Week 6 Lec. 1 (Week of 1st Exam)

Magnetic Forces on Moving Charges

Force on a Charged Particle in a Magnetic Field

The force experienced by a charged particle moving in a magnetic field is a fundamental concept in electromagnetism. The direction and magnitude of this force are determined by the charge, velocity, and the magnetic field.

• Magnetic Force Formula: The force on a charge q moving with velocity v in a magnetic field B is given by the vector cross product:

$\vec{F} = q\vec{v} \times \vec{B}$

• Direction: Determined by the right-hand rule; the force is perpendicular to both v and B.

• Magnitude: $F = |q|vB\sin\theta$, where θ is the angle between v and B.

• Uniform Circular Motion: If v is perpendicular to B, the particle moves in a circle with constant speed.

Radius of Circular Path

For a particle of mass m and charge q moving at speed v in a uniform magnetic field B:

$r = \frac{mv}{|q|B}$

• Application: This relationship is used in devices like cyclotrons and mass spectrometers to determine particle properties.

Example: Proton in a Cyclotron

Consider a proton with kinetic energy 11 MeV in a 1.2 T magnetic field. The following steps show how to calculate its speed and the radius of its orbit:

• Step 1: Convert kinetic energy to joules:

$K = (11 \times 10^6\ \text{eV}) \times \frac{1.60 \times 10^{-19}\ \text{J}}{1\ \text{eV}} = 1.76 \times 10^{-12}\ \text{J}$

• Step 2: Calculate speed using kinetic energy:

$v = \sqrt{\frac{2K}{m}} = \sqrt{\frac{2(1.76 \times 10^{-12}\ \text{J})}{1.67 \times 10^{-27}\ \text{kg}}} = 4.59 \times 10^7\ \text{m/s}$

• Step 3: Find the radius of the orbit:

$r = \frac{mv}{|q|B} = \frac{(1.67 \times 10^{-27}\ \text{kg})(4.59 \times 10^7\ \text{m/s})}{(1.60 \times 10^{-19}\ \text{C})(1.2\ \text{T})} = 0.40\ \text{m}$

Cyclotrons

Principle and Operation

A cyclotron is a type of particle accelerator that uses a constant magnetic field and a rapidly alternating electric field to accelerate charged particles in a spiral path.

• Dees: The D-shaped electrodes where particles are accelerated.

• Magnetic Field: Keeps particles in a circular path.

• Alternating Voltage: Accelerates particles each time they cross the gap between the Dees.

• Frequency: The frequency of the alternating voltage matches the orbital frequency of the particles:

$f = \frac{qB}{2\pi m}$

• Energy Gain: Each crossing of the gap increases the particle's kinetic energy by $\Delta T = qV$.

Mass Spectrometers

Principle and Application

A mass spectrometer is a device that separates ions according to their mass-to-charge ratio (m/q) using electric and magnetic fields. It is widely used in physics, chemistry, and biology for identifying isotopes and molecules.

• Velocity Selector: Uses perpendicular electric and magnetic fields to select particles of a specific velocity:

$qE = qvB \implies v = \frac{E}{B}$

• Magnetic Separation: After velocity selection, particles enter a region with only a magnetic field, causing them to follow circular paths. The radius depends on their mass:

$r = \frac{mv}{|q|B}$

• Detection: Particles with different masses strike the detector at different positions, allowing for mass identification.

Summary Table: Key Equations and Concepts

Concept	Equation	Description
Magnetic Force	$\vec{F} = q\vec{v} \times \vec{B}$	Force on a moving charge in a magnetic field
Radius of Path	$r = \frac{mv}{|q|B}$	Radius of circular motion in a uniform B field
Kinetic Energy	$K = \frac{1}{2}mv^2$	Relates speed and energy of a particle
Velocity Selector	$v = \frac{E}{B}$	Only particles with this velocity pass through
Cyclotron Frequency	$f = \frac{qB}{2\pi m}$	Frequency for resonance acceleration

Additional info:

• These concepts are foundational for understanding the behavior of charged particles in magnetic fields, which is essential in both experimental and applied physics.

• Applications include particle accelerators, medical imaging (cyclotrons for PET scans), and analytical chemistry (mass spectrometry).