#2 Lecture Notes

Velocity and Magnetic Field of a Charged Particle

• The concept of velocity of a particle and its relation to magnetic fields.
  • Understanding the angle between the direction of the velocity and the point of interest in the magnetic field.

Basic Principles of Magnetism

• Any moving charged particle acts as a source of a magnetic field without needing a permanent magnet.
  • The magnitude of the magnetic field created depends on the charge, the velocity of the charge, and the distance from the charge.
  • Mathematical representation:
  • $ext{Magnetic Field} = q extbf{v} imes extbf{r}$ where ( ext{v} ) is velocity and ( ext{r} ) is position vector.

Finding the Magnetic Field

• To determine the magnetic field at a specific point, consider:
  • The current direction vs. particle velocity direction, and the angle between them.
  • Use vector representation:
  • $ext{Magnetic Field} = \frac{q extbf{v} imes extbf{r}^ ext{cap}}{r^2}$
• Emphasis on applying the right-hand rule to determine the direction of the resulting magnetic field based on current or charge movement.

Current-Carrying Conductors and Magnetic Fields

• A current-carrying conductor generates a magnetic field.
  • Relation between current and moving charged particles.
  • The magnetic field generated can be found with the formula:
  • $B = \frac{ ext{u}_0 I}{2 ext{pi} r}$ where ( I ) is current and ( r ) is the distance from the wire.

Ampere's Law

• Magnetic field around current-carrying conductors described by Ampere’s Law:
  • The law states that the line integral of the magnetic field around a closed path is proportional to the total electric current flowing through the enclosed path:
  • $ext{Integral of } ext{B} ext{ d} ext{l} = ext{u}<em>0 I</em>{ ext{enc}}$
• Apply Ampere's Law to derive expressions for magnetic fields, particularly for solenoids.

Magnetic Field Inside a Solenoid

• Magnetic fields inside solenoids can be calculated using Ampere's Law.
  • For a solenoid of length ( l ), carrying current ( I ), and with ( n ) turns per unit length:
  • The field inside the solenoid:
  • $B = ext{u}_0 n I$ (constant inside the solenoid)
  • Outside of the solenoid, the magnetic field is considered negligible.

Effects of Magnetic Fields on Moving Charges

• Importance of the interaction between moving charges and magnetic fields:
  • Formula for magnetic force on a moving charge:
  • $F_m = q extbf{v} imes extbf{B}$ or in magnitude form:
  • $F_m = B q v ext{ sin } \theta$ where ( \theta ) is the angle between ( \textbf{v} ) and ( \textbf{B} ).
• Utilize the right-hand rule for determining the direction of force:
  • Thumb = velocity direction
  • Fingers = magnetic field direction
  • Palm = force direction.

Cyclotron Motion

• When a charged particle travels through a magnetic field:
  • This results in circular motion due to the perpendicular magnetic force acting on it.
  • Radius of circular motion is derived as:
  • $R = \frac{mv}{qB}$ where ( m ) is mass, ( v ) is velocity, ( q ) is charge, and ( B ) is magnetic field strength.
  • If the angle is not 90 degrees, it results in a helical path instead of a full circular path.

Hall Effect

• The Hall Effect refers to the development of a voltage across a conductive material when subjected to a magnetic field.
  • Magnetic force causes charge carriers to separate, leading to charge accumulation and creating voltage known as Hall voltage.
  • Formula to define Hall voltage:
  • $V_H = \frac{IB}{nq}$ where ( n ) is charge density, ( q ) is charge, and ( I ) is current.
  • Used in applications to measure charge carrier density and type (positive or negative).

Applications and Implications of Magnetism

• Applications of magnetism in technology, including electronics:
  • Understanding the behavior of conductors in magnetic fields is crucial for developing electrical devices and circuits.
• Explain the significance of current direction in relationships between multiple conductors, such as attraction or repulsion based on current flow.