Magnetic Fields

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Textbook

• University Physics Volume 2: Chapter 11 - Magnetic Fields

Ferromagnets and Electromagnets

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 use electric current to generate a magnetic field. The magnetic field strength is proportional to the current and the number of coils. All magnets have a north and south pole. Unlike poles attract and like poles repel. It is impossible to separate north from the south poles.

The magnetic properties of materials arise from:

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

Application of Ferromagnets and Electromagnets include:

• MRI machines
• Data Storage (Hard drives, Audio Tape)
• Electric motors

Magnetic Field Lines

Magnetic field lines represent the direction and strength of the magnetic field. Magnetic field lines go from the north pole to the south pole of a magnet (the direction a compass would point). The form loops and never cross.

Motion of a Charged Particle in a Magnetic Field

Force on a Moving Charge

Lorentz-Force Right-Hand-Rule for positive charges based on Cross-Product Right-Hand-Rule

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.

Cyclotron

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

Combined 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

Direction of the force on a wire RHR based on Cross-Product RHR

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.

Magnetic Dipole

A magnetic dipole consists of a closed loop of electric current or a pair of opposite magnetic poles separated by a distance. The magnetic moment ${\vec {\mu }}$ is a vector quantity that points perpendicular to the plane of the current loop and is proportional to the current and the area of the loop:

• Formula: ${\vec {\mu }}=N\cdot I\cdot A\cdot {\hat {n}}$

 Where:
 * ${\vec {\mu }}$ is the magnetic dipole moment,
* $N$ is number of turns,
 * $I$ is the current,
 * $A$ is the area of the loop,
 * ${\hat {n}}$ is a unit vector perpendicular to the loop, following the right-hand rule.

• • Right-hand rule**: Curl the fingers of your right hand in the direction of the current around the loop. Your thumb will point in the direction of the magnetic dipole moment ${\vec {\mu }}$.

Torque on a Magnetic Dipole in a Magnetic Field

Direction of the torque on a dipole RHR based on Cross-Product RHR

When a magnetic dipole is placed in an external magnetic field ${\vec {B}}$, it experiences a torque ${\vec {\tau }}$, which tends to align the dipole with the field:

• Formula: ${\vec {\tau }}={\vec {\mu }}\times {\vec {B}}$

 Where:
 * ${\vec {\tau }}$ is the torque,
 * ${\vec {\mu }}$ is the magnetic dipole moment,
 * ${\vec {B}}$ is the magnetic field.

Potential Energy of a Magnetic Dipole

The potential energy $U$ of a magnetic dipole in a magnetic field is given by:

• Formula: $U=-{\vec {\mu }}\cdot {\vec {B}}$

 Where:
 * $U$ is the potential energy,
 * ${\vec {\mu }}$ is the magnetic dipole moment,
 * ${\vec {B}}$ is the magnetic field.

The potential energy is minimized when the dipole is aligned with the magnetic field, and maximized when it is opposite to the magnetic field.

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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