⚡️College Physics III – Thermodynamics, Electricity, and Magnetism Unit 12 – Magnetic Field Sources

Key Concepts and Definitions

• Magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials
• Magnetic flux density ($\vec{B}$) measured in teslas (T) or webers per square meter (Wb/m²) quantifies the strength and direction of the magnetic field
• Magnetic permeability ($\mu$) is a measure of how easily a material can be magnetized (vacuum permeability: $\mu_0 = 4\pi \times 10^{-7}$ N/A²)
• Magnetic dipole moment ($\vec{m}$) is a vector quantity that characterizes the torque experienced by a closed current loop or a bar magnet in an external magnetic field
• Diamagnetism is a weak form of magnetism that opposes an applied magnetic field (water, copper, bismuth)
• Paramagnetism is a weak form of magnetism that aligns with an applied magnetic field (aluminum, platinum, oxygen)
• Ferromagnetism is a strong form of magnetism that aligns with an applied magnetic field and can retain magnetization (iron, nickel, cobalt)

Magnetic Field Fundamentals

• Magnetic fields are produced by moving electric charges or currents and by magnetic dipoles
• Magnetic field lines represent the direction of the magnetic field at each point in space
  • Field lines originate from the north pole and terminate at the south pole of a magnet
  • Field lines never cross each other and are more concentrated where the field is stronger
• Magnetic fields exert forces on moving charges according to the Lorentz force law: $\vec{F} = q\vec{v} \times \vec{B}$
• Magnetic fields interact with other magnetic fields, resulting in attractive or repulsive forces between magnetic dipoles
• Magnetic fields are conservative, meaning the work done by a magnetic force on a moving charge is always zero
• Magnetic fields are solenoidal, meaning they have no divergence ($\nabla \cdot \vec{B} = 0$)
• Magnetic fields are affected by the presence of magnetic materials, which can enhance or reduce the field strength

Sources of Magnetic Fields

• Electric currents produce magnetic fields according to the Biot-Savart law: $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2}$
  • The magnetic field at a point is proportional to the current and inversely proportional to the square of the distance
• Magnetic dipoles, such as bar magnets or current loops, create magnetic fields similar to electric dipole fields
  • The magnetic field of a dipole decays with the cube of the distance: $\vec{B} \propto \frac{1}{r^3}$
• Electromagnets are devices that use electric currents to generate strong, controllable magnetic fields (solenoids, Helmholtz coils)
• Earth's magnetic field is primarily generated by convection currents in its liquid outer core, creating a dipole-like field
• Magnetization ($\vec{M}$) is the net magnetic dipole moment per unit volume of a material, contributing to the total magnetic field
• Ampère's circuital law relates the magnetic field to the current enclosed by a closed loop: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$
• Displacement current, introduced by Maxwell, accounts for the change in electric flux and completes Ampère's law

Magnetic Field Equations and Calculations

• Biot-Savart law: $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2}$ calculates the magnetic field due to a current element $Id\vec{l}$
  • Integrate over the current distribution to find the total magnetic field: $\vec{B} = \int d\vec{B}$
• Ampère's circuital law: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$ relates the magnetic field to the enclosed current
  • Useful for calculating the magnetic field of symmetrical current distributions (infinite wire, solenoid, toroid)
• Magnetic dipole moment: $\vec{m} = NIA\hat{n}$ for a planar current loop with $N$ turns, current $I$, and area $A$
• Magnetic field of a dipole: $\vec{B} = \frac{\mu_0}{4\pi} \frac{3(\vec{m} \cdot \hat{r})\hat{r} - \vec{m}}{r^3}$ at a distance $r$ from the dipole
• Force on a current-carrying wire in a magnetic field: $\vec{F} = I\vec{l} \times \vec{B}$
• Torque on a current loop in a magnetic field: $\vec{\tau} = \vec{m} \times \vec{B}$
• Magnetic flux: $\Phi_B = \int \vec{B} \cdot d\vec{A}$ measures the total magnetic field passing through a surface

Experimental Techniques and Observations

• Magnetic compasses use the torque on a small magnetic needle to align with the local magnetic field (Earth's magnetic field)
• Hall effect sensors measure the voltage difference across a conductor in a magnetic field, proportional to the field strength
• Superconducting Quantum Interference Devices (SQUIDs) are highly sensitive magnetometers that use superconducting loops to detect minute changes in magnetic flux
• Magneto-optical imaging uses the Faraday effect to visualize magnetic field patterns in materials
• Zeeman effect is the splitting of atomic energy levels in the presence of a magnetic field, used to measure field strengths
• Stern-Gerlach experiment demonstrated the quantization of angular momentum by observing the deflection of silver atoms in a non-uniform magnetic field
• Magnetic resonance imaging (MRI) uses strong magnetic fields and radio waves to create detailed images of the human body

Applications in Technology and Science

• Electric motors and generators rely on the interaction between magnetic fields and electric currents to convert between electrical and mechanical energy
• Transformers use magnetic coupling between coils to step up or step down AC voltages
• Magnetic levitation (maglev) trains use strong magnetic fields to lift and propel the train, reducing friction and increasing efficiency
• Magnetic confinement fusion reactors use powerful magnetic fields to contain and control high-temperature plasmas for nuclear fusion
• Magnetohydrodynamics (MHD) studies the behavior of electrically conducting fluids in the presence of magnetic fields (plasma physics, astrophysics)
• Magnetic data storage devices (hard drives, magnetic tapes) use the magnetization of materials to store and retrieve digital information
• Biomagnetism involves the study of magnetic fields produced by living organisms (magnetoreception in animals, magnetoencephalography)

Common Misconceptions and FAQs

• Magnetic monopoles, isolated north or south poles, have not been observed in nature, despite theoretical predictions
• Magnetic fields do not perform work on moving charges, as the force is always perpendicular to the velocity
• Magnetic fields are not affected by stationary charges or non-magnetic materials (wood, plastic)
• The Earth's magnetic field is not perfectly aligned with its geographic poles, and the poles can drift or even reverse over time
• Magnetic fields do not have a direct physiological effect on the human body, but strong fields can interact with medical devices (pacemakers, cochlear implants)
• Superconductors are perfect diamagnets, expelling magnetic fields from their interior (Meissner effect)
• Magnetic fields do not decay with time, unlike electric fields in the presence of conductors

Practice Problems and Solutions

1. Calculate the magnetic field at the center of a circular current loop of radius $R$ carrying a current $I$.

   • Solution: $B = \frac{\mu_0 I}{2R}$

2. A proton with velocity $\vec{v} = (2 \times 10^6 \hat{i}) \text{ m/s}$ enters a uniform magnetic field $\vec{B} = (0.5 \hat{k}) \text{ T}$. Find the magnitude and direction of the force on the proton.

   • Solution: $F = qvB\sin\theta = (1.6 \times 10^{-19} \text{ C})(2 \times 10^6 \text{ m/s})(0.5 \text{ T})\sin 90° = 1.6 \times 10^{-13} \text{ N}$, direction: $-\hat{j}$

3. An infinitely long straight wire carries a current $I$. Derive an expression for the magnetic field at a distance $r$ from the wire using Ampère's circuital law.

   • Solution: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$, $B(2\pi r) = \mu_0 I$, $B = \frac{\mu_0 I}{2\pi r}$

4. A bar magnet with a magnetic dipole moment of $\vec{m} = (0.1 \hat{k}) \text{ A·m²}$ is placed in a uniform magnetic field $\vec{B} = (0.2 \hat{i}) \text{ T}$. Calculate the torque experienced by the magnet.

   • Solution: $\vec{\tau} = \vec{m} \times \vec{B} = (0.1 \hat{k}) \text{ A·m²} \times (0.2 \hat{i}) \text{ T} = (0.02 \hat{j}) \text{ N·m}$

5. A rectangular current loop with dimensions $l \times w$ and current $I$ is placed in a uniform magnetic field $\vec{B}$ such that the plane of the loop is perpendicular to the field. Find the magnetic flux through the loop and the torque on the loop.

   • Solution: $\Phi_B = \vec{B} \cdot \vec{A} = BА\cos 0° = BА = Blw$, $\vec{\tau} = \vec{m} \times \vec{B} = NIA\vec{B} \times \vec{B} = 0$