Electromagnetism II

🔋Electromagnetism II Unit 7 – Magnetostatics & Magnetic Materials

Magnetostatics and magnetic materials form a crucial part of electromagnetic theory. This unit explores the behavior of magnetic fields, their sources, and interactions with matter. From the fundamental Lorentz force law to the complexities of magnetic materials, students gain insights into the principles governing magnetic phenomena. The study covers key concepts like the Biot-Savart law, Ampère's law, and magnetic susceptibility. It also delves into practical applications, including MRI machines, electric motors, and magnetic data storage. Understanding these principles is essential for grasping the role of magnetism in modern technology and natural phenomena.

Got a Unit Test this week?

we crunched the numbers and here's the most likely topics on your next test

Key Concepts and Fundamentals

  • Magnetostatics deals with time-independent magnetic fields and their interactions with matter
  • Magnetic fields are generated by moving electric charges (electric currents) and magnetic dipoles
  • Magnetic fields exert forces on moving charges and magnetic dipoles according to the Lorentz force law: F=qv×B\vec{F} = q\vec{v} \times \vec{B}
    • qq represents the charge of the particle
    • v\vec{v} represents the velocity of the particle
    • B\vec{B} represents the magnetic field
  • Magnetic fields are represented by field lines, which form closed loops and never intersect
  • The magnetic flux ΦB\Phi_B through a surface is the total number of magnetic field lines passing through that surface: ΦB=BdA\Phi_B = \int \vec{B} \cdot d\vec{A}
  • Magnetic fields obey the principle of superposition, meaning that the total magnetic field at a point is the vector sum of all individual magnetic fields at that point
  • Magnetic fields are divergence-free (B=0\nabla \cdot \vec{B} = 0), implying that magnetic monopoles do not exist

Magnetic Fields and Sources

  • Magnetic fields can be generated by current-carrying conductors, such as wires, coils, and solenoids
  • The magnetic field around a long, straight current-carrying wire is given by: B=μ0I2πrB = \frac{\mu_0 I}{2\pi r}
    • μ0\mu_0 is the permeability of free space (4π×107 Tm/A4\pi \times 10^{-7} \text{ T} \cdot \text{m/A})
    • II is the current flowing through the wire
    • rr is the distance from the wire
  • The magnetic field inside a solenoid is uniform and given by: B=μ0nIB = \mu_0 n I, where nn is the number of turns per unit length
  • Permanent magnets, such as bar magnets and horseshoe magnets, generate magnetic fields due to the alignment of magnetic dipoles within the material
  • Earth's magnetic field is generated by the motion of molten iron in its outer core, creating a magnetic dipole field
  • The magnetic field due to a magnetic dipole at a distance rr is given by: B=μ04πr3(3(mr^)r^m)\vec{B} = \frac{\mu_0}{4\pi r^3}(3(\vec{m} \cdot \hat{r})\hat{r} - \vec{m})
    • m\vec{m} is the magnetic dipole moment
    • r^\hat{r} is the unit vector pointing from the dipole to the point of interest

Magnetic Forces and Torques

  • Magnetic fields exert forces on moving charges according to the Lorentz force law: F=qv×B\vec{F} = q\vec{v} \times \vec{B}
  • The direction of the magnetic force is perpendicular to both the velocity of the charge and the magnetic field, as determined by the right-hand rule
  • Magnetic fields do not perform work on charged particles, as the force is always perpendicular to the particle's motion
  • Current-carrying conductors experience a force in a magnetic field given by: F=IL×B\vec{F} = I\vec{L} \times \vec{B}, where L\vec{L} is the length vector of the conductor
  • Magnetic dipoles experience a torque in a magnetic field given by: τ=m×B\vec{\tau} = \vec{m} \times \vec{B}
    • The torque tends to align the magnetic dipole with the external magnetic field
  • Charged particles moving in a uniform magnetic field follow a circular path with a radius given by: r=mvqBr = \frac{mv}{qB}
    • This principle is used in devices such as cyclotrons and mass spectrometers

Biot-Savart Law and Applications

  • The Biot-Savart law calculates the magnetic field generated by a current-carrying conductor at a point in space
  • The general form of the Biot-Savart law is: dB=μ04πIdl×r^r2d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2}
    • II is the current flowing through the conductor
    • dld\vec{l} is an infinitesimal length element of the conductor
    • r^\hat{r} is the unit vector pointing from the conductor element to the point of interest
    • rr is the distance between the conductor element and the point of interest
  • To find the total magnetic field, integrate the Biot-Savart law over the entire current distribution: B=dB\vec{B} = \int d\vec{B}
  • The Biot-Savart law can be used to calculate the magnetic field of various current configurations, such as:
    • Straight wires
    • Circular loops
    • Solenoids
    • Toroidal coils
  • The magnetic field at the center of a circular loop of radius RR carrying a current II is given by: B=μ0I2RB = \frac{\mu_0 I}{2R}
  • The Biot-Savart law is particularly useful when dealing with complex current distributions or geometries where Ampère's law is difficult to apply

Ampère's Law and Its Uses

  • Ampère's law relates the magnetic field around a closed loop to the electric current passing through the loop
  • The integral form of Ampère's law is: Bdl=μ0Ienc\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}
    • \oint represents the closed line integral around the loop
    • dld\vec{l} is an infinitesimal length element along the loop
    • IencI_{enc} is the total current enclosed by the loop
  • Ampère's law is particularly useful for calculating the magnetic field in situations with high symmetry, such as:
    • Infinite straight wires
    • Infinite solenoids
    • Toroidal coils
  • The differential form of Ampère's law, known as the Ampère-Maxwell law, includes the displacement current term: ×B=μ0J+μ0ε0Et\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t}
    • J\vec{J} is the current density
    • ε0\varepsilon_0 is the permittivity of free space
    • Et\frac{\partial \vec{E}}{\partial t} is the time-varying electric field (displacement current)
  • The displacement current term is essential for maintaining the continuity of the magnetic field in time-varying situations, such as in electromagnetic waves

Magnetic Materials and Properties

  • Magnetic materials can be classified into three main categories based on their response to an external magnetic field:
    • Diamagnetic materials (weakly repelled by magnetic fields, e.g., water, copper, and bismuth)
    • Paramagnetic materials (weakly attracted to magnetic fields, e.g., aluminum, platinum, and oxygen)
    • Ferromagnetic materials (strongly attracted to magnetic fields, e.g., iron, nickel, and cobalt)
  • Diamagnetic materials have no unpaired electrons and generate a weak magnetic field that opposes the applied field
  • Paramagnetic materials have unpaired electrons that align with the applied magnetic field, resulting in a weak attraction
  • Ferromagnetic materials have a strong alignment of magnetic dipoles, leading to a large net magnetic moment and strong attraction to magnetic fields
  • Ferromagnetic materials exhibit hysteresis, meaning their magnetization depends on the history of the applied magnetic field
    • The hysteresis loop shows the relationship between the applied field (H) and the resulting magnetization (M)
    • Key points on the hysteresis loop include the saturation magnetization, remanent magnetization, and coercive field
  • Curie temperature is the critical temperature above which a ferromagnetic material becomes paramagnetic
    • At the Curie temperature, thermal energy overcomes the alignment of magnetic dipoles

Magnetization and Magnetic Susceptibility

  • Magnetization M\vec{M} is the magnetic dipole moment per unit volume of a material: M=dmdV\vec{M} = \frac{d\vec{m}}{dV}
  • The total magnetic field B\vec{B} in a material is the sum of the applied field H\vec{H} and the magnetization: B=μ0(H+M)\vec{B} = \mu_0 (\vec{H} + \vec{M})
  • Magnetic susceptibility χ\chi is a dimensionless quantity that describes a material's response to an applied magnetic field: M=χH\vec{M} = \chi \vec{H}
    • Diamagnetic materials have a small, negative susceptibility (χ<0\chi < 0)
    • Paramagnetic materials have a small, positive susceptibility (χ>0\chi > 0)
    • Ferromagnetic materials have a large, positive susceptibility (χ1\chi \gg 1)
  • Relative permeability μr\mu_r relates the magnetic permeability of a material to the permeability of free space: μ=μrμ0\mu = \mu_r \mu_0
    • For linear materials, μr=1+χ\mu_r = 1 + \chi
  • The magnetic field inside a material is given by: B=μH\vec{B} = \mu \vec{H}
  • Demagnetization factors describe the effect of a material's shape on its internal magnetic field
    • The demagnetization factor is highest along the shortest dimension of the material

Practical Applications and Real-World Examples

  • Magnetic Resonance Imaging (MRI) uses strong magnetic fields and radio waves to generate detailed images of the human body
    • The magnetic field aligns the protons in the body, and radio waves are used to manipulate and detect their resonance
  • Electromagnetic induction, the basis for transformers and generators, relies on time-varying magnetic fields to induce electric currents
    • Faraday's law describes the relationship between the change in magnetic flux and the induced electromotive force (emf)
  • Electric motors and generators convert between electrical and mechanical energy using magnetic fields
    • In motors, current-carrying coils experience a torque in a magnetic field, causing rotation
    • In generators, the motion of conductors in a magnetic field induces an electric current
  • Magnetic levitation (Maglev) trains use strong magnetic fields to lift and propel the train, reducing friction and enabling high-speed travel
  • Magnetic data storage devices, such as hard disk drives and magnetic tape, use the magnetization of ferromagnetic materials to store and retrieve digital information
  • Particle accelerators, such as cyclotrons and synchrotrons, use magnetic fields to guide and accelerate charged particles to high energies
    • The Large Hadron Collider (LHC) at CERN uses superconducting magnets to accelerate and collide protons for high-energy physics experiments
  • Earth's magnetic field protects the planet from harmful solar wind particles and cosmic rays, deflecting them towards the poles and creating the auroras (Northern and Southern Lights)


© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.

© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Glossary
Glossary