9: Motion of the charged particle in electric and magnetic field

Example number 9.1, Page number 230

In [3]:
#importing modules
import math
from __future__ import division

#Variable declaration
L=1.33*10**-22;   #angular momentum(kg m**2/sec)
B=0.025;    #magnetic field(Wb/m**2)
m=6.68*10**-27;   #mass of alpha particle(kg)
q=3.2*10**-19;   #charge of alpha particle(c)
e=1.6*10**-19;   #charge of electron(c)

#Calculation
w=(B*q)/m;   #angular velocity
E=0.5*L*w;   #KE of particle(J)
E=E/e;     #KE of particle(eV)

#Result
print "KE of particle is",round(E,2),"eV"

KE of particle is 497.75 eV

Example number 9.2, Page number 230

In [7]:
#importing modules
import math
from __future__ import division

#Variable declaration
R=0.35;   #radius of cyclotron(m)
n=1.38*10**7;   #frequency(Hz)
m=1.67*10**-27;  #mass of proton(kg)
q=1.6*10**-19;   #charge of proton(c)

#Calculation
B=(2*math.pi*n*m)/q;   #magnetic field induction(Wb/m**2)
E=((B**2)*(q**2)*(R**2))/(2*m);   #maximum energy of proton(J)
E=E/q;    #maximum energy of proton(eV)

#Result
print "magnetic field induction is",round(B,3),"Wb/m**2"
print "maximum energy of proton is",round(E/10**6,1),"MeV"

magnetic field induction is 0.905 Wb/m**2
maximum energy of proton is 4.8 MeV

Example number 9.3, Page number 231

In [16]:
#importing modules
import math
from __future__ import division

#Variable declaration
m=9.1*10**-31;   #mass of electron(kg)
e=1.6*10**-19;   #charge of electron(c)
V=1000;   #potential difference(V)
B=1.19*10**-3;   #magnetic field of induction(Wb/m**2)

#Calculation
#due to potential difference V, electron is accelerated
#eV=0.5*m*(v^2)
#due to transverse magnetic field B electron moves in circular path of radius R
#(m*(v^2))/R=BeV
v=math.sqrt((2*e*V)/m);   #velocity(m/sec)
R=(m*v)/(B*e);   #radius of electron trajectory(m)
L=m*v*R;    #angular momentum(kg m**2/sec)

#Result
print "radius of electron trajectory is",round(R*100,3),"cm"
print "angular momentum of electron is",round(L/10**-28,2),"*10**-28 kg m**2/sec"
print "answer for angular momentum varies due to rounding off errors"

radius of electron trajectory is 8.962 cm
angular momentum of electron is 15294.12 *10**-28 kg m**2/sec
answer for angular momentum varies due to rounding off errors

Example number 9.4, Page number 231

In [23]:
#importing modules
import math
from __future__ import division

#Variable declaration
vx=1.7*10**7;   #horizontal velociy(m/sec)
Ey=3.4*10**4;   #electric field(V/m)
x=3*10**-2;     #horizontal displacement(m)
m=9.1*10**-31;   #mass of electron(kg)
e=1.6*10**-19;   #charge of electron(c)

#Calculation
t=x/vx;   #time(sec)
#y=0.5*ay*(t^2)
ay=(e*Ey)/m;    #acceleration(m/sec**2)
y=0.5*ay*(t**2);   #vertical displacement(m)
Bz=Ey/vx;   #magnitude of magnetic field(Wb/m**2) 

#Result
print "vertical displacement of electron is",round(y*100,4),"cm"
print "answer varies due to rounding off errors"
print "magnitude of magnetic field is",Bz,"Wb/m**2"
print "direction of field is upward as Ey is downward"

vertical displacement of electron is 0.9308 cm
answer varies due to rounding off errors
magnitude of magnetic field is 0.002 Wb/m**2
direction of field is upward as Ey is downward

Example number 9.5, Page number 232

In [31]:
#importing modules
import math
from __future__ import division

#Variable declaration
m=1.67*10**-27;      #mass of proton(kg)
q=1.6*10**-19;    #charge of proton(c)
B=0.5;     #magnetic field(Wb/m**2)
R=1;     #radius of cyclotron(m)


#Calculation
n=((B*q)/(2*math.pi*m));   #frequency of oscillation voltage(Hz)
E=((B**2)*(q**2)*(R**2))/(2*m);    #maximum energy of proton(J)
E=E/q;    #maximum energy of proton(eV)

#Result
print "frequency of oscillation voltage is",round(n/10**6,3),"MHz"
print "maximum energy of proton is",round(E/10**6,3),"MeV"

frequency of oscillation voltage is 7.624 MHz
maximum energy of proton is 11.976 MeV

Example number 9.6, Page number 232

In [4]:
#importing modules
import math
from __future__ import division

#Variable declaration
q=3.2*10**-19  #charge of a9lpha particle(c)
m=6.68*10**-27;  #mass(kg) 
B=1.5;   #magnetic field(Wb/m**2)
v=7.263*10**6;   #velocity(m/s) 

#Calculation
F=B*q*v;    #force on particle(N)
T=(2*math.pi*m)/(B*q);   #periodic time(sec)
n=1/T;    #resonance frequency(Hz)

#Result
print "force on particle is",round(F*10**13,2),"*10**-13 N"
print "periodic time is",round(T*10**8,3),"*10**-8 sec"
print "answer for periodic time varies due to rounding off errors"
print "resonance frequency is",round(n/10**6,2),"MHz"

force on particle is 34.86 *10**-13 N
periodic time is 8.744 *10**-8 sec
answer for periodic time varies due to rounding off errors
resonance frequency is 11.44 MHz

Example number 9.7, Page number 233

In [42]:
#importing modules
import math
from __future__ import division

#Variable declaration
n=1.2*10**7;   #frequency(Hz)
mp=1.67*10**-27;   #mass of proton(kg)
qp=1.6*10**-19;    #charge of proton(c)
R=0.5;   #radius(m)
malp=6.68*10**-27;   #mass of alpha particle(kg)

#Calculation
Bp=(2*math.pi*mp*n)/qp;   #flux density for proton(Wb/m**2)
Ep=((Bp**2)*(qp**2)*(R**2))/(2*mp);   #energy of proton(J)
Ep=Ep/qp;          #energy of proton(eV)
qalp=2*qp;    #charge of alpha particle(c)
Balp=(2*math.pi*malp*n)/qalp;    #flux density of alpha particle(Wb/m**2)
Ealp=((Balp**2)*(qalp**2)*(R**2))/(2*malp);   #energy of alpha particle(J)
Ealp=Ealp/qp;    #energy of alpha particle(eV)

#Result
print "flux density for proton is",round(Bp,5),"Wb/m**2"
print "flux density for alpha particle is",round(Balp,4),"Wb/m**2"
print "energy of proton is",round(Ep/10**6,2),"MeV"
print "energy of alpha particle is",round(Ealp/10**6,2),"MeV"

flux density for proton is 0.78697 Wb/m**2
flux density for alpha particle is 1.5739 Wb/m**2
energy of proton is 7.42 MeV
energy of alpha particle is 29.67 MeV

Example number 9.8, Page number 233

In [45]:
#importing modules
import math
from __future__ import division

#Variable declaration
e=1.6*10**-19;   #charge of electron(c) 
me=9.1*10**-31;  #mass of electron(kg)
malp=6.68*10**-27;  #mass of alpha particle(kg)
B=0.05;  #magnetic field(Wb/m**2)
V=20*10**3;    #potential difference(V)

#Calculation
q=2*e;    #charge of alpha particle(c)
#v=sqrt((2*q*V)/m)
#R=(1/B)*sqrt((2*m*V)/q)
Re=(1/B)*math.sqrt((2*me*V)/e);   #radius of electron(m)
Ralp=(1/B)*math.sqrt((2*malp*V)/q);   #radius of alpha particle(m)
S=2*Ralp-2*Re;  #linear separation between two particles(m)

#Result
print "linear separation between two particles on common boundary wall is",round(S*100,1),"cm"

linear separation between two particles on common boundary wall is 113.7 cm

Example number 9.9, Page number 234

In [46]:
#importing modules
import math
from __future__ import division

#Variable declaration
V1=200;    #potential difference(V)
i=60;   #angle(degrees)
r=45;   #angle(degrees)

#Calculation
#electrostatic focusing condition (sini/sinr)=(v2/v1)=sqrt(V2/V1)
#0.5mv2=eV
i=i*(math.pi/180);  #angle(radian)
r=r*(math.pi/180);  #angle(radian)
V2=V1*((math.sin(i)/math.sin(r))**2);   #potential difference(V)
pd=V2-V1;   #potential difference(V)

#Result
print "potential difference between two regions is",pd,"V"

potential difference between two regions is 100.0 V

Example number 9.10, Page number 235

In [7]:
#importing modules
import math
from __future__ import division

#Variable declaration
E=250;    #electric field(V/m)
R=10**-8;  #radius of drop(m)
rho=10**3;   #density of water(kg/m**3)

#Calculation
#F=mg=qE
m=(4/3)*math.pi*(R**3)*rho;    #mass of water drop(kg)
W=m*9.8;   #weight of drop
q=W/E;     #charge on water drop(C)

#Result
print "charge on water drop is",round(q*10**21,3),"*10**-21 C"

charge on water drop is 0.164 *10**-21 C

Example number 9.11, Page number 235

In [58]:
#importing modules
import math
from __future__ import division

#Variable declaration
e=1.6*10**-19;   #charge of electron(c)
v=5*10**5;   #velocity(m/s)
B=0.3;    #flux density(Wb/m**2)
N=6.025*10**26;   #avagadro number
M72=72/N;    #mass(kg)
M74=74;    #mass(kg)

#Calculation
R72=(M72*v)/(B*e);    #radius(m)
R74=(R72/72)*M74;    #radius(m)
S=2*(R74-R72);   #linear separation of two lines(m)

#Result
print "linear separation of two lines is",round(S,3),"m"
print "answer given in the book is wrong"

linear separation of two lines is 0.069 m
answer given in the book is wrong

Example number 9.12, Page number 236

In [8]:
#importing modules
import math
from __future__ import division

#Variable declaration
l=5*10**-2;  #length(m)
d=0.3;   #distance of screen from end of magnetic field(m)
y=0.01;   #deflection on screen(m)
m=9.1*10**-31;  #mass of electron(kg)
e=1.6*10**-19;  #charge of electron(C)
Va=1000;   #anode voltage(V)

#Calculation
D=d+(l/2);   #distance(m)
B=(y/(D*l))*math.sqrt((2*m*Va)/e);   #flux density(Wb/m**2)

#Result
print "flux density is",round(B*10**6,1),"*10**-6 Wb/m**2"

flux density is 65.6 *10**-6 Wb/m**2

Example number 9.13, Page number 236

In [11]:
#importing modules
import math
from __future__ import division

#Variable declaration
e=1.6*10**-19;     #charge of electron(C)
Va=150;    #potential difference(V)
m=9.1*10**-31;  #mass of electron(kg)
V=20;     #potential(V)
D=1/2;
d=10**-2;   #distance of seperation(m)
l=10*10**-2;    #length(m)

#Calculation
vx=math.sqrt((2*e*Va)/m);    #velocity of electron reacting the field(m/s)
ay=(e/m)*(V/d);    #acceleration due to deflecting field(m/s**2)
vy=ay*(l/vx);    #final velocity attained by deflecting field(m/s)
theta=math.atan(vy/vx);    #angle of deflection(radian)
thetaD=theta*(180/math.pi);   #angle of deflection(degrees)
Y=D*math.tan(theta);     #deflection on screen(m)
S=(Y/V);     #deflection senstivity(m/V)


#Result
print "velocity of electron reacting the field is",round(vx/10**6,2),"*10**6 m/s"
print "acceleration due to deflecting field is",round(ay*10**-14,3),"*10**14 m/s**2"
print "final velocity attained by deflecting field is",round(vy/10**6,1),"*10**6 m/s"
print "angle of deflection is",round(thetaD,2),"degrees"
print "answer varies due to rounding off errors"
print "deflection on screen is",round(Y,2),"m"
print "deflection senstivity is",round(S,4),"m/V"
print "answer varies due to rounding off errors"

velocity of electron reacting the field is 7.26 *10**6 m/s
acceleration due to deflecting field is 3.516 *10**14 m/s**2
final velocity attained by deflecting field is 4.8 *10**6 m/s
angle of deflection is 33.69 degrees
answer varies due to rounding off errors
deflection on screen is 0.33 m
deflection senstivity is 0.0167 m/V