#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.

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.

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} = rac{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.

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 = rac{ ext{u}_0 I}{2 ext{pi} r} where ( I ) is current and ( r ) is the distance from the wire.

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}0 I{ ext{enc}}
Apply Ampere's Law to derive expressions for magnetic fields, particularly for solenoids.

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.

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.

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.

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 = rac{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 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.