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Transcript of Unit 1
EDC
Unit 1
INTRODUCTION: Electronics means electron mechanics. Electronics deals with the movement of electrons under the influence of
externally applied electric & magnetic field. IEEE defines electronics as “that field of science and engineering which deals
with the electron devices and their utilization”. Electron device may be defined as “a device in which conduction takes place by
movement of electros through vacuum, gas or a semiconductor”. Applications
1. communication 2. entertainment3. defense4. mics5. instrumentation6. medical science7. industry
CHARGED PARTICLE:In physics, a charged particle is a particle with an electric charge. Particles either
have a positive, negative or no charge (being neutral). An electron is a subatomic
particle that carries a negative electric charge. In many physical phenomena, such as
electricity, magnetism, and thermal conductivity, electrons play an essential role. An
electron generates a magnetic field while moving, and it is deflected by external
magnetic fields. When an electron is accelerated, it can absorb or radiate energy in the
form of photons. Electrons, together with atomic nuclei made of protons and neutrons,
make up atoms.
Electron Charge------ 1.6x10-19 coulombs
Mass------- 9.11x10-19 Kg
Radius----- 10-15 meters
Motion of charged particle is due to either of electric or magnetic or gravitational fields.
When electron is subjected to these E or M or G fields it get acceleration and its
trajectory is determined by Newton’s laws provided that the force acting on the particle
is known. Their are two models of electron, for large scale phenomena classical model
is used and for small scale phenomena wave model is used.
MOTION OF CHARGED PARTICLE IN ELECTRIC FIELD:The force on a unit positive charge at any point in an electric field is the electric
field intensity at that point. Field intensity is represented by ‘E’. The force on a positive
charge q in an electric field of intensity E is given by ‘q E’.
f q = q E---------------------------------------------------------------------------------- 1
Where f q is in Newton’s, q is in coulombs, E is in volts/meter.
In order to calculate the path of a charged particle in on electric field the force, given by
equ.1, must related to the mass and the acceleration of the particle by Newton’s second
law of motion. Hence
f q = q E = m a = m dv /dt.------------------------------------------------------------- 2
Note: ‘f’,’E’,’a’,‘v’ represents scalar quantities, ‘f’,’E’,’a’,‘v’ represents vector
quantities. Where
m is mass in Kgs
a is acceleration in m/sec2
v is velocity in m/sec.
The solution of equ.2 subject to approximate initial conditions, gives the path of the
particle resulting from the action of the electric force.
Force on electron
f = - e E --------------------------------------------------------------------------------- 3
The minus sign denotes that the force is in the direction opposite to the field.
Constant electric field:
Electron is situated between the two plates of a parallel plate capacitor which are
contained in an evacuated envelop.
Figure: 1 the two dimensional electric field between the parallel plates of capacitor.
A difference of potential is applied between the plates, and E is in –X –axis direction. If
d is small compared to dimensions of plates, E.F is considered as uniform. In initial
conditions, characteristics of the motion are as follows.
vx = vox, x=xo when t=0------------------------------------------------------------ 4
This means that initial velocity vox is chosen along E. the lines of force, and that the
initial position x0 of the electron is along the X-axis.
d
Y
E o
Z
X
_ +
As in Y-axis — f = 0, a = 0 (Newton’s law)
Z-axis — f = 0, a = 0
a = 0 means velocity is constant, since initial velocity along these axes is zero the
particle will not move along these directions.
Newton‘s law applies to the X – axis direction yields
e E = max-------------------------------------------------------------------------------- 5
ax = e E / m constant velocity in constant E.F
In case of freely falling body in the uniform gravitational field of the earth.
vx = vox +ax t, x = xo + vox t + ½ ax t2
------------------------------------- 6
Here ax constant and independent of time.
Motion is determined by differentiating equ 6
d vx / dt = ax, d x /dt = vx
These are definitions of acceleration and velocity
Potential:
Ex need not be uniform and is function of distance, not function of time from Newton’s
2nd law
-e Ex / m = d vx / dt multiply with d x = vx dt and integrating we get,
- e/m ∫ xox Ex dx = ∫vxo
vx vx d vx ----------------------------------------- 8
The definite integral ∫xox Ex dx is an expression for the work done by the field in
carrying a unit +ve charge from the xo to the point x.
The potential V (in volts) of point x w.r.t point xo is the work done against the field in
taking a unit +ve charge from xo to x.
V= - ∫ xox Ex dx ------------------------------------------------------------------- 9
From equ 8 & 9
eV = ½ m (vx 2- vox 2
) ------------------------------------------------------ 10
V is independent of variation of filed distribution and dependent on magnitude of field
above equ indicates law of conservation of energy valid for multidimensional. Let two
points A&B, B at a higher potential than A in general form.
q VAB = ½ m (vA 2- vB 2
) --------------------------------------------------- 11
q – Coulombs, qVAB – joules, vA &, vB – initial & final velocities.
The potential energy between two points equals to the potential multiplied by the
charge.
The eV unit of energy:
Energy ---joule- mks,
When joule is small convert to watt, watt =103 (KW), 106 (MW)
When joule is large convert to erga= joule x 10-7
1 eV = 1.6x 10-19 j used for electrical, mechanical, thermal etc.
MeV= million eV, BeV = billion eV.
Relation between field intensity and potential.
V= - ∫ xox Ex dx only when E.F is uniform with distance.
Ex (x - xo) = V
Ex = V/ (x - xo)
= -V/d volts / meter
When Ex = f (d) correct result is obtained by
Ex = - dV/dx
Minus sign shows that the electric field is directed from the region of higher potential to
the region of lower potential.
Two- dimensional motion:
Figure 2: two dimensional electric motion in a uniform field.
Initial conditions when t = 0
vx = vox x =0
vy = 0 y =0
l
d Vd
vox
-
+
Y
X
vz = 0 z = 0
Since there is no force in Z- axis direction acceleration in that direction is zero. Motion
is only in X-Y plane. In X – axis direction velocity is, vox, constant from which it
follows that
x = vox t. -------------------------------------------------------------------------- 17
For a constant velocity in Y – direction
Velocity vy = ay t ( as voy = 0 ) --------------------------- 18
Displacement y = ½ ay t2 ( as voy = 0 )------------------------------ 19
ay = - e E / m
= e V / m d
vy varies from point to point,
vx constant throughout the path.
By combining equ 17 & 18
We get the path of the particle as
y = (½ ay /vox2) x 2 ------------------------------------------------------------- 20
This shows that the particle moves in a parabolic path in the region between the plates.
MOTION OF CHARGED PARTICLE IN MAGNETIC FIELD:
Motion of a charged particle in magnetic field is characterized by the change in
the direction of motion. It is expected also as magnetic field is capable of only changing
direction of motion. In order to keep the context of study simplified, we assume
magnetic field to be uniform. This assumption greatly simplifies the description and lets
us easily visualize the motion of a charged particle in magnetic field.
Lorentz magnetic force law is the basic consideration here. Hence, we shall first take a
look at the Lorentz magnetic force expression:
F = q (v x B) ----------------------------------------------------------------------1
We briefly describe following important points about this expression:
1: There is no magnetic force on a stationary charge (v=0). As such, our study here
refers to situations in which charge is moving with certain velocity in the magnetic field.
This condition is met when the charge is released with certain velocity in the magnetic
field.
2: The magnetic field (B) is a uniform stationary magnetic field for our consideration in
the module. It means that the magnitude and direction of magnetic field do not change
during motion. The charged particle, however, is subjected to magnetic force acting side
way. The direction of motion of charged particle, therefore, changes. In turn, the
direction of magnetic force being perpendicular to velocity also changes. Important
point to underline here is that this loop of changing directions of velocity and magnetic
force is continuous. In other words, the directions of both velocity and magnetic force
keep changing continuously with the progress of motion.
Force in a magnetic field:
Force on a moving charge in a magnetic field is given by motor law. It gives
fm = BIL--------------------------------------------------------------------------------- 1
fm – force (Newton), B - M.F intensity (weber/m2), I-current (A), L-length (m) directions
of B & I are perpendicular. F perpendicular (I &B) like screw .If I not perpendicular B
only component of I perpendicular to B contributes to the force. Direction of current
depends on selection of particle.
Figure 3: pertaining to the determination of the direction of the fore on a charged
particle in a magnetic field.
N-total e- in L length, takes T sec to travel a distance of L then the total number of
electrons passing through any cross section of wire in unit time is N/T. Then
Current I = N e / T.
The force in Newton’s on a length Lm is
fm = BIL = B eN/T.L or
fm = BeNv.
Where v. average or drift speed m/sec. force per electron
fm = eBv
To summarize:
The force on a negative charge e (coulombs) moving with a component of
velocity v (meters per sec’), normal to a field B (Weber’s per square meter) is given by
eBv (Newton’s) and in a direction perpendicular to the plane of B and v.
v
B
fm
I 90o
o
CURRENT DENSITY:
Figure 4: Pertaining to the determination of the magnitude of the force fm on a charged
particle in a M. field.
Current density J = current per unit area of the conducting medium assuming a uniform
current distribution.
J ≡ I/A
J-A/m2, A (m) from equ.
J = Ne/TA = Nev/LA
Electron concentration n= N/LA (per cubic meter) and we get
J = nev = pv
Where p-charge density coulombs/cubic meter v-m/sec.
Independent of conducting medium. ρ & v need not be constant vary with time &
distance.
MOTION IN A MAGNETIC FIELD:
Path of a charged particle.
If the particle is at rest, fm = 0 and the particle remains at rest.
If the initial velocity of the particle is along the lines of the magnetic flex, there is no
force acting on the particle. (As fm = eBv)
Hence a particle whose initial velocity has no component normal to a uniform magnetic
field will continue to move with constant speed along the lines of flex.
L
A
N electrons
Figure 5: Circular motion of an e in a transverse M-field.
Initial speed v0, constant M.F. Since the force fm is perpendicular to v and so to the
motion at every instant, no work is done on the electron. So no increase in K.E. So no
change in speed.
Since v & B are constant in magnitude, then fm is constant in magnitude and
perpendicular to the direction of motion of the particle. So this type of force results in
motion in a circular path with constant speed. To find the radius of circle, it is recalled
that a particle moving in a circle path with a constant speed v, has an acceleration
toward the center of the circle of magnitude v2 / R. where R is the radius of the path in
meters then
m v2 / R = eBv
R = mv/eB
The corresponding angular velocity in radians/sec is given by
w = v / R = e B / m.
The time in sec for one complete revolution, called the period, is
T = 2∏ / w= 2∏ m/eB
For an electron T = 2∏ m/B
It is noted that radius of the path is directly perpendicular to the speed of the particle.
The period and the angular velocity are independent of speed or radius. This mean of
course, that faster moving particles will traverse larger circles in the same time that a
slower particle mores in its smaller circles.
Ex: Cyclotron and magnetic focusing apparatus.
x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x
R
Magnetic field into paper
P
vo
Field free region
O
P
CATHOD RAY OSCILLOSCOPE (CRO):
The block diagram of CRO is as shown in:-
Figure 6: CRO block diagram
Attenuator: It is a potential divider ckt that attenuates the amplitude of the input signal
to the required amplitude.
Vertical Amplifier: To deflect the beam of the CRO, voltages, in the range of 100V are
required. Generally, the input signal will be in the range of mV. Hence, the vertical
amplifier amplifies the weak input signal to a level required by CRT. Gain of the
amplifier should be large.
Delay line: To be able to observe the complete waveform of the input signal. The
horizontal of vertical deflection of the e- beam should start simultaneously.
Signal processing in horizontal amplifier system takes a finite amount of
time. Such a delay doesn’t occur in vertical deflection system. Hence, the signal given
to the vertical input should be delayed to observe the waveform property.
CRT: It is the heart of the CRO. It provides a high velocity e- beam that passes through
a set of deflection plates. One set of plates is oriented to deflect the e - beam vertically
when appropriate voltage is applied between them. The 2nd set of plates deflects the
beam horizontally.
Time base Generator:
Need for time base: When we apply voltage across the vertical plates, the spot moves up
and down and we can observe a vertical line. This is due to the fact that we have not
provided a time axis.
To provide a time axis, we have to move the spot horizontally from one
end of the screen to other end at a ray proportional to the time
attenuatorVertical amplifier
Delay line
Triggering circuit
Time base generator Horizontal
amplifier
In put
Signal
Vertical deflection plates screen
Electron gun
Horizontal deflectionplates
Time Base Ckt:
The simplest ckt to generate time base is a
Capacitor that is charged by a constant current source.
Measurement of frequency:
The signal for which the frequency ‘f’’ is to be measured is given to the vertical
input. The no. of divisions occupied by 1 complete cycle of the wave form is measured.
The no. of divisions multiplied by the time base setting in sec. is equal to the time
period (T) of one cycle. The frequency of wave form is inverse of the time period. ‘T’.
Measurement of phase Difference:
The phase difference between 2 sinusoidal signals of same frequency can be
calculated from the amplitude of the 2 signals as when A&B
are the amplitudes of 2 signals.
To measure the phase difference of two signals, the 2 sine waves are applied
simultaneously to the vertical & horizontal deflecting plate of CRO.
Horizontal amplifier: This amplifies the output of the time base generator. Some times,
it amplifies the horizontal input signal.
Trigger Ckt: It is a ckt that synchronies horizontal & vertical inputs. It makes sure that
the horizontal sweep and vertical input always start at the same point on the input signal.
Application of CRO:
It is used to measure the voltage, current, frequency & phase difference
of the given signals.
Measurement of voltage:
If the signal is applied to the vertical deflection plate only a vertical line
appears on the screen. The height of the line is proportional to peak voltage of the
Ø = 0o
Ø = 30o
Ø = 90o
Ø = 150o
Ø = 180o
applied signal. The vertical scale on the CRT screen is marked in Cms. Each centimeter
is further sub-divided into 5 parts, so that each part represents 0.2cm.
Measurement of current:
When a current is to be measured it is passed through a known resistance
and voltage across it is measured.
CRO: OSCILLOSCOPES DEFINITION:
Waveforms having a frequency as low as approximately 1 hertz (Hz) or as high
as several megahertz (MHz). High-end oscilloscopes can display signals having
frequencies up to several hundred gigahertzes (GHz). The display is broken up into so-
called horizontal divisions (hor div) and vertical divisions (vert div). Time is displayed
from left to right on the horizontal scale. Instantaneous voltage appears on the vertical
scale, with positive values going upward and negative values going downward.
CATHODE-RAY TUBE
Power and Scale Illumination: Turns instrument on and controls illumination of the
graticule.
Focus: Focus the spot or trace on the screen.
Intensity: Regulates the brightness of the spot or trace
ELECTROSTATIC focusing:
Figure: electrostatic focusing system of a CRT
The accelerating beam would be scattered now because of variation in energy and
would produce a brad ill-defined spot on the screen. This electron beam is focused on
the screen by an electrostatic lens consisting of two more accelerating anode.
Figure 7: electrostatic focusing
Preaccelerating anode
Focusing anode
Equipotential surfaces
focuse
Accelerating anode
High voltage supply
Control grid
Va
Anode voltage
Screen
C
Electron
gun
At equipotential surfaces the electron changes its direction. These equipotential surfaces act as electrostatic lens.
MAGNETIC FOCUSING:
Axis of tube along M. Field lines
Figure 8: The helical path of an electron introduced at an angle (not 90o) with a constant
M. Field.
Velocity of origin is v0.
Initial transverse velocity clue to repulsion v0x
Velocity = vy + vθ along & transverse to the M.F
Since F perpendicular B no ay = 0 as vy is constant and equal to v0y
F = eBvθ perpendicular to path exist, resulting from transverse velocity. This
force gives rise to circular motion. The radius of the circle is
R = mvθ /eB where vθ = v0x.
The pitch of the helix, defined as the distance traveled along the direction of the
magnetic field in one revolution is given by
P = v0yT T-time for one revolution = period
P = 2Πm/eB v0y
When applied M.F = 0 smudge is seen on screen different transverse velocities v0x as
different points on screen at M.F. increase e- follow helix path with diff R at v0x is diff
period is independent of v0x so the period will be same for all electrons. If then the
distance from the anode to the screen is made equal to one pitch all the electrons will be
brought back to the y axis (the point o| ) here only one spot.
Critical field smallest points here distance between A & S is P at critical field increase P
decrease and e- travels more than one revolution.
Electronic path
f = e B vox
ZR= m vox/ e B
X
Y O’
B
vo
voy vox
O
The current rating of the solenoid is the factor that generally furnishes a practical
limitation to the order of the focus. In General
If the screen is perpendicular to the y axis at a distance L from the point of emergence of
the electron beam from the anode, then, for an anode cathode potential equal to v a, the
electron beam will come to a focus at the center of the screen provided that L is an
integral multiple of P. under these conditions, we can write.
e/m = 8Π2 Van2/L2 B2 where n is an integer representing the order of the focus.
Here eVa = ½ mv0y2 the only effect of the anode potential is to accelerate the electron
along the tube axis this implies that the transverse velocity v0x, which is variable and
unknown, is negligible in comparison with v0y.
A short Focusing Coil:
Longitudinal M.Field over the entire length of a commercial tube is not too practical.
A short coil is wound around the neck of the tube.
Because of the fringing of the magnetic lines of flux a radial component of B exists in
addition to the component along the tube axis.
Two components of force on electron.
f = f (axial com of v + radial c of field) + f (axial com of field + radial c of v).
The motion will be a rotation about axis of the tube and if conditions are correct, the
electron on leaving the region of the coil may turn sufficiently so as to move in a line
toward the center of the screen.
A rough adjustment of focus is done by positioning of coil along neck. A fine
adjustment of focus is done by controlling the coil current.
ELECTROSTATIC DEFLECTION IN A CATHODE RAY
TUBE:
Those e- which are not collected by anode pass through the tiny anode hole and
strike the end of the glass envelope.
Figure 9: Electrostatic deflection in a cathode-ray tube.
D depends on Vd (defection plate’s potential)
V0x = (2e Va/m) ½ where initial velocity is negligible.
Between plate e- move in parabolic path given by
Y = ½ (ay/v0x2)x2
From point M at the edge of the plate path is straight line towards screen as it is field
free. This path is tangent to parabola at the point M slope of the line is
Tan θ = dy/dx where x=l
= ayl/v0x2
Straight line, is (from figure)
Y = ayl / v0x2 (x-l/2) --------- 1
Since x=l & y= ½ ay l2/v0x2 at the point M
AT POINT O | e- move towards p| in straight line path regardless of va & vd.
AT POINT P | Y=D, x=1+ ½ l equ 1 reduces to
D = = aylL/v0x2
Substituting equ ay & v0x
We get D = lLvd/2dva ------------ 2
D α Vd
Mean CRT can be used at a linear – voltage indicating device.
The electrostatic deflection sensitivity of a CRT is defined at the deflection (in meters)
on the screen per volt of deflecting voltage.
S ≡ D/vd = lL/2dVa. --------------- 3
S is independent of Vd & e/m & S α 1/vd.
Correction is needed in measured value of CRT.
Magnetic Deflection in a cathode – ray tube:
P’Vertical deflection plates
+ Vd + DCathode M θ d X O O’
P_ _ L
Va l/2 l/2AnodeVoltage
L flurorescent screen
Figure 10: Magnetic deflection in a cathode ray tube.
B is uniform in l region & zero outside.
Cathode to O: Straight line, in magnetic field: force of magnitude eBv, v.velocity. The
OM is arc of circle whose center is at Q the speed of e- is constant and equal to
v = v0x = (2eVa/m)½
The angle φ is by definition of radian measure, equal to the length of the arc OM/R, R
radius of circle. We assume a small angle of deflection then
φ ≈ l/R
We know R = mv/eB
If MP| projected backward will pass through the center O| of the region of the M.F. then
D ≈ L tan φ ≈ L φ = lL/R = lLeB/mv = lLB/(va)½ (e/2m)½
The deflection per unit M.F. intensity, D/B given by
D/B = lL/(va)½ (e/2m)½ is called magnetic deflection sensitivity of the tube.
MDS is independent of B.
MDS α 1/(va) ½ , MDS α (e/m) ½
MDS α L, magnetic coils are placed as far as down the neck of the tube as possible.
Usually directly affect the accelerating anode.
Deflection in a TV tube:
If magnetic deflection coil is driven by a saw tooth current waveform, the deflection of
the beam on the face of the tube will not be linear with time for such wide angle
deflection tubes, special linearity correcting n/w’s must be added. When the video signal
is applied to the electron gun, it modulates the intensity of the beam and thus forms the
TV picture.
Parallel Electric and Magnetic Fields:
Q P’ ** * * * * Magnetic field out of paper
θ * * * * * * * * * * * * DCathode * * * * * * * M θ * * * * * * * * * * * * * * X
O * * O’* * * P * * * * * * * * * * * * * * L
Va * * * * * * AnodeVoltage l/2 l/2
l flurorescent screen
If the initial velocity of the electron either is zero or is directed along the fields
the magnetic field exerts no force on the electron and the resulting motion depends
solely upon the electric field intensity E.
When e- parallel E & M.F with constant a.
If fields are selected as follow the motion of e- will be.
vy = v0y –at
y = v0y t -1/2 at2
‘-‘ indicate e- motion in opposite to E. Field.
If, initially, a component of velocity v0x perpendicular to the M.F. exists this component,
together with the M.F., will give rise to circular motion, the radius of the circular path
being independent of E.
E v various along the field with time.
The resulting path is helical with a pitch that changes with the time. i.e., the distance
traveled along the y axis per revolution increase with each revolution.
Perpendicular Electric and Magnetic field:
Assume E is in –x, B is in –y, force (due to B) perpendicular B is in xz plane.
ay = 0. Hence fy = 0, vy = v0y y = v0yt.
If the initial velocity component parallel to B is zero, the path lies entirely in a plane
perpendicular to B.
When e- starting at rest at the origin fMF = 0 since vy = 0. and since E is in –x fEM is in +x
e- moves, as e- move force due to Magnetic field is not 0 and that force will exist in +z
direction on e-.Therefore e- turns from +x to +z this will result to cycloid.
Force due to E.F. is eE along +x direction.
Force due to M.F. is as follows.
Due to vx the force is eBvx in +z direction
Due to vz the force is eBvz in –x direction
From Newton’s law
fx = m dvx/dt = eE -eBvz.
Figure 11:Parallel E.F & M.F
Y
E B
X
Z
fz = m dvz/dt = eBvx
we can write w ≡ eB/m and u≡E/B
we get dvx/dt = wu - wvz ------------- 1
dvz/dt = + wvx ---------------- 2
Differentiating equation 1 and combining with equation 2 we get
dvx2 / dt2 = -w dv2 / dt = -w2 vx -------------- 3
vx, vz = 0 initial conditions
vx = u sin wt, vz = u – u cos wt ------------ 4
In order to find coordinates x & z from these expressions each equation must be
integrated subject to the initial condition x=z=0.
X= u (1- cos wt)/w, z = ut- u/w sin wt -------------- 5
Θ = wt and Q = u/w
Then x = Q (1- cos Θ), z = Q (Θ- sin Θ)
Cycloidal path:
x = Q (1- cos Θ) ,, z = Q (Θ- sin Θ) are the parametric equ of a common cycloid ,
defined as the path generated by a point on the circumference of a circle of radius Q
which rolls along a straight line the Z-axis.
Figure 12: the cycloid path of an electron in perpendicular electric and magnetic fields
when the initial velocity is zero
oc’ is circumference that has already come in contact with Z-axis
oc’= pc’, arc = Q Θ
Θ gives number of radians through which the circle has rotated. We get from diagram
x = Q (1- cos Θ),
z = Q (Θ- sin Θ)
This path is cycloid.
W – Angular velocity of rotation of the rolling circle.
Θ – Number of radians through which the circle has rotated
Q – Radius of the rolling circle.
U = wQ is velocity of translation of the circle of center of the rolling circle.
Max displacement of e- along x is zQ (diameter of r circle).
Distance between cusps along z is 2Π Q (circumference).
At each cusp v=0 since v change its direction & same V. Therefore the electron has
gained no energy from the E.F. and its speed must again be zero.
If an initial velocity exists that is directed parallel to the magnetic field the projection of
the path on the xz plane will still be cycloid but the particle will now have a constant
Q x Qcosθ
O X
p Qθ rolling Qsinθ circle
C’ C
2ΠQ cycloidal path 2Q
velocity normal to the plane. This path may be called a “cycloidal helical motion” is
given by x = Q (1- cos Θ), z = Q (Θ- sin Θ) and vy = v0y, y = v0yt.
Straight line path:
When e- is perpendicular to both E.F. and M.F.
And V0x = v0y = 0 and v0x ≠ 0
The E.F. is eE along +x, M.F. is eBv0x along –x so net force is zero; it will continue to
move along the z axis with the constant speed v0z. This conditions is realized when
eE = eBv0z
v0z = E/B = u is independent of the charge or mass of the ions.
This system of perpendicular field will act as a ‘velocity filter’ and allow only those
particles whose velocity is given by the ratio E/B to be selected.
Trochoidal Paths:
If initial velocity component in the direction perpendicular to the M.F. is not zero, it can
be shown that the path is a trochoid. This curve is the locus of a point a “spoke” of a
wheel rolling on a straight line.
a. Q| (spoke length) > Q (radius) of rolling circle – prolate cycloid
b. Q| = Q common cycloid
c. Q| < Q curtate cycloid
Figure 13: The trochoidal paths of e- in perpendicular E & M.F.
X
b B magnetic a Field c ZInto paper
E