Lecture Notes on Magnetic Forces and Charged Particles

Physics 30 Lesson 20: Magnetic Forces – Charged Particles

I. Charged Particles in External Magnetic Fields

  • Understanding motion of charged particles in magnetic fields:

    • Charged particles create induced magnetic fields around them when in motion, which are perpendicular to the particle's motion.

    • When projected through an existing magnetic field (e.g., between two bar magnets), the induced magnetic field interacts with the external magnetic field.

    • This interaction results in a net force acting on the charged particle.

    • Induced Magnetic Field Representation:

      • The direction of the induced magnetic field is represented with an arrow:

      • Arrowhead indicates an induced north pole; tail indicates an induced south pole.

    • Interaction of induced and external fields:

      • Above the particle, the induced field is attracted to the magnets (downward force).

      • Below the particle, the fields repel each other (again resulting in a downward force).

    • Conclusion: A charged particle in a magnetic field experiences a deflecting force perpendicular to both its motion direction and the external magnetic field.

II. Third Hand Rule – Direction of the Magnetic Force

  • Third Hand Rule: Used to find the direction of force between charged particles and external fields.

    • Three quantities involved:

      • v: Direction of particle's motion

      • B: Direction of external magnetic field

      • F: Direction of resulting force

    • These quantities are mutually perpendicular, resembling x, y, and z coordinates in mathematics:

      • Fingers: Point in direction of external magnetic field (B) from north to south.

      • Thumb: Points in direction of the particle’s motion (v).

      • Palm: Indicates direction of the resulting force on the particle (F).

    • Usage:

      • Right hand for positive charges, left hand for negative charges.

  • Example 1: Determine deflecting force direction for:

    • A. An electron:

      • Left hand; fingers point right, thumb into page → force is upwards (out of page).

    • B. An alpha particle:

      • Right hand; fingers point right, thumb into page → force is downwards (into page).

III. Magnitude of the Deflecting Force

  • Formula for Deflecting Force:
    F = q v B ext{ sin } \theta

    • Where:

      • F = deflecting force (N)

      • B = magnetic flux density or field strength (Tesla, T)

      • q = charge of moving particle (C)

      • v = speed of particle (m/s)

      • \theta = angle between v and B

    • Maximum deflecting force occurs when v and B are perpendicular (i.e., \theta = 90^\circ).

      • Note: ext{sin } 90^\circ = 1

  • Example 2:

    • Given:

      • Particle mass = 20 mg (0.020 g)

      • Charge = +2.0 µC (2.0 x 10^-6 C)

      • Magnetic field = 0.020 T

      • Speed = 40 m/s

    • Acceleration experienced by particle:

      • Use right-hand rule: deflects upwards out of the page.

  • Example 3:

    • Finding magnetic field strength for an electron moving at 400 m/s with a deflecting force of 2.0 x 10^-19 N:

      • B = \frac{F}{qv}

      • B = \frac{2.0 \times 10^{-19} N}{(1.6 \times 10^{-19} C)(400 m/s)} = 3.1 \times 10^{-3} T

IV. Particles in Magnetic Fields

  • When a charged particle enters a magnetic field at 90º, it experiences a force perpendicular to its velocity that changes only the direction, not the speed, thus leading to uniform circular motion.

    • The magnetic force acts centrally:

      • F = m \frac{v^2}{r}

      • This identifies the magnetic force as a centripetal force.

      • Important relationship: derived during studying particle motion in magnetic fields.

  • Example 5: Determine the radius of an electron's path:

    • Given: Speed = 6.0 x 10^6 m/s; Magnetic field = 40 mT (0.040 T).

    • Use radius formula:

    • F = qvB = \frac{mv^2}{r}

      • Leads to radius computation.

V. Velocity Selectors

  • A velocity selector measures the speed of charged particles, consisting of a cylindrical tube within a magnetic field B and a parallel plate system creating an electric field E.

  • Forces acting: Magnetic force (FB) and electric force (FE). When these forces are equal in magnitude, the particle moves undeflected through the fields.

  • Particles not at the specific speed will be deflected up or down based on force imbalances.

  • Example 6:

    • An electron enters an electric field (4.5 x 10^5 N/C) and a perpendicular magnetic field (2.5 x 10^-2 T); determine its speed when passing undeflected.

VI. Mass Spectrometer (Mass Spectrograph)

  • Mass Spectrometer: Utilizes both centripetal motion and magnetic deflection for separating particles by mass.

    • Process:

    1. Ion generation (by heating/discharge)

    2. Acceleration through potential difference

    3. Passing through the spectrometer.

    • Results: Ions with equal charge but varying mass travel different paths.

    • Important consideration: Show derivations for velocity selectors and magnetic separation problems.

  • Basic Parts of a Mass Spectrometer:

    1. Ion source and accelerator

    2. Velocity selector

    3. Ion separator

  • Ion Generation and Acceleration:

    • Ions are accelerated through a potential difference, gaining kinetic energy.

  • Velocity Selector Function:

    • Ensures ions pass through at the same speed to allow for accurate separation.

    • The balancing of magnetic and electric forces occurs within this chamber.

  • Ion Separator:

    • Ions enter a magnetic field that determines their circular paths; radius provides mass information.

  • Example 7: Separation of uranium isotopes (U-235 vs U-239): Identifies differences in radius by mass and charge settings in mass spectrometer.

    • Given charge of +2 results in q = +3.20 x 10^-19 C.

VII. Van Allen Radiation Belts

  • Charged particles entering Earth's magnetic field spiral towards poles, a result of mixed velocity components.

  • These spiraling particles may lead to phenomena such as the Aurora Borealis and Aurora Australis.

  • The area containing these particles is known as the Van Allen Radiation Belt.

VIII. Black and White Television

  • Magnetic fields used to direct electrons in TV picture tubes:

    • Electrons accelerated through 50,000 V potential difference and directed by coils towards luminescent screen.

  • Each electron creates a brief glow upon impact, producing images through rapid sequences — multiple images create an illusion of motion (30 framed images scanned per second).

  • The particles produce a single complete image across 525 lines — 15,750 horizontal passes per second.

IX. Practice Problems

  • Tasked with solving problems involving force direction, magnetic field strength, and particle paths.

    • Example Problem 1: Identifying deflecting force direction for different charged particles (proton, fluoride ion).

    • Example Problem 3: Establish magnetic field strength with provided conditions.

X. Hand-In Assignment

  • Required tasks concerning the difference between magnets, particle charge movement through magnetic fields, understanding particle motion speed, and challenges with magentic field interactions.

  • Example Task 1: Differences between permanent magnets and electromagnets.

  • Example Task 2: Direction of resultant force on a negatively charged object in a magnetic field.

  • Example Task 4: Determining relative speeds of identical charge and mass particles in a magnetic field.

  • Example Task 10: Radius of circular path for a magnesium ion in a mass spectrometer.

30–B3.5k: Uniform Magnetic Field Effects on Moving Charges

  • Qualitative Interaction: A moving electric charge generates its own induced magnetic field. When this charge enters an external uniform magnetic field, the interaction between these fields results in a deflecting force (F). According to the Third Hand Rule, the force is always perpendicular to both the velocity (v) of the charge and the magnetic field lines (B).

  • Quantitative Analysis: The magnitude of the magnetic force is calculated using:

    • F = qvB \sin \theta

    • Maximum force occurs when motion and field are mutually perpendicular (\theta = 90^\circ), resulting in F = qvB.

    • Because the magnetic force acts perpendicular to the velocity, it changes only the particle's direction, not its speed, acting as a centripetal force (F_c):

    • qvB = \frac{mv^2}{r}

30–B3.6k: Perpendicular Magnetic and Electric Fields

  • Velocity Selectors: When a charge moves through a space with both a uniform electric field (E) and a perpendicular magnetic field (B), it experiences an electric force (Fe = qE) and a magnetic force (Fm = qvB).

  • Quantitative Relationship: For a particle to move undeflected, the forces must be balanced (Fe = Fm):

    • qE = qvB

    • v = \frac{E}{B}

    • Only particles with this specific speed will pass through the selector in a straight line; others will deflect towards the stronger force.

30–B3.7k: Interaction with Moving Charges and Conductors

  • Moving Charges: The deflection is essentially the result of the magnetic field from the moving charge 'pushing' against the external field lines. The direction depends on the charge type: use the Right Hand for positive charges and the Left Hand for negative charges.

  • Current-Carrying Conductors: A current is a stream of moving charges within a wire. When a conductor is placed in a magnetic field, the external field interacts with the individual magnetic fields of every moving charge in the wire. The sum of these individual magnetic forces results in a macroscopic force on the entire conductor, known as the Motor Effect.