Magnetic Field

Magnetic field is defined by the equation: $\vec{F} = q\vec{v} \times \vec{B}$

Where:

• $\vec{F}$ is the magnetic force on a charged particle
• $q$ is the charge
• $\vec{v}$ is the velocity

Direction determination:

1. There is one line through any point where a charged particle experiences no force when moving along it
2. This line defines the direction of the magnetic field
3. The force is perpendicular to both velocity and magnetic field (right-hand rule)

Unit: Tesla (T) = newton/ampere·meter = weber/meter²

(a) When velocity is perpendicular to field:

• Particle moves in a circular path
• Radius: $r = \frac{mv}{qB}$
• Force provides centripetal acceleration
• Speed remains constant (force perpendicular to velocity)

(b) When velocity is at an angle θ to field:

• Motion follows a helical path
• Velocity component parallel to field ($v_\parallel$) remains unchanged
• Velocity component perpendicular to field ($v_\perp$) creates circular motion
• Resulting path is a helix with radius $r = \frac{mv_\perp}{qB}$
• The particle advances along field lines while circling around them

The magnetic force on a current-carrying wire is: $\vec{F} = I \vec{l} \times \vec{B}$

Derivation:

1. Force on a single electron: $\vec{f} = -e\vec{v}_d \times \vec{B}$
2. Number of electrons in segment dl: $nAdl$
3. Total force: $d\vec{F} = (nAdl)(-e\vec{v}_d \times \vec{B})$
4. Using $I = neAv_d$ and defining $d\vec{l}$ along current direction: $d\vec{F} = I d\vec{l} \times \vec{B}$

Direction:

• Determined by right-hand rule
• Place fingers of right hand along current direction
• Curl fingers toward field
• Thumb indicates force direction
• Force is perpendicular to both current and field

For a straight wire of length l: $F = IlB\sin\theta$, where θ is angle between wire and field.

Magnetic field variation with distance:

1. Long straight current-carrying wire:

   • $B \propto \frac{1}{r}$ (inversely proportional to distance)
   • $B = \frac{\mu_0 I}{2\pi r}$

2. Current loop (far from the loop):

   • $B \propto \frac{1}{r^3}$ (inverse cube relationship)
   • Similar to electric dipole field

Comparison with electric field:

• Point charge electric field: $E \propto \frac{1}{r^2}$ (inverse square law)
• No magnetic monopoles exist in nature (fundamental difference)
• Magnetic fields always form closed loops while electric field lines begin/end on charges
• Both fields can be uniform in certain configurations

Reference frame transformation reveals the fundamental connection between electric and magnetic fields:

1. In a stationary frame (S):

   • A current-carrying wire produces magnetic field only
   • A moving charge experiences magnetic force perpendicular to velocity

2. In a moving frame (S') moving with the charge:

   • The charge appears stationary
   • No magnetic force can act on a stationary charge
   • Must experience electric force instead
   • The wire's charges appear to have different densities due to length contraction

Key insights:

• Electric and magnetic fields are aspects of a single electromagnetic field
• Whether observed as electric or magnetic depends on observer's reference frame
• Special relativity unifies these as a single entity
• A purely magnetic field in one frame appears as combination of electric and magnetic fields in another

The Lorentz force law combines electric and magnetic forces on a charged particle: $\vec{F} = q\vec{E} + q\vec{v} \times \vec{B}$

Components:

• Electric force ($q\vec{E}$):

  • Parallel to electric field for positive charges
  • Opposite to electric field for negative charges
  • Independent of particle velocity
  • Can change particle's kinetic energy

• Magnetic force ($q\vec{v} \times \vec{B}$):

  • Perpendicular to both velocity and magnetic field
  • Zero when velocity parallel to field
  • Cannot change particle's kinetic energy (no work done)
  • Magnitude: $F_B = qvB\sin\theta$

Applications:

• Particle accelerators
• Mass spectrometers
• Hall effect devices
• Cyclotrons and synchrotrons

The force on the wire is given by:

• F = i × L × B
• Direction: perpendicular to both the current direction and magnetic field
• In a sliding wire setup with parallel rails, this force causes the wire to move along the rails
• Magnitude depends on:
  • Current strength (i)
  • Length of wire between rails (L)
  • Magnetic field strength (B)

Example: A 10 cm wire carrying 2A current perpendicular to a 0.5T magnetic field experiences a force of 0.1N along the rails.

Cyclotron frequency is:

• The frequency at which a charged particle completes one circular orbit in a uniform magnetic field
• Formula: $\nu = \frac{qB}{2\pi m}$ or $\omega = \frac{qB}{m}$ (angular frequency)
• Measured in Hz (cycles per second)

Key properties:

• Independent of the particle's speed or radius of orbit
• Depends only on charge-to-mass ratio and field strength
• Time period: $T = \frac{2\pi m}{qB}$

Applications:

• Forms the operating principle of cyclotron particle accelerators
• Used in mass spectrometry to separate ions by their charge-to-mass ratios
• Important in understanding charged particle motion in magnetic fields like Earth's magnetosphere

Note: While faster particles move in larger orbits, they travel greater distances at greater speeds, resulting in the same orbital period regardless of velocity.

• In a stationary reference frame (S): A charge moving parallel to a current-carrying wire experiences a magnetic force but no electric force
• In a reference frame (S₁) moving with the charge: The charge is at rest and experiences an electric force but no magnetic force
• This demonstrates that electric and magnetic fields are frame-dependent aspects of a single electromagnetic field
• The acceleration of the charge is the same in both reference frames, but the explanation for this acceleration differs
• Key principle: What appears as a pure magnetic field in one frame can manifest as a combination of electric and magnetic fields in another frame

• Conservation of physical reality: If a charge experiences acceleration in one reference frame, it must experience equivalent acceleration in all reference frames
• In the stationary frame (S): The charge is at rest and experiences no magnetic force (as magnetic forces only act on moving charges)
• In the moving frame (S₁): Since the charge still accelerates but is now stationary relative to this frame, a force must exist
• This force can only be electric since magnetic forces cannot act on stationary charges
• This necessitates the existence of an electric field in the moving frame that wasn't present in the stationary frame
• The transformation follows Lorentz transformation equations, preserving the invariance of Maxwell's equations across different inertial frames