6.3.7 Motion of Charged Particles in an E Field

• Permittivity is the measure of how easy it is to generate an electric field in a certain material
• The relativity permittivity ε_r is sometimes known as the dielectric constant
• For a given material, it is defined as:

The ratio of the permittivity of a material to the permittivity of free space

• This can be expressed as:

  

• Where:
  • ε_r = relative permittivity
  • ε = permittivity of a material (F m⁻¹)
  • ε₀ = permittivity of free space (F m⁻¹)

• The relative permittivity has no units because it is a ratio of two values with the same unit

• When the polar molecules in a dielectric align with the applied electric field from the plates, they each produce their own electric field
  • This electric field opposes the electric field from the plates

The electric field of the polar molecules opposes that of the electric field produced by the parallel plates

• The larger the opposing electric field from the polar molecules in the dielectric, the larger the permittivity
  • In other words, the permittivity is how well the polar molecules in a dielectric align with an applied electric field

• The opposing electric field reduces the overall electric field, which decreases the potential difference between the plates
  • Therefore, the capacitance of the plates increases

• The capacitance of a capacitor can also be written in terms of the relative permittivity:

• Where:
  • C = capacitance (F)
  • A = cross-sectional area of the plates (m²)
  • d = separation of the plates (m)
  • ε_r = relative permittivity of the dielectric between the plates
  • ε₀ = permittivity of free space (F m⁻¹)

• Capacitor plates are generally square, therefore, if it has a length of L on all sides then the cross-sectional area will be A = L² 

A parallel plate capacitor consists of conductive plates each with area A, a distance d apart and a dielectric ε between them

Step 1: Write down the relative permittivity equation

Step 2: Rearrange for permittivity of the material ε

ε = ε_rε₀

Step 3: Substitute in the values

ε = (4.5 × 10¹¹) × (8.85 × 10⁻¹²) = 3.9825 = 4 F m⁻¹

• A charged particle in an electric field will experience a force on it that will cause it to move
• If a charged particle remains still in a uniform electric field
  • It will move parallel to the electric field lines (along or against the field lines depending on its charge)
• If a charged particle in motion travels initially perpendicular through a uniform electric field (e.g. between two charged parallel plates)
  • It will experience a constant electric force and travel in a parabolic trajectory

The parabolic path of charged particles in a uniform electric field

• The direction of the parabola will depend on the charge of the particle
  • A positive charge will be deflected towards the negative plate
  • A negative charge will be deflected towards the positive plate

• The force on the particle is the same at all points and is always in the same direction
• Note: an uncharged particle, such as a neutron experiences no force in an electric field and will therefore travel straight through the plates undeflected

• The amount of deflection depends on the following properties of the particles:
  • Mass – the greater the mass, the smaller the deflection and vice versa
  • Charge – the greater the magnitude of the charge of the particle, the greater the deflection and vice versa
  • Speed – the greater the speed of the particle, the smaller the deflection and vice versa

Step 1: Compare the charge of the boron nucleus to the proton

• • Boron has 5 protons, meaning it has a charge 5 × greater than the proton
  • The force on boron will therefore be 5 × greater than on the proton

Step 2: Compare the mass of the boron nucleus to the proton

• • The boron nucleus has a mass of 11 nucleons meaning its mass is 11 × greater than the proton
  • The boron nucleus will therefore be less deflected than the proton

Step 3: Draw the trajectory of the boron nucleus

• • Since the mass comparison is much greater than the charge comparison, the boron nucleus will be much less deflected than the proton
  • The nucleus is positively charged since the neutrons in the nucleus have no charge
    • Therefore, the shape of the path will be the same as the proton