Magnetic Fields

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Textbook

• University Physics Volume 2: Chapter 11 - Magnetic Fields

Magnets (Ferromagnets and Electromagnets)

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Ferromagnets

Materials such as iron, cobalt, and nickel exhibit strong magnetic properties due to the alignment of their atomic magnetic moments.

• Ferromagnetic domains align to create a net magnetic field.

Electromagnets

Electromagnets use electric current to generate a magnetic field. The magnetic field strength is proportional to the current and the number of coils.

• Formula: $B=\mu _{0}\cdot n\cdot I$

 Where:
 * $B$ is the magnetic field,
 * $\mu _{0}$ is the permeability of free space,
 * $n$ is the number of turns per unit length,
 * $I$ is the current.

Models for the Source of Magnetism

The magnetic properties of materials arise from:

• The spin of electrons,
• The orbital motion of electrons around the nucleus.

Application of Ferromagnets and Electromagnets

• MRI machines (Electromagnets),
• Hard drives (Ferromagnets),
• Electric motors (Electromagnets),
• Transformers (Electromagnets).

Magnetic Field Lines

Magnetic field lines represent the direction and strength of the magnetic field.

• Magnetic field lines flow from the north pole to the south pole of a magnet.

Motion of a Charged Particle in a Magnetic Field

Force on a Moving Charge

The force ${\vec {F}}$ on a moving charge $q$ in a magnetic field ${\vec {B}}$ is given by the Lorentz force:

• Formula: ${\vec {F}}=q\cdot ({\vec {v}}\times {\vec {B}})$

 Where:
 * ${\vec {v}}$ is the velocity of the charge,
 * $\times$ denotes the cross product.

Applications

Northern Lights

Charged particles from the solar wind are guided by the Earth's magnetic field, exciting atmospheric gases and producing visible light.

Van Allen Belt

Regions of trapped charged particles in the Earth's magnetic field. These particles spiral along magnetic field lines.

Mass Spectrometer

Used to measure the mass-to-charge ratio of ions.

• Formula: $r={\frac {mv}{qB}}$

 Where:
 * $r$ is the radius of the ion's path,
 * $m$ is the ion's mass,
 * $v$ is the ion's velocity,
 * $q$ is the charge,
 * $B$ is the magnetic field strength.

Cyclotron

A type of particle accelerator that uses a perpendicular magnetic field to accelerate charged particles in a circular path.

• Formula: $r={\frac {mv}{qB}}$

Crossed Fields

Force on a Moving Charge due to Electric and Magnetic Fields

The force on a charge moving in both electric ${\vec {E}}$ and magnetic ${\vec {B}}$ fields is:

• Formula: ${\vec {F}}=q\cdot ({\vec {E}}+{\vec {v}}\times {\vec {B}})$

Thomson Experiment (Discovery of the Electron)

Used crossed electric and magnetic fields to determine the charge-to-mass ratio of the electron.

Hall Effect

The creation of a voltage difference across an electrical conductor, transverse to the electric current and the applied magnetic field.

Magnetic Force on a Current-Carrying Wire

The force on a wire of length $L$ carrying current $I$ in a magnetic field $B$ is:

• Formula: ${\vec {F}}=I\cdot ({\vec {L}}\times {\vec {B}})$

Torque on a Current Loop - the Electric Motor

A current loop in a magnetic field experiences torque, which is the basis for electric motors.

• Formula: $\tau =n\cdot I\cdot A\cdot B\cdot \sin(\theta )$

 Where:
 * $\tau$ is the torque,
 * $n$ is the number of loops,
 * $I$ is the current,
 * $A$ is the area of the loop,
 * $B$ is the magnetic field,
 * $\theta$ is the angle between the magnetic field and the normal to the loop.

Simulations

Understanding the earth's magnetic field:

• Bar Magnet and compass
• Earth's magnetic Field
• demo of magnetic field lines with iron filings (one magnet)
  
• demo of magnetic field lines with iron filings (two magnets)
  
  

Vector Product:

• Cross product of two vectors

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