Magnetic Field

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.

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

Starting with radius equation: $r = \frac{mv}{qB}$

Changes in radius:

1. Velocity doubles: $r_{new} = 2r$

   • Radius directly proportional to velocity

2. Mass doubles: $r_{new} = 2r$

   • Radius directly proportional to mass

3. Charge doubles: $r_{new} = \frac{r}{2}$

   • Radius inversely proportional to charge

4. Field strength doubles: $r_{new} = \frac{r}{2}$

   • Radius inversely proportional to field strength

5. Kinetic energy doubles: $r_{new} = \sqrt{2}r$

   • Since $K = \frac{1}{2}mv^2$, doubling K means $v_{new} = v\sqrt{2}$
   • Therefore $r_{new} = \frac{mv_{new}}{qB} = \frac{m(v\sqrt{2})}{qB} = \sqrt{2}r$

Note: For particles with equal kinetic energy but different mass/charge ratios (like proton vs. deuteron), radius scales as $r \propto \sqrt{\frac{m}{q}}$

Derivation showing cyclotron frequency independence:

1. Magnetic force provides centripetal acceleration: $qvB = \frac{mv^2}{r}$

2. Solve for radius: $r = \frac{mv}{qB}$

3. Time period for one revolution: $T = \frac{2\pi r}{v} = \frac{2\pi(mv/qB)}{v} = \frac{2\pi m}{qB}$

4. Cyclotron frequency: $\nu = \frac{1}{T} = \frac{qB}{2\pi m}$ Angular frequency: $\omega = 2\pi\nu = \frac{qB}{m}$

Explanation:

• As velocity increases, radius increases proportionally
• Increased distance (circumference) is exactly balanced by increased speed
• Time to complete one revolution remains constant
• Only depends on charge-to-mass ratio ($q/m$) and field strength ($B$)
• Fundamental principle behind cyclotron particle accelerator design

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.

The kinetic energy of a charged particle in a uniform magnetic field:

• Remains constant throughout its motion
• Is not changed by the magnetic force

This occurs because:

• The magnetic force ($\vec{F} = q\vec{v} \times \vec{B}$) is always perpendicular to the velocity
• Work done by the force is zero: $W = \vec{F} \cdot \vec{d} = 0$ (since force and displacement are perpendicular)
• The force changes only the direction, not the magnitude of velocity

Consequences:

• The speed of the particle remains constant
• The radius of the circular path is constant
• For non-perpendicular motion, the particle follows a helical path with constant pitch

Example: An electron moving through a bubble chamber in a magnetic field travels in a perfect circle or helix with unchanging speed, allowing scientists to calculate its momentum from the radius of curvature.

• 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

• Electric and magnetic fields are not independent entities but different manifestations of a unified electromagnetic field
• When changing reference frames:
  • Magnetic fields can transform partly or entirely into electric fields
  • Electric fields can transform partly or entirely into magnetic fields
• The laws of physics remain invariant, but the fields transform according to relativistic principles
• Example: A charge experiencing purely magnetic attraction toward a current-carrying wire in one frame will experience purely electric attraction in a frame moving with the charge
• This relativistic behavior is a direct consequence of Einstein's Special Theory of Relativity

• 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

The relationship between velocity orientation and magnetic force:

1. Parallel orientation ($\vec{v}$ parallel to $\vec{B}$):

   • Force is zero ($F = qvB\sin(0°) = 0$)
   • No deflection occurs
   • Example: Proton moving directly along field lines experiences no magnetic force

2. Perpendicular orientation ($\vec{v}$ perpendicular to $\vec{B}$):

   • Maximum force ($F = qvB\sin(90°) = qvB$)
   • Produces circular motion with radius $r = \frac{mv}{qB}$
   • Force always perpendicular to velocity, causing uniform circular motion

3. Arbitrary angle $\theta$:

   • Force magnitude is $F = qvB\sin\theta$
   • Force perpendicular to both velocity and field
   • Results in helical motion: circular motion in plane perpendicular to field + constant velocity along field
   • Pitch of helix determined by component of velocity parallel to field

The general vector equation: $\vec{F} = q\vec{v} \times \vec{B}$