In-Depth Notes on Magnetism and Charged Particles

Charged Particles in Magnetic Fields

  • A moving charged particle, such as an electron, experiences a force when it enters a magnetic field (B-field).
  • Direction of B-field is typically defined as into the screen.
  • This magnetic force is always oriented towards the center of the trajectory, acting as a centripetal force.
    • Causes the charged particle to move in a circular path.

Applications of Magnetic Fields

  • Traditional screens, such as CRTs (Cathode Ray Tubes), use magnetic fields to manipulate electron paths created by an "electron gun".
    • The deflected electrons illuminate phosphor on the screen, creating images.
    • Rapid succession of images (every 1/24th of a second) gives the illusion of smooth motion.

Centripetal and Magnetic Force Equations

  • Centripetal Force (

    Fc=mv2RF_c = \frac{mv^2}{R}

    )

    • Where:
      • $m$: mass of the particle
      • $v$: velocity of the particle
      • $R$: radius of the circular path

  • Magnetic Force Equation:

    Fm=BqvF_m = Bqv

    • Where:
      • $B$: magnetic field strength
      • $q$: charge of the particle
      • $v$: speed of the particle

  • For an electron moving in a magnetic field, we can equate forces:

    mv2R=Bqv\frac{mv^2}{R} = Bqv

    by rearranging yields:

    R=mvBqR = \frac{mv}{Bq}

Example Problems

  • Example with a mass spectrometer:

    • Distance from the entrance slit ($R1 = 24$ cm, $R2 = 25.37$ cm)
    • Given the mass of the first ion ($m_1 = 34.980$ amu), one can calculate the mass of the second isotope using the formula:

    R<em>12R</em>22=m<em>2m</em>1\frac{R<em>1^2}{R</em>2^2} = \frac{m<em>2}{m</em>1}

  • Radius Calculation Example:

    • Charge with mass $m = 4 \times 10^{-15}$ kg, charge $q = 2.0 \times 10^{-9}$ C, and speed $v = 4 \times 10^6$ m/s.
    • Inward magnetic field strength $B = 5.0$ T leads to radius:

    R=mvBq=(4x10−15)(4x106)(2.0x10−9)(5)=1.6 mR = \frac{mv}{Bq} = \frac{(4x10^{-15})(4x10^6)}{(2.0x10^{-9})(5)} = 1.6 \text{ m}

The Thompson Tube and Electron Mass

  • Inside the Thompson Tube:
    • Streams of electrons pass through slits into combined electric (E) and magnetic (B) fields set perpendicular to each other.
    • Equal forces ($E = BqV$) imply that the electrons follow a straight trajectory.
  • When electric fields are turned off, electrons follow a circular path under the influence of the magnetic field.
Formula for Electron Velocity Calculation

  • Thompson devised:

    v=EBv = \frac{E}{B}

  • Empirical observations led Thompson to determine:

    • Mass of an electron: 9.11×10−31 kg9.11 \times 10^{-31} \text{ kg}
    • Mass of a proton: 1.67×10−27 kg1.67 \times 10^{-27} \text{ kg}

Characteristics of Mass Spectrometry Techniques

  • Mass spectrometers study isotopes by:

    • Ionizing a vaporized sample, followed by acceleration and deflection through electric/magnetic fields.

  • The separation and measurement relate to the charge-to-mass ratio:

    qm=2VB2R2\frac{q}{m} = \frac{2V}{B^2R^2}

Mass Calculation Example in Mass Spectrometry

  • For a doubly ionized neon atom:

    • Given potential difference $V = 34$ V and $B = 0.05$ T, radius $r = 0.053$ m, use:

    m=qB2r22Vm = \frac{qB^2r^2}{2V}

  • Repeat calculations for other ions with varying charge states.