CHPTR 27 Notes: Magnetic Fields
Structure of Magnets and Magnetic Forces
Every magnet has two poles referred to as north and south poles.
Like poles repel each other and unlike poles attract.
The force between two magnetic poles varies inversely with the square of their distance:
Formula: F ext{ } ext{varies as} ext{ } rac{1}{d^2} where d is distance between poles.
Properties of Magnetic Poles
Unlike electric charges, magnetic poles are always observed in pairs.
The existence of magnetic monopoles is a postulate of some theoretical frameworks, although no monopoles have been observed.
Magnetic Fields
Magnetic fields exist around magnetic materials.
A moving charged particle is also a source of magnetic field in addition to being a source of an electric field (referenced in Chapter 28).
Magnetic fields are denoted by B.
Field lines around a bar magnet can be traced using a small compass.
Definition of Magnetic Fields
The magnetic field at a point can be defined in terms of the force it exerts on a moving charged particle.
Force Exerted by Magnetic Fields
The formula for the force F on a charged particle in a magnetic field is:
F = q v B ext{ sin} heta
Key details:
The force is zero on a charged particle that is at rest.
If the particle moves parallel to the magnetic field, the force on it is also zero.
Comparing Electric and Magnetic Forces
The electric force on a charged particle is given by the formula:
F = q E
It acts in the same direction as the electric field, while the magnetic force acts in a perpendicular direction.
Direction of Forces in Magnetic Fields
To find directions for forces associated with E , B , or v , one can use the following procedure:
Step 1: Point your thumb in the direction of v (velocity of the charged particle).
Step 2: Point your fingers in the direction of the magnetic field B .
Note: (Fingers = Field)
Step 3: Your palm will indicate the direction of the force on the particle (Palm = Push).
Note:
The magnetic force on a negative charge is the opposite direction compared to the force on a positive charge.
Therefore, the back of your hand should point in the direction for negative charges.
Work Done by Magnetic Fields
Magnetic fields do no work on particles and consequently do not alter their kinetic energy.
Units of Magnetic Fields
In the mks (meter-kilogram-second) system, the unit of magnetic field is Tesla.
In the cgs (centimeter-gram-second) system, the unit of magnetic field is Gauss.
Relation between Tesla and Gauss:
1 ext{ T} = 10^{4} ext{ G}
Earth's Magnetic Field
The Earth itself behaves as a magnet and possesses a magnetic field.
The Earth's geographic North Pole is actually its magnetic south pole, and vice versa.
Interactions between charged particles from the Sun and Earth's magnetic field lead to the formation of Auroras (Aurora Borealis and Aurora Australis).
Combined Electric and Magnetic Fields
When both electric and magnetic fields are present, a moving charged particle is affected by both fields:
ext{F} = q(E + ext{v} imes B)
Magnetic Flux
Magnetic flux through a surface is defined as:
The number of field lines threading through the surface.
Given by the expression:
d ext{Φ}_B = B ullet dA
Units of magnetic flux include:
ext{Tm}^2 or Webers (Wb).
Gauss’s Law in Magnetism
Gauss's Law in magnetism states:
The net flux through any closed surface is always zero:
ext{ ∫ } B ullet dA = 0
Implication: This supports the notion that monopoles do not exist, as they have yet to be observed.
Motion of Charged Particles in Magnetic Fields
If the initial velocity of a charged particle is perpendicular to the magnetic field, the particle will move in a circular path at constant speed.
The magnetic force provides the centripetal force necessary for circular motion.
The cyclotron frequency is defined as:
f = rac{qB}{2 ext{π}}
The variables represent:
m = mass of particle,
q = charge of particle,
B = magnetic field strength.
Helical Motion of Charged Particles
When the velocity vector of a charged particle is at an angle (neither perpendicular nor co-linear) with respect to the magnetic field, the particle will move in a helical path.
Dynamics in a Non-Uniform Magnetic Field
In non-uniform magnetic fields, charged particles exhibit spiraling motion along magnetic field lines and can be reflected back by strong fields.
Such field configurations, depicted in diagrams, are termed magnetic bottles, commonly utilized in fusion reactors.
Velocity Selector
Many experiments, including medical research and forensics, require charged particle beams with consistent speeds.
A velocity selector is a device used to provide a beam of charged particles with the same speed:
F = q( ext{E} + ext{v} imes B) = 0
Mass Spectrometer
A mass spectrometer is a device that separates ions based on their mass-to-charge ratios.
It combines the function of a velocity selector with circular motion in a magnetic field.
Force on Current-Carrying Conductors
A conductor carrying a current experiences a force when placed in a magnetic field.
For a straight current-carrying conductor in a uniform magnetic field B , the force is given as:
F = I l imes B
Or:
F = B I l ext{ sin} heta
Torque on Current-Loops in Magnetic Fields
The net force on a current loop in a uniform magnetic field is zero; however, it experiences torque ( τ ) .
Torque can be expressed as:
τ = I A B ext{ sin} θ
Where:
I = current,
A = area vector of the loop,
B = magnetic field vector.
Application of Magnets
Magnets play a role in various applications, such as:
Motors,
Generators,
Loudspeakers.
Magnetic Dipole Moment
The magnetic dipole moment μ of a current loop with N turns is defined as:
μ = N I A
The torque on a current-carrying loop concerning its magnetic moment can be indicated as:
τ = μ imes B
Potential Energy of Magnetic Dipoles
The potential energy of a magnetic dipole in a magnetic field is given by:
U = -μ ullet B
Torquers and Satellite Positioning
Many satellites employ devices called torquers to adjust their orientations.
Torquers interact with the Earth’s magnetic field, using solar energy instead of thruster fuel, which is advantageous.
Typical values for these systems include:
Magnetic moments: approximately 250 ext{ A} ext{ m}^2
Magnetic fields: around 3 imes 10^{-5} ext{ T}
Torques: approximately 7.5 imes 10^{-3} ext{ N m}
Hall Effect
When a current-carrying conductor is positioned in a magnetic field, a potential difference occurs perpendicular to both the current and the magnetic field, known as the Hall effect.
This accumulated charge leads to the generation of a potential difference termed Hall voltage.
Applications of Hall Voltage
The Hall voltage can be utilized to measure one of the following parameters:
(a) Magnetic field strength,
(b) Charge type (applicable in semiconductors),
(c) Carrier density, using the relation:
ext{ΔV}_H = rac{nqt}{IB}
Magnetic Resonance Imaging (MRI)
Intense magnetic fields, typically ranging from 0.5 to 2.0 Tesla, are used in Magnetic Resonance Imaging (MRI) devices.
MRI scanners can distinguish tissue types at a scale as small as 0.5 mm in each dimension, creating detailed 3D maps of the body.
MRI Fields Strength Comparison
Magnetic field strengths in MRI devices range from 5,000 to 20,000 Gauss (i.e., 0.5 to 2.0 Tesla).
The Earth's magnetic field strength is approximately 0.5 Gauss (notably, 1 ext{ Tesla} = 10,000 ext{ Gauss} ).
Research can use magnetic fields up to 60 Tesla.
Superconducting Magnets in MRI
Superconducting magnets are commonly utilized in MRI devices, where coils of wires are submerged in liquid helium at 4.2 K.
While superconductive systems remain costly, they can easily generate fields needed for higher-quality imaging (between 0.5 to 2.0 Tesla).