phy4

Definition: A magnet is defined as a material capable of interacting with other materials, easily influenced by magnetism.
Etymology: The term 'magnet' originates from the ancient Greek place Magnesia, where most of the known lodestones were found in ancient times.

Definition: The magnetic pole of a magnet is the region that exhibits polar characteristics, meaning it has both a north and a south pole.

Definition: A magnetic field is the field of force that surrounds a magnet.
Creation: It is created from a changing electric field, which stems from the movement of electric charges.
Direction: The orientation and direction of the magnetic field are determined by the right-hand rule, which provides a way to visualize the direction of the magnetic field relative to the current direction.

Definition: The Lorentz force is the total force acting on a charged particle in an electromagnetic field, influenced by both electric and magnetic forces.
Formula: The Lorentz force is represented by the equation:
$extbf{F}_L = q extbf{E} + q extbf{v} imes extbf{B}$
Special Case (Magnetic Force): Eliminating the electric field component leads to the expression for the magnetic force:
$extbf{F}_B = q extbf{v} imes extbf{B}$

Electric Force:
- Projection: The electric force acts in a direction parallel to the electric field.
- Effect on Charges: It causes a moving charge to change its direction in accordance with the electric field's projection.
Magnetic Force:
- Projection: The magnetic force operates perpendicular to the directions of both the electric field and the magnetic field.
- Effect: It changes the trajectory of a moving charge in a manner perpendicular to the magnetic field direction.

Definition: The cyclotron radius represents the radius of the circular motion of a charged particle in a magnetic field.
Formula:
$r<em>g = \frac{mv</em>{\bot}}{qB}$

Definition: The cyclotron frequency defines the angular frequency of a charged particle's circular motion in a magnetic field.
Behavioral States:
- When the Lorentz force equals the magnetic force:
1. If qvB > rac{mv^2}{r}, the charged particle moves in a spiral path, trapped within the magnetic field.
2. If qvB < rac{mv^2}{r}, the movement is an arc as the particle escapes the field.
3. If $qvB = \frac{mv^2}{r}$ , the movement is circular.
Formula:
$ext{Angular Frequency: } ext{w}_g = \frac{qB}{m}$

Definition: Magnetic flux quantifies the strength of a magnetic field passing through a specific area.
Unit: The unit for magnetic flux is the Weber.
Conversion:
- $1 ext{ Wb} = 1 ext{ V} ext{s}$
Formula:
$ext{Φ}_B = BA ext{cos}( heta)$
Implication: In a closed surface, the total magnetic flux is equal to zero:
$ext{Total Magnetic Flux} = 0$

Definition: A solenoid is a conducting wire coiled in a helical shape, functioning similarly to multiple single-loop conductors arranged in parallel.

Definition: This law describes how a magnetic field is generated by changing electric fields and is consistent with both Ampere’s and Gauss’s Laws.
Formula:
$ext{d} extbf{B} = \frac{ ext{μ}_0 I}{4 ext{π}} \frac{ ext{d} extbf{l} ext{sin}( heta)}{r^2}$

Definition: Ampere's Law relates the current flowing through a conductor to the magnetic field it generates.
Formula:
$B = \frac{ ext{μ}_0 I}{ ext{enc}}$

Point in the Magnetic Field
- Long Straight Conductor:
  $B = \frac{ ext{μ}_0 I}{2 ext{π} r}$
- Long Cylindrical Conductor:
  - Inside the conductor:
    $B = \frac{ ext{μ}_0 I}{2 ext{π} r R^2}$
  - Outside the conductor:
    $B = \frac{ ext{μ}_0 I}{2 ext{π} r}$
- Long Solenoid (nearly ideal), with number of loops per length:
  - Inside the solenoid:
    $B = ext{μ}_0 n I$
  - Outside the solenoid:
    $B = 0$
- Moving Charge:
  - At distance:
    $B = \frac{ ext{μ}_0}{4 ext{π}} \frac{q v}{r^2}$
- Circular loop with a specific radius:
  - Loop as an arc with central angle $φ$ (in radians):
    $B = \frac{ ext{μ}_0 I}{2 ext{π} r R^2}$
  - Center of complete loop at origin:
    $B = \frac{ ext{μ}_0 I}{2r}$
  - Distance given (a) above or below origin:
    $B = \frac{ ext{μ}_0 I}{4 ext{π} r}$

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