the velocity increases linearly because there is a constant force on the charge as it moves. The trajectory of the particle viewed in 3D shows that the motion is only two dimensional (xy-plane) as the effects of fringing are ignored.

4.1 Conclusion
The model enables definition of position of the particle at time when launched into plates. The electric field decays more exponentially for large inclination angles. The inclining of the plates causes most significant factor of trajectory of particle. The configuration coincides with Newtonian motion for parallel plate configuration. Particle moves in an almost straight line when the plates are inclined.

References
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