46Cathode rays, Positive rays and Electrons

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MILLIKAN'S OIL DROP EXPERIMENT
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▢ Use: Determination of charge of electron
▢ Based On:
  • •
    Stokes' law of viscosity
  • •
    Quantization of charge
▢ Importance: Strong evidence of quantization of charge

Table 1: Millikan's Oil Drop Formulae

Quantity
Formula
Viscous falling relation
\(\frac{v}{r^2g}=\frac{2}{9}\frac{\rho-\sigma}{\eta}\)
Charge relation
\(qE=6\pi\eta r(v_1+v_2)\)
Charge on oil drop
\(q=\pm ne\)
Electronic charge
\(e=1.6\times10^{-19}C\)
▢ Symbols:
  • •
    \(v_1\) = velocity of drop with no electric field
  • •
    \(v_2\) = velocity of drop with electric field
  • •
    \(\rho\) = density of oil
  • •
    \(\sigma\) = density of medium
  • •
    \(\eta\) = coefficient of viscosity
  • •
    \(r\) = radius of oil drop
  • •
    \(E\) = electric field
  • •
    \(n=1,2,3,...\)
▢ Conclusion: Charge on every droplet is integral multiple of electronic charge; quantum nature of charge established
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GASEOUS DISCHARGE
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▢ Basic Point: Dry gas at NTP is bad conductor but becomes good conductor at low pressure
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Table 1: Discharge Tube Phenomena at Different Pressures

Pressure
Observation
\(10\ mmHg\)
Crackling sound and luminous streaks / blue streamers
\(5\ mmHg\)
  1. Geissler discharge
  2. Colour of positive column depends on gas
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    1. Neon → Red colour
    2. Hydrogen → Blue colour
\(1\ mmHg\)
  1. Positive column near anode and bluish negative glow near cathode
  2. Faraday dark space between positive column and negative glow
\(0.5\ mmHg\)
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  1. Cathode glow and Crookes dark space appear
▢ Conduction: Due to positive ions, negative ions and electrons
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CATHODE RAYS
▢ Discovered By: Crookes
▢ Nature: Stream of fast-moving electrons

Table 1: Cathode Ray Properties

Property
Answer
Nature
Particle nature
Not
Electromagnetic waves
Deflection
Deflected by electric and magnetic fields
Heat effect
Produce heat on metal surface
Fluorescence
Produce fluorescence
Photographic plate
Affect photographic plate
Energy
Possess kinetic energy
Mechanical effect
Exert mechanical pressure
Ionization
Ionize gas through which they pass
Ionizing power
Less than positive rays
Speed
\(\frac{1}{30}\) to \(\frac{1}{10}\) of speed of light

Table 2: Electron Constants

Quantity
Value
Charge of electron
\(e=1.6\times10^{-19}C\)
Mass of electron
\(m=9.1\times10^{-31}kg\)
Specific charge
\(\frac{e}{m}=1.76\times10^{11}C/kg\)
▢ Specific Charge:
❖ Determined By: J.J. Thomson
❖ Method: Crossed electric and magnetic fields
❖ Independent Of:
  • •
    Nature of gas used in discharge tube
  • •
    Material of cathode
📚
POSITIVE RAYS
▢ Discovered By: Goldstein
▢ Nature: Streams of positive ions moving towards cathode in discharge tube

Table 1: Positive Ray Properties

Property
Answer
Constituents
Positive ions
Direction
Towards cathode
Place of production
Produced at different places in discharge tube
Velocity
Different ions acquire different velocities
Deflection
Deflected by electric and magnetic fields
Mass
Heavier than cathode rays
Speed
Much less than cathode rays
Ionization
Ionize gas through which they pass
Ionizing power
More than cathode rays
📚
EFFECT OF ELECTRIC FIELD ON CHARGED PARTICLE
▢ Condition: Charged particle of charge \(q\), mass \(m\), velocity \(\vec v\) enters uniform electric field \(\vec E\)

Table 1: Charged Particle in Electric Field

Quantity
Formula / Result
Force
\(\vec F=q\vec E\)
Positive charge
Force along \(\vec E\)
Negative charge
Force opposite to \(\vec E\)
Acceleration
\(\vec a=\frac{\vec F}{m}=\frac{q\vec E}{m}\)
If \(\vec v\parallel\vec E\) or anti-parallel
Path is straight line
If angle is other than \(0^\circ\) or \(180^\circ\)
Path is parabolic
Time inside field of length \(l\)
\(t=\frac{l}{v}\)
Deflection when field is perpendicular to initial motion
\(y=\frac{1}{2}at^2\)
Deflection
\(y=\frac{1}{2}\frac{qE}{m}\frac{l^2}{v^2}\)
▢ Changes In:
  • •
    Speed
  • •
    Velocity
  • •
    Momentum
  • •
    Kinetic energy
📚
EFFECT OF MAGNETIC FIELD ON CHARGED PARTICLE
▢ Condition: Charged particle of charge \(q\), mass \(m\), velocity \(\vec v\) enters magnetic field \(\vec B\)

Table 1: Charged Particle in Magnetic Field

Quantity
Formula / Result
Magnetic force
\(\vec F=q(\vec v\times\vec B)\)
Direction of force
Perpendicular to \(\vec v\) and \(\vec B\)
If \(\vec v\perp\vec B\)
\(F=qvB\), maximum force
Path for \(\vec v\perp\vec B\)
Circular path
Radius of circular path
\(r=\frac{mv}{qB}\)
If \(\vec v\parallel\vec B\) or anti-parallel
\(F=0\), velocity unchanged
Acceleration
\(a=\frac{qvB}{m}\)
Time inside magnetic field of length \(l\)
\(t=\frac{l}{v}\)
Magnetic deflection
\(y=\frac{1}{2}at^2\)
Magnetic deflection
\(y=\frac{1}{2}\frac{qBl^2}{mv}\)
If angle \(\theta\ne0^\circ,90^\circ,180^\circ\)
Path is helical
Undeviated motion in crossed fields
\(v=\frac{E}{B}\)
▢ Unchanged Quantities:
  • •
    Speed
  • •
    Magnitude of momentum
  • •
    Kinetic energy
📚
CATHODE RAYS VS POSITIVE RAYS

Table 1: Comparison

Feature
Cathode rays
Positive rays
Discovered by
Crookes
Goldstein
Constituents
Electrons
Positive ions
Direction in discharge tube
From cathode towards anode
Towards cathode
Mass
Very small
Large
Speed
High
Low
Deflection
More
Less
Ionizing power
Less
More
Nature
Same for all gases/cathodes
Depends on gas in tube
📚
HIGH-YIELD RECALL

Table 1: Charged Particles One-Liners

Fact
Answer
Millikan experiment
Determines charge of electron
Millikan experiment based on
Stokes' law
Quantization of charge
\(q=\pm ne\)
Electron charge
\(1.6\times10^{-19}C\)
Electron mass
\(9.1\times10^{-31}kg\)
Specific charge
\(1.76\times10^{11}C/kg\)
Dry gas at NTP
Bad conductor
Gas at low pressure
Good conductor
Geissler discharge
At about \(5\ mmHg\)
Cathode rays
Fast-moving electrons
Cathode rays discovered by
Crookes
Positive rays discovered by
Goldstein
Positive rays
Positive ions
Cathode ray speed
\(\frac{1}{30}\) to \(\frac{1}{10}\) of \(c\)
Electric force
\(\vec F=q\vec E\)
Electric acceleration
\(\vec a=\frac{q\vec E}{m}\)
Electric field straight path
\(\vec v\parallel\vec E\) or anti-parallel
Electric field parabolic path
\(\theta\ne0^\circ,180^\circ\)
Electric field deflection
\(y=\frac{1}{2}\frac{qE}{m}\frac{l^2}{v^2}\)
Magnetic force
\(\vec F=q(\vec v\times\vec B)\)
Maximum magnetic force
\(F=qvB\)
Magnetic circular radius
\(r=\frac{mv}{qB}\)
Magnetic acceleration
\(a=\frac{qvB}{m}\)
Magnetic deflection
\(y=\frac{1}{2}\frac{qBl^2}{mv}\)
Magnetic field does not change
Speed, KE, magnitude of momentum
Helical path
\(\theta\ne0^\circ,90^\circ,180^\circ\)
Undeviated crossed fields
\(v=\frac{E}{B}\)
Q1.
Proton accelerated through V volt has wavelength λ. Alpha particle needs same wavelength when accelerated through
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Q2.
A cathode ray is:
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Q3.
The kinetic energy of a proton accelerated by 1 V is:
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Q4.
Find the wavelength of an electron moving with a velocity of 300 km/s.
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Q5.
If the kinetic energy of a particle is 16 times its initial kinetic energy, then the percentage change in de Broglie wavelength will be:
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Q6.
A proton is 1836 times heavier than an electron. If the proton is accelerated through 1 V, its kinetic energy is:
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Q7.
For the same de Broglie wavelength of an electron and a proton, the kinetic energy is greater for:
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Q8.
Electrons are:
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Q9.
What is the formula of the de Broglie wavelength of a particle?
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Q10.
What is the de-Broglie wavelength of the electron accelerated through a potential difference of 150 volt?
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Q11.
2 kg of a moving object have 20 km/s velocity, then the de Broglie wavelength associated with it is:
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Q12.
Which radiation has the lowest ionizing power?
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Q13.
A proton and an electron enter the same uniform electric field with equal initial speeds perpendicular to the field. Comparing the magnitudes of their deflections over the same time interval, which statement is correct?
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Q14.
An electron enters a uniform electric field with initial velocity \(v\) perpendicular to the field. What is its trajectory while inside the field (neglecting gravity)?
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Q15.
Quantum theory of light introduced the concept of:
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Q16.
Which statement about cathode rays is correct?
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Q17.
An electron of charge \(-e\), where \(e\) is the positive elementary-charge magnitude, is placed between parallel plates separated by \(d\) with potential difference \(V\). What is its force component along the electric-field direction?
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Q18.
Arrange alpha, beta and gamma radiations in decreasing order of penetrating power.
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Q19.
Compared with alpha particles, the ionizing power of gamma rays is:
Q20.
Cathode rays are:
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Q21.
An electron moving in a uniform magnetic field experiences only the magnetic force. Its speed:
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Q22.
What kinetic energy does a proton gain when accelerated from rest through a potential difference of \(1\,\mathrm V\)?
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Q23.
An electron of mass \(m\) and charge magnitude \(e\) is accelerated from rest through potential difference \(V\) in vacuum. What is its nonrelativistic speed?
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Q24.
What is the magnitude of work done in moving one electron through a potential difference of \(10\,\mathrm V\)?
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Q25.
In Millikan's oil-drop experiment, a drop is suspended by a potential difference of \(400\,\mathrm V\). If a second drop has twice its radius but the same charge and density, with unchanged plate separation, what potential difference is needed to suspend it (neglect buoyancy)?
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Q26.
The ratio of the charge-to-mass ratio of a proton to that of an alpha particle is:
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Q27.
Cathode rays are produced in a discharge tube when the gas pressure is approximately:
Q28.
The colour of the positive column in a gas-discharge tube mainly depends on:
Q29.
A conventional cyclotron is primarily used to accelerate:
Q30.
Which observation directly establishes that cathode rays carry electric charge?
Q31.
The current–voltage behaviour of a gas-discharge tube is generally:
Q32.
An electron is accelerated from rest through \(200\,\mathrm V\). If its charge-to-mass ratio magnitude is \(1.6\times10^{11}\,\mathrm{C/kg}\), what speed does it acquire?
Q33.
Doubly ionized helium ions \(\mathrm{He}^{2+}\) and hydrogen ions \(\mathrm H^+\) are accelerated from rest through the same potential difference. What is the ratio of their final speeds \(v_{\mathrm{He}}:v_{\mathrm H}\)?
Q34.
A negatively charged oil drop of mass \(50\,\mathrm{mg}\) and charge \(-5\,\mu\mathrm C\) is suspended against gravity. Find the required electric field direction and magnitude. Take \(g=9.8\,\mathrm{m/s^2}\).
Q35.
A charged spherical dust particle of radius \(5\times10^{-7}\,\mathrm m\) moves at terminal speed \(0.01\,\mathrm{m/s}\) in a horizontal electric field of \(6.28\times10^5\,\mathrm{V/m}\). If air viscosity is \(\eta=1.6\times10^{-5}\,\mathrm{N\,s/m^2}\), how many elementary charges does it carry? Neglect other horizontal forces and take \(e=1.6\times10^{-19}\,\mathrm C\).
Q36.
A charged oil drop of mass \(m\) falls at terminal speed \(v\) in still air. What electric-field magnitude is needed to make it rise steadily at the same speed, if the charge magnitude is \(q\)? Assume Stokes' drag applies.
Q37.
In Millikan's oil-drop experiment, a charged drop of mass \(1.8\times10^{-14}\,\mathrm{kg}\) is stationary between horizontal plates \(0.90\,\mathrm{cm}\) apart at \(2\,\mathrm{kV}\). Approximately how many elementary charges does it carry? Use \(g=9.8\,\mathrm{m/s^2}\) and \(e=1.6\times10^{-19}\,\mathrm C\).
Q38.
Electric conduction in a gas-discharge tube can occur through the movement of:
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