Magnetism and Magnetic Fields

Magnetic Field: A vector quantity that describes the magnetic force exerted on moving charges, currents, or magnetic materials.
Magnetic Field Lines:
- Help visualize the magnetic field around a magnetic object.
- Always point from north to south.
Enclosed Surface: The net magnetic field through any closed surface is zero; this is due to the looping nature of magnetic field lines.
Bar Magnet: The magnetic field is connected by its north and south dipoles.

Creation: Magnetic dipoles are created by the rotational motion of charge carriers.
Spin: Can create a magnetic field that can be oriented up or down.
As a charge moves in a circle, it creates a magnetic dipole.
Torque: Magnetic dipoles can experience torque, causing alignment with the magnetic field.
- In non-magnetic materials, the magnetic dipoles point in random directions, resulting in no net magnetic field.
- In permanent magnets, all dipoles align and combine to form a net magnetic field.

Magnetic Permeability: Describes a substance's ability to form internal magnetic fields.
- Even empty space has magnetic permeability.

Moving Charges: Produce a magnetic field.
Magnitude of Magnetic Field:
- Right Hand Rule:
- Thumb: Direction of Velocity.
- Fingers curled: Direction of Magnetic Field.
- For Negatively Charged Particles: The magnetic field points in the opposite direction of your thumb.

Charged Object at Rest:
- Magnetic field does not exert a force on a stationary charged object.
Parallel/Antiparallel Velocity:
- (Positive charge moving in the same direction as magnetic field) results in no force exerted.
- Opposite direction yields no force as well.
Perpendicular Velocity:
- Exerts a force directed into the page, perpendicular to both the velocity and the magnetic field.
Velocity at an Angle:
- Only the perpendicular component of the velocity will experience a force; the velocity will curve accordingly.
- Magnitude of the Magnetic Force: Follow the Right Hand Rule.

Electric Force (E):
- Given by the equation F = qE , where ( q ) is the charge.
- It acts in the same direction as the electric field.
Crossed Fields:
- When a positively charged particle moves through both an electric and magnetic field, it experiences forces from both fields.
- If the forces are equal in magnitude but opposite in direction, the velocity remains constant, resulting in zero net force (no acceleration).
- Velocity can be determined by setting the forces equal.

Charge Movement: Charges in a conductor experience a magnetic force ( F_B = qv imes B ), causing one side of the conductor to acquire a buildup of charge (one side negative, the other side positive).
A magnetic field applied perpendicular to a current-carrying wire produces an electric field perpendicular to the current within the wire.

Force on Current-Carrying Wire:
- Given by F_B = ILB \, ext{sin} \, \theta , where ( I ) is the current and ( B ) is the magnetic field.
- The wire must be straight, and the magnetic field must be uniform in strength and direction.
Right Hand Rule for Direction of Force:
- Thumb: current direction.
- Fingers: direction of the magnetic field.

Used to determine the magnetic field of an infinitesimal segment of current-carrying wire.
Ampere’s Law:
- Connects the magnetic field created in a region of space to the current enclosed by an imaginary closed path (Amperian Loop).
- For a long current carrying wire, the magnetic field is constant along the loop, parallel or perpendicular to its direction.

Use Ampere's law to evaluate the magnetic field in various regions:
1. Inner Core (r < a): Calculation includes enclosed current contributions.
2. Gap between Wires (a < r < b): Similar calculations apply.
3. Outer Core (r > c): The magnetic field decreases as distance increases and can be determined by Ampere's Law.
4. Outside Everything (r > c): The magnetic field is zero due to symmetry.

Solenoid: A tightly wound helical coil of wire that acts as an electromagnet when current is applied.
Each loop contributes to the magnetic field inside the coil.
Right Hand Rule for Magnetic Field of a Solenoid:
- Fingers curl in the direction of the current flow, and the thumb indicates the direction of the magnetic field inside the solenoid.
Equation for Magnetic Field:
- Given by B{sol} = \mu0 n I , where ( n ) is the number of turns per unit length and ( I ) is the current.