46Cathode rays, Positive rays and Electrons

📚
CHARGED PARTICLES AND DISCHARGE
▢ Millikan's Oil Drop Experiment:
❖ 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
▢ Gaseous Discharge:
❖ Basic Point: Dry gas at NTP is bad conductor but becomes good conductor at low pressure

Table 1: Discharge Tube Phenomena at Different Pressures

Pressure
Observation
\(10\ mmHg\)
Crackling sound and luminous streaks / blue streamers
\(5\ mmHg\)
Geissler discharge
\(5\ mmHg\)
Colour of positive column depends on gas
Neon
Red colour
Hydrogen
Blue colour
\(1\ mmHg\)
Positive column near anode and bluish negative glow near cathode
\(1\ mmHg\)
Faraday dark space between positive column and negative glow
\(0.5\ mmHg\)
Cathode glow and Crookes dark space appear
❖ Conduction: Due to positive ions, negative ions and electrons
▢ 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
\(v=\frac{E}{B}\)
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Power of ionization of a gamma particle is
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Q2.
If electron, proton, neutron and alpha-particle are deflected in the same electric field with same velocity, what will be the deflection in them?
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An electron moving with velocity v enters a uniform electric field perpendicularly. Its trajectory within the field will be
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Q4.
Quantum theory gives the concept of
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Q5.
Which one of the following statements is true for cathode rays?
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Q6.
An electron of charge e is at rest in an electric field between two plates separated by distance d and with potential difference V. The force experienced by it is
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Penetrating power in decreasing order is
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Q8.
Ionizing power of gamma ray as compared to alpha particle is
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Cathode rays are
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When an electron moves through a uniform magnetic field, its speed
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The kinetic energy of a proton accelerated by 1 V is
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An electron of mass m and charge e is accelerated from rest through a potential difference of V volts in vacuum. The speed of electron will be
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Work done in carrying an electron across a potential difference of 10 V is
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Q14.
In Millikan's oil drop experiment, an oil drop is held stationary by a potential difference of 400 V. If another drop of double radius but same charge is to be held stationary, the required potential difference is
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Q15.
The ratio of charge-to-mass ratio of a proton to an alpha-particle equals
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Q16.
Cathode rays are produced when the pressure in the discharge tube is of the order of
Q17.
The colour of positive column in a gas discharge tube depends on
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Cyclotron is used to accelerate
Q19.
A strong argument for particle nature of cathode rays is
Q20.
The resistance of discharge tube is
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An electron is accelerated through a potential difference of 200 V. If e/m for electron is 1.6 × 10^11 C/kg, the velocity acquired by the electron will be
Q22.
Doubly ionised helium atom and hydrogen ions are accelerated from rest through the same potential difference. The ratio of final velocities of helium and hydrogen is
Q23.
An oil drop of mass 50 mg and charge -5 μC is just balanced in air against gravity. Calculate the electric field required to balance it. Take g = 9.8 m/s².
Q24.
A charged dust particle of radius 5 × 10^-7 m moves in horizontal electric field of intensity 6.28 × 10^5 V/m. If η = 1.6 × 10^-5 Nsm^-2 and speed is 0.01 m/s, the number of electrons on it is
Q25.
An oil drop carrying charge q has mass m kg. It is falling freely in air with terminal velocity v. The electric field required to make the drop move upwards with the same speed is
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