Project report plot the trajectory of electron in static electromagnetic field

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..

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: CC01

Group: 10

Members:

Ho Chi Minh City, November 2021

1 INTRODUCTION 1

2 THEORY 3

3 MATLAB Code and Explanation 5

4 Results and discussion 8

5 Conclusion 14

REFERENCES 14

1

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 𝐸⃗ Also, the charged particle will experience a magnetic force acting upon it when moving with a velocity 𝑣 in a magnetic field 𝐵⃗

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

is expressed by the Lorentz force:

𝐹

⃗⃗⃗ = 𝐹 𝐸 + 𝐹 𝐿 = q𝐸⃗ + q𝑣 × 𝐵⃗

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

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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

This project requires students to use MATLAB to calculate and simulation

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

3

2 Theory

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

to an electric field

𝐸 ⃗⃗⃗ (0,0, 𝐸𝑍) → 𝐸 ⃗⃗⃗ = 𝐸𝑍 𝑘̂

In newtons per coulomb and a magnetic field

𝐵 ⃗⃗⃗ (0,0, 𝐵𝑍) → 𝐵 ⃗⃗⃗ = 𝐵𝑍 𝑘̂

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

is expressed by the Lorentz force:

𝐹

⃗⃗⃗ = 𝐹 𝐸 + 𝐹 𝐿 = q 𝐸⃗ + q𝑣 × 𝐵⃗

𝐹

⃗⃗⃗ = 𝑚𝑎 ⃗⃗⃗

𝐹

⃗⃗⃗ = 𝑞( 𝐸 ⃗⃗⃗⃗ + 𝑣 ⃗⃗⃗ × 𝐵 ⃗⃗⃗ )

=> 𝑚𝑎 ⃗⃗⃗ = 𝑞( 𝐸 ⃗⃗⃗⃗ + 𝑣 ⃗⃗⃗ × 𝐵 ⃗⃗⃗ )

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

𝑚( 𝑎𝑥𝑖̂ + 𝑎𝑦𝑗̂ + 𝑎𝑧𝑘̂ ) = 𝑞[𝐸𝑍 𝑘̂ + (𝑣𝑥𝑖̂ + 𝑣𝑦𝑗̂ + 𝑣𝑧𝑘̂ ) × 𝐵𝑍 𝑘̂]

𝑚( 𝑎𝑥𝑖̂ + 𝑎𝑦𝑗̂ + 𝑎𝑧𝑘̂ ) = 𝑞𝐸𝑍 𝑘̂ + 𝑞(𝑣𝑥𝑖̂ + 𝑣𝑦𝑗̂ + 𝑣𝑧𝑘̂) × 𝐵𝑍 𝑘̂

=> {

=>

{

𝑥̈ = 𝑞𝐵 𝑚 𝑦̇𝑍

𝑦̈ = − 𝑞𝐵 𝑚 𝑥̇𝑍 𝑧̈ = 𝑞𝐸 𝑚𝑍

Projection in the direction of Ox

Differential equation

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Projection in the Oy direction

Differential equation

𝑦′′(𝑡) = − 𝑞𝐵 𝑚 𝑥′𝑍 (𝑡)

Projection in the Oz direction

Differential equation

𝑧′′(𝑡) = 𝑞𝐸 𝑚𝑍

With

𝑥(0) = 𝑥0

𝑧(0) = 𝑧0

𝑥′(0) = 𝑣𝑥0

𝑧′(0) = 𝑣𝑧0

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

𝑟0

⃗⃗⃗ (𝑥0, 𝑦0, 𝑧0) → 𝑟 ⃗⃗⃗ = 𝑥0 0𝑖̂ + 𝑦0𝑗̂ + 𝑧0𝑘̂

𝑣0

⃗⃗⃗⃗ (𝑣𝑥0, 𝑣𝑦0, 𝑣𝑧0) → 𝑣 ⃗⃗⃗⃗ = 𝑣0 𝑥0𝑖̂ + 𝑣𝑦0𝑗̂ + 𝑣𝑧0𝑘̂

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3 MATLAB Code and Explanation

% Motion of a electron in uniform cross B and E fields

clear; clc; clf;

syms x(t) y(t) z(t) ;

syms k k1 k2 vx0 vy0 vz0 x0 y0 z0 ;

format short ;

% -% SYMBOLIC OPERATION

Dx = diff(x,t);

Dy = diff(y,t);

Dz = diff(z,t);

% ODE function

ode1 = diff(x,t,2) == k1*diff(y,t);

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

ode3 = diff(z,t,2) == k2;

Eqn = [ode1, ode2, ode3];

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

x(0) == x0; y(0) == y0; z(0) == z0];

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,t)));

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)));

% -% OUTPUT FUNCTION

% -% Motion function

disp( 'Motion function on x-direction: x=' ); disp(x_func);

disp( 'Motion function on y-direction: y=' ); disp(y_func);

disp( 'Motion function on z-direction: z=' ); disp(z_func);

disp( 'ooooooooooooooooooooooooooooooooooooooooooooo' );

% 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( 'ooooooooooooooooooooooooooooooooooooooooooooo' );

% 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( 'ooooooooooooooooooooooooooooooooooooooooooooo' );

% Note

disp( 'with k1 = q*B/m' );

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

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% -% DISPLAY RECOMMENDED INPUT PARAMETERS

disp( 'ooooooooooooooooooooooooooooooooooooooooooooo' );

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' );

% -% INPUT PARAMETERS

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

% Enter initial position and velocity of electron

r0 = input( 'Enter the initial position of electron [x0 y0 z0] (m) - ' );

v0 = 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)); disp( ' ' ); disp( ' ' ); disp( ' ' ); disp( ' ' );

% -% OUTPUT FUNCTION

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,y0,z0,vx0,vy0,vz0,k1,k2],[r0(1),r0(2),r0(3),v0(1),v0(2),v0(3),k11, k22]);

h3=subs(S.z,[x0,y0,z0,vx0,vy0,vz0,k1,k2],[r0(1),r0(2),r0(3),v0(1),v0(2),v0(3),k11, k22]);

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disp( ' -Function after entering parameters -' );

% Motion function

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

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

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

disp( 'ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo' );

% 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( 'ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo' );

% 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([XMin, XMax, YMin, YMax, ZMin, ZMax]);

xlabel( 'x [m]' );

ylabel( 'y [m]' );

zlabel( 'z [m]' );

set(gca, 'fontsize' ,10);

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4 Results and discussion

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5 Conclusion

The project has completed plot the trajectory of electron in static electromagnetic field problem using MATLAB symbolic calculation With this tool we can plot more complex situations that cannot be plotted by the analytical method The trajectory is calculated by computationally solving differential equations The direction and magnitude of magnetic and electric field can be changed along with other attributes of motion

References

1) General Physics A1, General Physics Exercises A1

2) “Motion of Charged particle in E and B” YouTube, uploaded by For the Love of Physics, 5th May, 2019,

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