faradays and more magnitude force notes

Overview of Charged Particles in Magnetic Fields

  • Discussion on the effects of charged particles, such as protons or electrons, when they move through a magnetic field.

Right Hand Rule

  • Two right hand rules:
    • One for current-carrying wires.
    • One for charged particles in magnetic fields.
  • Application of the right hand rule involves:
    • Thumb pointing in the direction of velocity (v).
    • Index finger pointing in the direction of magnetic field (b).
    • Palm or middle finger pointing in the direction of force (F) for positive charges.

Magnetic Force Calculation

  • Magnetic force is calculated using the formula: F=qbbextsin(heta)F = q b b ext{sin}( heta) where:
    • $F$ = Magnetic force
    • $q$ = Charge of the particle (absolute value)
    • $b$ = Magnetic field strength
    • $ heta$ = Angle between velocity vector and magnetic field vector.
Example Problem
  1. Given a charged particle in a magnetic field:
    • Charge (q) = Charge of an electron = -1.6 × 10^{-19} C (absolute value used)
    • Magnetic field (b) = 0.7 T
    • Angle ($ heta$) = 90 degrees
  2. Calculating the Magnitude:
    • F=qbbextsin(90exto)F = q b b ext{sin}(90^{ ext{o}})
    • Sine of 90 degrees = 1
    • Resulting in: F=∣qbb∣F = |q b b|
    • Calculation:
    • Substitute known values: F=1.6imes10−19Cimes0.7Times1F = 1.6 imes 10^{-19} C imes 0.7 T imes 1
    • Final Result: F=2.8imes10−12NF = 2.8 imes 10^{-12} N
Understanding the Angles
  • Clarification of angles in three-dimensional space:
    • When analyzing angles in a coordinate system, the 90-degree angle between the vectors can be visualized using a top-down approach.

Force Direction with Right Hand Rule

  • For a positive charge: F direction is upward (palm points up).
  • For a negative charge, the force direction flips, pointing downward (negative z-direction).
Relation to Circular Motion
  • Forces acting on charged particles moving in a magnetic field are always perpendicular to their velocity, causing circular motion.
  • Magnetic force ($F_{magnetic}$) = centripetal force ($F_{centripetal}$):
    • F=mv2rF = \frac{m v^2}{r}
    • Equating magnetic and centripetal force gives:
      qbb=mv2rq b b = \frac{m v^2}{r}
  • Rearranging the equation helps find the radius (r) of the path that the particle travels:
    r=mvqbr = \frac{m v}{q b}

Cyclotron Principle

  • A cyclotron accelerates charged particles to high speeds using alternating electric fields:
    • Structure includes two D-shaped electrodes (called D's) where protons can gain speed and energy between the D's.
  • The protons spiral outward, each loop increasing their speed as they gain energy in the magnetic field created by the cyclotron.
  • Outcome: Protons exit and bombard isotopes for applications such as medical imaging (e.g., PET scans).
Example Problem on Cyclotron
  • Given a magnetic field strength (B) = 1.2 T, and neutron kinetic energy:
  1. Calculate velocity from kinetic energy:
    • Kinetic Energy (KE) = 11 MeV = 11imes106imes1.6imes10−19J11 imes 10^6 imes 1.6 imes 10^{-19} J = 1.76imes10−12J1.76 imes 10^{-12} J
    • Using kinetic energy formula: v=extsqrt(2KEm)v = ext{sqrt}(\frac{2 KE}{m})
    • Substitute values:
    • Mass of a proton ($m$) = 1.67imes10−27kg1.67 imes 10^{-27} kg
  2. Solve for v:
    • Result: v=4.6imes107m/sv = 4.6 imes 10^{7} m/s
  3. Calculate radius:
    • r=mvqbr = \frac{m v}{q b}, known values substituted yield: rext(approx)=0.4extmetersr ext{ (approx)} = 0.4 ext{ meters}.

Earth's Magnetic Field and Charged Particles

  • Charged particles moving in the Earth's magnetic field exhibit helical motion due to the combination of perpendicular and parallel components of velocity.
  • Results in phenomena like the northern lights (aurora borealis) as charged particles collide with atmospheric gases.

Magnetic Flux Concept

  • Magnetic flux ($ ext{Φ}$) is defined as the quantity of magnetic field passing through a given area:
    • extΦ=BimesAimesextcos(heta)ext{Φ} = B imes A imes ext{cos}( heta) where:
    • $B$ = Magnetic field
    • $A$ = Effective area of the loop
    • $ heta$ = Angle between magnetic field lines and the normal to the surface area.
  • Unit of flux is the Weber:
    -1extWeber=1extTeslaimes1extm21 ext{ Weber} = 1 ext{ Tesla} imes 1 ext{ m}^2.
Simulation of Induced Current
  • Demonstration of induced current generated by moving a magnet near a coil of wire, showing the interrelation of magnetism and electric current generation.

Clarification of Terms

  • Flux in the context of physics refers to a quantified value and does not imply change, unlike its everyday use as something being in change or transition.