Project report plot the trajectory of electron in static electromagnetic

VIETNAM NATIONAL UNIVERSITY HO CHI MINH CITY HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY PROJECT REPORT Plot the trajectory of electron in static electromagnetic field Instructor: Pro

VIETNAM NATIONAL UNIVERSITY HO CHI MINH CITY

HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY

PROJECT REPORT Plot the trajectory of electron in static electromagnetic field

Instructor: Prof Huynh Quang Linh

Course code: PH1003

Class: CCO1 Group: 10 Members:

Ho Chi Minh City, November 2021

I1 000400900009) 07 1

PM 9è 3

3 MATLAB Code and ExplanafiOII - Ăn nh nn 5

4 Results and CISCUSSIOI - 5 5 1 9 90 9.0 họ Thi n 8

5 Conclusion n5 14

;585)95)0105S0777 14

1 Introduction

Electromagnetic theory is concerned with the study of charges at rest and

in motion Electromagnetic principles are fundamental to the study of electrical engineering It is also required for the understanding, analysis and design of various electrical , electromechanical and electronic systems Electromagnetic theory can be thought of as generalization of circuit theory Electromagnetic theory deals directly with the Ielectric and magnetic field vectors where as circuit theory deals with the voltages and currents Voltages and currents are integrated effects of electric and magnetic fields respectively

The Electromagnetic field problems involve three space variables along with the time variable and hence the solution tends to become correspondingly complex

A charged particle of mass m and charge q will experience a force acting

upon it in an electric field # Also, the charged particle will experience a

magnetic force acting upon it when moving with a velocity v in a magnetic

field B

The equation of the electron when its moves in static electromagnetic field

is expressed by the Lorentz force:

E =E+E =q# +qux#Ẻ With the initial position and velocity, we can determine the kinetic motion equations of electron x (t), y (t) and z (t) After that, we can determine the acceleration of the electron

Subsequently, eliminating t from mentioned motion equations, we can

derive f (x, y, z) = const, which is the orbital equation of electron

If the charged particle is stationary ( v = 0), the force depends only of the electric field The direction of the electric force is in the same direction as the electric field if q > 0 and the electric force is in the opposite direction to the electric field if q <0

When a charged particle is moving only in a magnetic field, the direction

1

of the magnetic force is at right angles to both the direction of motion and the direction of the magnetic field as given by the right hand palm rule

+q

Vv

J

out of page

palm face

positive chargeina

fingers magnetic field

‘

‘vy, (+q) thumb

B fingers

= motion of a

( 4) negative charge in

= a magnetic field

F palm face This project requires students to use MATLAB to calculate and simulation

of the trajectory of a particle in electric and magnetic field (electromagnetic field).

2 Theory

Consider a particle of charge q coulombs and mass m kilograms subjected

to an electric field

In newtons per coulomb and a magnetic field

B (0.0.Bz) ~ B = Bf

The equation of the electron when its moves in static electromagnetic field

is expressed by the Lorentz force:

F =K+ER=qE +quxB

E =mữ

E =q( +t xB)

=> md =q(E-+& x B)

With & is the acceleration vector Expressing by component in the Cartesian coordinates reference, we can obtain following differential equations:

m( a,0101 + a,ETI + a;kl ) = q[Ez kÌ + (u„TTI+ u,ETT + u;kl ) x B; kỉ]

m( a„LTTI + aETI+ azkl ) = gE, ke + q(u,ET] + u,ETI + u;kl) x B„ kì

ma, = qBz Uy

=> { may = —qBz Uy

ma, = qEz

_ qBz

=

qBz

=> JO = — = 30

E,

w= q1z

Projection in the direction of Ox

Differential equation

ney Bz,

x”(Ð = a ()

Projection in the Oy direction

Differential equation

qBz ,

" t =—'“ x t

y'@== —xŒ

Projection in the Oz direction

Differential equation

2"(t) =

With

x(0) = Xp

y(0) = Yo

z(0) = Zọ

x/(0) = Uzo

z'(0) = Vz

These are coupled second-order ordinary differential equations that can be solved by either analytical or numerical methods

Numerically, as done in this demonstration, the solution needs initial

conditions for the velocity and the position, given by

3 (Xo.¥0.Zo) > BH = Xo + yj + Zoke

By (Vx0.Vy9sVz9) —> Ups = UyoFTT+ vy OID + vzolel

3 MATLAB Code and Explanation

% Motion of a electron in uniform cross B and E fields clear; cle; clf;

syms x(t) y(t) 2(0);

syms k kl k2 vx0 vy0 vz0 x0 y0 z0;

format short;

coh

% SYMBOLIC OPERATION

coh

Dx = diff(x,t);

Dy = dif(y,0;

Dz = diff(z.t);

% \k1 = q*B/m and k2 = q*E/m

% ODE function

odel = diff(x,t,2) == k1*diff(y.t);

ode2 = diff(y,t,2) == -k1*diff(x,t);

ode3 = diff(z,t2) == k2;

Eqn = [ode1, ode2, ode3];

Cond = [Dx(0) = vx0; Dy(0) == vy0; Dz(0) == vz0;

x(0) == x0; y(0) == yO; z(0) == 70];

S = dsolve(Eqn Cond);

x_func = collect(simplify(S.x));

y_func = collect(simplify(S.y));

z_func = collect(simplify(S.z));

vx_func = collect(simplify(diff(S.x,0)));

vy_func = collect(simplify(diff(S-y,t));

vz_func = collect(simplify(diff(S.z,t)));

ax_func = collect(simplify(diff(S.x.t/2)));

ay_func = collect(simplify(diff(S -y,t,2)));

az_func = collect(simplify(diff(S.z t.2)));

coh

% OUTPUT FUNCTION

coh

% Motion function

dispCMotion function on x-direction: x='); disp(x_func); dispCMotion function on y-direction: y='); disp(y_func); disp('‘Motion function on z-direction: z="); disp(z_func); disp('000000000000000000000000000000000000000000000');

% Velocity function

disp('Velocity function on x-direction: vx ='); disp(vx_func); disp('Velocity function on y-direction: vy ='); disp(vy_func); disp('Velocity function on z-direction: vz ='); disp(vz_func); disp('000000000000000000000000000000000000000000000');

% Acceleration function

disp('‘Acceleration function on x-direction: ax ='); disp(ax_func); disp('‘Acceleration function on x-direction: ay ='); disp(ay_func); disp('‘Acceleration function on x-direction: az ='); disp(az_func); disp('000000000000000000000000000000000000000000000');

% Note

disp('with kl = q*B/m');

disp('with k2 = q*E/m');

% DISPLAY RECOMMENDED INPUT PARAMETERS

%

disp('000000000000000000000000000000000000000000000');

disp(‘Recommended parameters for you to enter/input')

disp(‘Recommended initial position of electron: [0 0 0] ');

disp('‘Recommended initial position of electron [2 3 -5]');

disp(Recommended static magnetic field parallel to z-axis: 2e-11');

disp(‘Recommended static electric field parallel to z-axis: 5e-12');

coh

% INPUT PARAMETERS

%

dispC ); dispC 3; disp( 3; dispC );

% Enter initial position and velocity of electron

r0 = input('Enter the initial position of electron [x0 yO z0] (m) - );

vO = input('Enter the initial velocity of electron [vx0 vy0 vz0] (m/s) - ');

% Enter magnitude of uniform B and E fields

B = input(‘Enter static magnetic field parallel to z-axis [0 0 B] (T) - ');

E = input('Enter static electric field parallel to z-axis [0 0 E] (V/m) - ');

% Parameter of electron

m = 9.10939e-31;

q = 1.602177e-19;

k11= q*B/m;

k22= q*E/m;

disp(''); disp(''); disp(''); disp( ');

%

% CALCULATE THE ELECTROMAGNETIC FORCE ACTING ON THE ELECTRON

%

Fx=subs(m*ax_func,[x0,y0,z0,vx0,vy0,vz0,k1 ,k2],[r0(1).r0(2),r0(3),v0(1),v0(2),v0(3

)k11,k22));

Fy=subs(m*ay_ func,[x0,y0,z0,vx0,vy0,vz0,k1 ,k2],[r0(1).r0(2),r0(3),v0(1),v0(2),v0(3

)k11,k22));

Fz=subs(m#az_func,[x0,y0,z0,vx0,vy0,vz0,k1 ,k2],[r0(1).r0(2),r0(3),v0(1).v0(2).v0(3

)k11,k22));

disp('Force acting on the electron on x-direction: Fx='); pretty(Fx);

disp('Force acting on the electron on y-direction: Fy='); pretty(Fy);

disp('Force acting on the electron on z-direction: Fz='); disp(double(Fz));

dispC 3; disp(''); disp(''); dispC );

Œ%

% OUTPUT FUNCTION

coh ‘0

h1=subs(S.x,[x0,y0.z0,vx0,vy0,vz0,k1,k2] [r0(1) r0(2) r0(3).v0(1),v0(2).v0(3),k11,

k22)):

h2=subs(S.y,[x0,y0z0,vx0,vy0,vz0,k1,k2][r0(1).r0(2).r0(3),v0(1),v0(2),v0(3),k11,

k22)):

h3=subs(S.z,[x0,y0z0,vx0,vy0,vz0,k1,k2][r0(1) r0(2).r0(3) ,v0(1),v0(2),v0(3),k11,

k22)):

dispCMotion function on x-direction: x='); pretty(h1);

dispCMotion function on y-direction: y='); pretty(h2);

disp('Motion function on z-direction: z="); pretty(h3);

disp('0000000000000000000000000000000000000000000000000000000000000000000000000');

% Velocity function

disp('Velocity function on x-direction: vx ='); pretty(diff(h1 ,t));

disp('Velocity function on y-direction: vy ='); pretty(diff(h2,t));

disp('Velocity function on z-direction: vz ='); pretty(diff(h3 ,t));

disp('0000000000000000000000000000000000000000000000000000000000000000000000000');

% Acceleration function

disp('‘Acceleration function on x-direction: ax ='); pretty(diff(h1 ,t,2));

disp('‘Acceleration function on x-direction: ay ='); pretty(diff(h2 ,t,2));

disp('‘Acceleration function on x-direction: az ='); pretty(diff(h3 ,t,2));

%

% PLOT THE TRAJECTORY OF ELECTRON

%

figure(1)

XMax=5 ;XMin= -XMax;

YMax = XMax ; YMin = -YMax;

ZMax = 20 ; ZMin = -20;

fplot3(h1 h2,h3,[0 50],','LineWidth' ,1);

grid on

axis equal

box on

axis([X Min, XMax, YMin, YMax, ZMin, ZMax]);

xlabel('x [m]');

ylabel('y [m]');

zlabel('z [m]');

set(gca,'fontsize',10);

4 Results and discussion

15 |

y [m] lở 5 x [m]

y

Motion function on x-direction: x=

x0 + (vv0 - vy0*cos(kl*t) + vxz0*sin(kl*t))/kl

Motion function on y-direction: y=

v0 + (vx0*cos(kl*t) - vx0 + vy0*sin(kl*t))/kl

Motion function on z-direction: z=

(k2*t^2)/2 + vz0*t + z0

[o ololololololololololoiololololololololololololololoiolololololololololololoioloioioloiol

Velocity function on x-direction: vx =

cos (k1*t)*vx0 + vy0*sin(kl*t)

Velocity function on y-direction: vy

(-sin(k1*t))*vx0 + vy0*cos(k1*t)

Velocity function on z-direction: vz =

vz0 + k2*t

©GØØØÒOOOOOOOOOODOOOOOOOOOOOOOOOOOOOOOOOODOOOOO

Acceleration function on x-direction: ax =

(T-kl*sin (kl*t))*vx0 + kl*vy0*cos (kl*t)

Acceleration function on x-direction: ay =

(-k1*cos (k1*t))*vx0 - ki*vy0*sin(k1*t)

Acceleration function on x-direction: az =

k2

©OOOOOOOOOOOOOOOCOCOOOOOOOOOOOOOOOOCOCOOOOOOOOO

with kl = q*B/m

with k2 = q*E/m

000000000000000000000000000000000000000000000

Recommended prameters for you to enter/input

Recommended initial position of electron: [0 0 0]

Recommended initial position of electron [2 3 -5]

Recommended static magnetic field parallel to z-axis: 2e-11

Recommended static electric field parallel to z-axis: 5e-12

Enter the initial position of electron [x0 y0 z0] (m) - [0 0 0] Enter the initial velocity of electron [vx0 vy0 vz0] (m/s) - [2 3 -5]

Enter static magnetic field parallel to z-axis [0 0 B] (Tesla) - 2e-11 Enter static electric field parallel to z-axis [0 0 E] (V/m) - 5e-12

Force acting on the electron on x-direction: Fx=

/ 7921017477769179 t \

CoS] sso esse ess esas | 61790380198481666882465389812213

\ 2251799813685248 /

ie 64277521770359611021 67848 36936465041 0088811975131171341 205504

10

Force acting on the electron on x-direction: Fx=

/ 792101 7477769179 -t: \

COS!) | aS | 61790380198481666882465389812213

\ 2251799813685248 /

6427752177035961102167848369364650410088811975131171341205504

/ 7921017477769179 t \

sin| -— - | 20596793399493888960821796604071

\ 2251799813685248 /

3213876088517980551083924184682325205044405987565585670602752 Force acting on the electron on y-direction: Fy=

/ 7921017477769179 t \

Gog] ==—=========ằ=-=~= | 20596793399493888960821796604071

\ 2251799813685248 /

3213876088517980551083924184682325205044405987565585670602752 / 7921017477769179 t \

sin| - | 61790380198481666882465389812213

\_ 2251799813685248 /

6427752177035961102167848369364650410088811975131171341205504 Force acting on the electron on z-direction: Fz=

8.0109e-31

Motion function on x-direction: x=

/ 7921017477769179 t \

sin] - | 4503599627370496

\ 2251799813685248 /

7921017477769179

/ 7921017477769179 t \

Cos| —-—-—-—-—~—-—~ —~ | 2251799813685248

\ 2251799813685248 / 2251799813685248

2640339159256393 2640339159256393 Motion function on y-direction: y=

/ T921017477769179 t \

gøgÌ -—— | 4503599627370496

\ 2251799813685248 /

7921017477769179

II

Motion function on x-direction: x=

/ 7921017477769179 t \

sig) ——=———===—r=-==== | 4503599627370496

\ 2251799813685248 /

7921017477769179

/ 7921017477769179 t \

Cas] ==—==========—m=——=— | 2251799813685248

\ 2251799813685248 / 2251799813685248

Se By easrenesaieeesosenenastietee

2640339159256393 2640339159256393 Motion function on y-direction: y=

/ 7921017477769179 t \

cos| - | 4503599627370496

\ 2251799813685248 /

7921017477769179

/ 7921017477769179 t \

\ 2251799813685248 / 4503599627370496

# "“ ˆ _, a

2640339159256393 7921017477769179

Motion function on z-direction: z=

Z

7921017477769179 t

18014398509481984

09909000000000000000000000000000000000000000000000000000000000000099000000

Velocity function on x-direction: vx =

/ 7921017477769179 t \ / 7921017477769179 t \

CG8| ceases sess | 2+ gin) -— - | 3

\ 2251799813685248 / \ 2251799813685248 /

Velocity function on y-direction: vy =

/ 7921017477769179 t \ / 7921017477769179 t \

gØB] ===—==—=>~—=—==—==~ l 3= g7ml “—>>====>~===~—= | 2

\ 2251799813685248 / \ 2251799813685248 /

Velocity function on z-direction: vz =

7921017477769179 t

9007199254740992

0090900000000000000000000000000000000000000000000000000000000000000000000

Acceleration function on x-direction: ax =

/ 7921017477769179 t \

gös| m>—-~ -~=—-z=—~ | 23763052433307537

Se \X_ 2251799813685248 /

12

Velocity function on x-direction: vx =

/ 7921017477769179 t \ / 7921017477769179 t \

Cos| =-~~-~-~~ —~———~—— | 2 + sin| -~ ~-~—~ | 3

\ 2251799813685248 / \ 2251799813685248 /

Velocity function on y-direction: vy =

/ 7921017477769179 t \ / 7921017477769179 t \

COS sno sss SEs Sse == [ So= SQn) saeesssssssrsS==s= 12

\ 2251799813685248 / \ 2251799813685248 /

Velocity function on z-direction: vz =

7921017477769179 t

9007199254740992

2000000000000000000000000000000000000000000000000000000000000000000 Acceleration function on x-direction: ax =

/ 7921017477769179 t \

cos| - | 23763052433307537

\_ 2251799813685248 /

2251799813685248

/ 7921017477769179 t \

sin| - | 7921017477769179

1125899906842624 Acceleration function on x-direction: ay =

/ 7921017477769179 t \

608] ss ocse tor ssos SoS | 23763052433307537

\ 2251799813685248 /

2251799813685248

/ 7921017477769179 t \

sin| - | 7921017477769179

\ 2251799813685248 /

1125899906842624

Acceleration function on x-direction: az =

/ 7921017477769179 t \

ces) =H oss seas | 23763052433307537

\ 2251799813685248 /

2251799813685248

/ 7921017477769179 t \

sin| - | 7921017477769179

\ 2251799813685248 /

1125899906842624

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13