Topic: More Applications of Ampere’s Law - Motion of charged particle in magnetic field

  • Recap of Ampere’s law
  • Introduction to the motion of a charged particle in a magnetic field
  • Definition of Lorentz force
  • Explanation of motion due to Lorentz force
  • Relation between force and magnetic field strength

Lorentz Force Equation:

The force experienced by a charged particle moving in a magnetic field can be given by the Lorentz force equation: $$\mathbf{F} = q(\mathbf{v} \times \mathbf{B})$$ where:

  • $\mathbf{F}$ is the force experienced by the charged particle,
  • $q$ is the charge of the particle,
  • $\mathbf{v}$ is the velocity of the particle, and
  • $\mathbf{B}$ is the magnetic field.

Direction of Motion:

  • The direction of the magnetic force $\mathbf{F}$ experienced by a charged particle is always perpendicular to both its velocity $\mathbf{v}$ and the magnetic field $\mathbf{B}$.
  • The motion of the charged particle is defined by the right-hand rule.

Circular Motion:

  • When a charged particle moves perpendicular to a uniform magnetic field, it experiences a force that causes it to move in a circular path.
  • The centripetal force required for circular motion is provided by the magnetic force.

Radius of Circular Path:

  • The radius of the circular path followed by a charged particle in a magnetic field can be determined using the formula: $$r = \frac{mv}{qB}$$ where:
    • $r$ is the radius of the circular path,
    • $m$ is the mass of the particle,
    • $v$ is the velocity of the particle,
    • $q$ is the charge of the particle, and
    • $B$ is the magnetic field strength.

Cyclotron Motion:

  • Cyclotron is a device that accelerates charged particles using a combination of electric and magnetic fields.
  • The charged particles move in a spiral path, gradually increasing in radius due to acceleration and crossing a gap between two D-shaped electrodes.

Cyclotron Equation:

  • The equation that relates the magnetic field strength, radius of motion, and kinetic energy of particles in a cyclotron is given by: $$B = \frac{mv}{qR}$$ where:
    • $B$ is the magnetic field strength,
    • $m$ is the mass of the particle,
    • $v$ is the velocity of the particle,
    • $q$ is the charge of the particle, and
    • $R$ is the radius of the circular path.

Velocity Selector:

  • A velocity selector is a device that allows charged particles with a specific velocity to pass through while deflecting those traveling with a different velocity.
  • It consists of a combination of electric and magnetic fields that balance the forces acting on the particles.

Velocity Selector Equation:

  • The equation that defines the condition for particles to pass through the velocity selector is given by: $$v = \frac{E}{B}$$ where:
    • $v$ is the velocity of the charged particle,
    • $E$ is the electric field strength, and
    • $B$ is the magnetic field strength.

Examples of Applications:

  • Cathode ray tubes (CRTs) in televisions and computer monitors
  • Particle accelerators
  • Mass spectrometers
  • Magnetic resonance imaging (MRI)
  • Hall effect sensors for measuring current
  • Magnetic confinement in fusion reactors

Magnetic Force on a Current-Carrying Wire:

  • A current-carrying wire placed in a magnetic field experiences a force.
  • The magnitude of the force can be calculated using the formula: $$F = ILB\sin(\theta)$$ where:
    • $F$ is the force on the wire,
    • $I$ is the current flowing through the wire,
    • $L$ is the length of the wire within the magnetic field,
    • $B$ is the magnetic field strength, and
    • $\theta$ is the angle between the wire and the magnetic field.

Applications:

  • Electric motor operation
  • Loudspeakers
  • Galvanometers

Magnetic Torque on a Current Loop:

  • A current loop placed in a magnetic field experiences a torque.
  • The magnitude of the torque can be calculated using the formula: $$\tau = IAB\sin(\theta)$$ where:
    • $\tau$ is the torque on the loop,
    • $I$ is the current flowing through the loop,
    • $A$ is the area of the loop,
    • $B$ is the magnetic field strength, and
    • $\theta$ is the angle between the normal of the loop and the magnetic field.

Applications:

  • Electric motors
  • Electromagnetic switches
  • Magnetic resonance imaging (MRI)

Magnetic Field due to a Straight Current-Carrying Wire:

  • The magnetic field at a point due to a straight current-carrying wire can be calculated using the formula: $$B = \frac{\mu_0 I}{2\pi r}$$ where:
    • $B$ is the magnetic field strength,
    • $\mu_0$ is the permeability of free space,
    • $I$ is the current flowing through the wire, and
    • $r$ is the distance from the wire to the point.

Applications:

  • Solenoids
  • Transformers
  • Circuit breakers

Magnetic Field due to a Circular Loop:

  • The magnetic field at the center of a circular loop carrying current can be calculated using the formula: $$B = \frac{\mu_0 I}{2R}$$ where:
    • $B$ is the magnetic field strength,
    • $\mu_0$ is the permeability of free space,
    • $I$ is the current flowing through the loop, and
    • $R$ is the radius of the loop.

Applications:

  • Magnetic resonance imaging (MRI)
  • Inductors
  • Magnetic sensors

Magnetic Field due to a Solenoid:

  • The magnetic field at the center of a long solenoid carrying current can be considered uniform and can be calculated using the formula: $$B = \mu_0 n I$$ where:
    • $B$ is the magnetic field strength,
    • $\mu_0$ is the permeability of free space,
    • $n$ is the number of turns per unit length of the solenoid, and
    • $I$ is the current flowing through the solenoid.

Applications:

  • Electromagnets
  • Transformers
  • Inductors

Magnetic Properties of Materials:

  • Some materials can be magnetized and exhibit magnetic behavior.
  • There are three types of materials based on their magnetic properties:
    1. Ferromagnetic materials: Examples include iron, nickel, and cobalt. These materials can be strongly magnetized and retain the magnetization even after the external magnetic field is removed.
    2. Paramagnetic materials: Examples include aluminum and platinum. These materials are weakly magnetized when subjected to an external magnetic field but lose their magnetization when the field is removed.
    3. Diamagnetic materials: Examples include copper and gold. These materials are weakly repelled by a magnetic field and do not retain magnetization.

Magnetic Domains:

  • In ferromagnetic materials, the atomic magnetic dipoles align in small regions called magnetic domains.
  • When external magnetic fields are applied, the domains can align and create a net magnetization for the material.

Hysteresis:

  • Hysteresis is the phenomenon where the magnetization of a material lags behind the applied magnetic field.
  • Hysteresis loops are graphical representations of the relationship between the magnetization and the magnetic field for a given material.

Magnetic Field of Earth:

  • The Earth has a magnetic field that aligns approximately along the North-South axis.
  • The Earth’s magnetic field can be represented by a magnetic dipole.
  • It influences various natural phenomena such as compass needles aligning with the magnetic field and the formation of the Northern and Southern Lights.

Review:

  • Recap of the main concepts covered:
    • Motion of a charged particle in a magnetic field
    • Magnetic force on a current-carrying wire
    • Magnetic torque on a current loop
    • Magnetic field due to straight wire, circular loop, and solenoid
    • Magnetic properties of materials and domains
    • Hysteresis and the Earth’s magnetic field

Slide 21

  • Applications of Ampere’s Law:
    • Magnetic field inside a solenoid
    • Magnetic field outside a solenoid
    • Magnetic field inside a toroid
    • Magnetic field outside a toroid
    • Magnetic field due to a straight current-carrying wire

Slide 22

  • Magnetic Field inside a Solenoid:
    • A solenoid is a long, cylindrical coil of wire.
    • Inside a solenoid, the magnetic field is uniform and parallel to the axis of the solenoid.
    • The magnitude of the magnetic field can be given by the equation: $$B = \mu_0 n I$$ where:
      • $B$ is the magnetic field strength,
      • $\mu_0$ is the permeability of free space,
      • $n$ is the number of turns per unit length of the solenoid,
      • $I$ is the current flowing through the solenoid.

Slide 23

  • Magnetic Field outside a Solenoid:
    • Outside a solenoid, the magnetic field is similar to that of a bar magnet.
    • The field lines are curved and point from the north pole to the south pole.
    • The magnetic field outside a solenoid is much weaker compared to inside the solenoid.
  • Magnetic Field inside a Toroid:
    • A toroid is a doughnut-shaped coil of wire.
    • Inside a toroid, the magnetic field is uniform and parallel to the axis of the toroid.
    • The magnitude of the magnetic field can be given by the equation: $$B = \frac{\mu_0 n I}{2\pi r}$$ where:
      • $B$ is the magnetic field strength,
      • $\mu_0$ is the permeability of free space,
      • $n$ is the number of turns per unit length of the toroid,
      • $I$ is the current flowing through the toroid, and
      • $r$ is the distance from the center of the toroid to the point.

Slide 24

  • Magnetic Field outside a Toroid:
    • Outside a toroid, the magnetic field is very weak.
    • The field lines are almost negligible due to the canceling effect of the currents in the opposite directions.
  • Magnetic Field due to a Straight Current-Carrying Wire:
    • The magnetic field at a point due to a long straight wire can be given by the equation: $$B = \frac{\mu_0 I}{2\pi r}$$ where:
      • $B$ is the magnetic field strength,
      • $\mu_0$ is the permeability of free space,
      • $I$ is the current flowing through the wire, and
      • $r$ is the distance from the wire to the point.

Slide 25

  • Magnetic Field due to a Circular Loop:
    • The magnetic field at the center of a circular loop carrying current can be given by the equation: $$B = \frac{\mu_0 I}{2R}$$ where:
      • $B$ is the magnetic field strength,
      • $\mu_0$ is the permeability of free space,
      • $I$ is the current flowing through the loop, and
      • $R$ is the radius of the loop.
  • Magnetic Field due to a Solenoid:
    • The magnetic field at the center of a long solenoid carrying current can be considered uniform and can be given by the equation: $$B = \mu_0 n I$$ where:
      • $B$ is the magnetic field strength,
      • $\mu_0$ is the permeability of free space,
      • $n$ is the number of turns per unit length of the solenoid, and
      • $I$ is the current flowing through the solenoid.

Slide 26

  • Applications of Ampere’s Law:
  • Examples:
    • Magnetic field inside a solenoid is used in transformers and inductors.
    • Magnetic field outside a solenoid is utilized in magnetic sensors and MRI machines.
    • Magnetic field inside a toroid is used in particle accelerators.
    • Magnetic field due to a straight current-carrying wire is used in electric motors and loudspeakers.
    • Magnetic field due to a circular loop is used in magnetic sensors and electric generators.
    • Magnetic field due to a solenoid is used in electromagnets and transformers.

Slide 27

  • Magnetic Properties of Materials:
  • Ferromagnetic materials:
    • Examples: iron, nickel, and cobalt.
    • Properties: strong magnetization, retain magnetization even after the external field is removed.
  • Paramagnetic materials:
    • Examples: aluminum and platinum.
    • Properties: weak magnetization, lose magnetization when the field is removed.
  • Diamagnetic materials:
    • Examples: copper and gold.
    • Properties: weakly repelled by a magnetic field, do not retain magnetization.

Slide 28

  • Magnetic Domains:
  • Ferromagnetic materials have atomic magnetic dipoles that align in small regions called magnetic domains.
  • When an external magnetic field is applied, the domains can align either parallely or anti-parallely, resulting in a net magnetization for the material.

Slide 29

  • Hysteresis:
  • Hysteresis is the phenomenon where the magnetization of a material lags behind the applied magnetic field.
  • Hysteresis loops are graphical representations of the relationship between the magnetization and the magnetic field for a given material.
  • The area of the hysteresis loop represents the energy loss during a complete magnetization cycle.

Slide 30

  • Magnetic Field of Earth:
  • The Earth has a magnetic field that aligns approximately along the North-South axis.
  • The magnetic field protects the Earth from harmful cosmic rays and solar wind.
  • The magnetic field influences various natural phenomena such as compass needles aligning with the magnetic field and the formation of the Northern and Southern Lights.
  • End of Lecture.