Magnetic Fields - ch. 28

ucla | PHYSICS 1C | 2023-01-17T13:05

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Table of Contents

• Definitions
• Big Ideas
  • Magnetic Fields of a Moving Charge
  • due to a Straight Current Element
    • Biot-Savart Law (current elements)
    • Infinitely Long Wire
    • Between 2 parallel conductors (wires)
  • due to a Circular Current Loop(s)
    • Current Loop
    • Coil
  • Ampere’s Law
  • due to Cylindrical Conductor
  • due to a straight Solenoid
  • due to a Toroidal Solenoid
  • Bohr Magneton
    • Quantized Angular Momentum
  • Paramagnetism and Diamagnetism
    • Diamagnetism
    • Magnetism Table at 20 ºC
  • Ferromagnetism
    • Hysteresis
• Resources

Definitions

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Big Ideas

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Magnetic Fields of a Moving Charge

• magnetic fields are determined at a point (distance) away from the source and depends on the velocity vector

$\vec B=\frac{\mu_0}{4\pi}\frac{q\vec v\times \hat r}{r^2}=\frac{\mu_0}{4\pi}\frac{q\vec v\times \vec r}{r^3}\quad\text{st.}\quad \hat r=\frac{\vec r}{

\vec r

}$

• ${\mu }_0$ is the permeability of free space given as the magnetic constant

${\mu }_0=4\pi \times {10}^{-7}[T\cdot \frac{m}{A}]$

• Field Lines:
  the magnetic field lines form concentric circles around the velocity vector of the charge given by the “curled right hand rule” $\therefore \overrightarrow{B}=0⟺\overrightarrow{r}\times \overrightarrow{v}=0$ (i.e. point of measurement is on the velocity vector)

due to a Straight Current Element

• for a current-carrying conductor segment with constant cross-sectional area $A$, length $dl$, and $n$ charges per unit volume of charge $q$
• given total charge in the segment

$dQ=nqAdlQ=\int nqAdl$

• the magnitude of current through the wire

$I=n

q

v_d\cdot A=J\cdot A\quad\text{st.}\quad J=nqv_d$

• Field Lines

Biot-Savart Law (current elements)

$d\overrightarrow{B}=\frac{{\mu }_0}{4\pi }\frac{dQ({\overrightarrow{v}}_d\times \hat{r})}{r^2}=\frac{{\mu }_{0}I}{4\pi }\frac{d\overrightarrow{l}\times \hat{r}}{r^2}$

$\overrightarrow{B}=\frac{{\mu }_0}{4\pi }\int \frac{Id\overrightarrow{l}\times \hat{r}}{r^2}$

• Using Biot-Savart for a wire of length $l$ at a distance $r$ away is shown dependent to $r_{\perp }$

$B=\frac{{\mu }_{0}I}{4\pi }\frac{l}{r_{\perp }\sqrt{r_{\perp }^2+(l/2{)}^2}}$

Infinitely Long Wire

• a rigorous solution provided by Biot-Savart law
• field strength at a perpendicular distance $r$ from the wire

$B=\frac{{\mu }_{0}I}{2\pi r_{\perp }}$

Between 2 parallel conductors (wires)

• given 2 infinitely long wires → the direction of their currents tell us thheir interaction
• when in the same direction → the wires attract each other through exerted attractive forces on each other
• we can find the force exerted by any given wire as:

$\vec F=I_1\vec L\times\vec B$

• this allows us to find force pr unit length of the wire:

$\frac{F}{L}=\frac{{\mu }_0{I}_1{I}_2}{2\pi r}$

• Field Lines

  Current in SAME direction

  Current in OPPOSITE direction

due to a Circular Current Loop(s)

Current Loop

• we can use Biot-Savart to find the differential parts of the field due to a loop with radius $a$ on its normal (central) axis $x$ distance away from the center

$dB=\frac{{\mu }_{0}I}{4\pi }\frac{dl}{x^2+a^2}$

• the radial components cancel out and we are left with only the field exerted in the direction of the normal vector

$dB_x=dB\cos\theta=\frac{\mu_0I}{4\pi}\frac{dl}{x^2+a^2}\frac{a}{\sqrt{x^2+a^2}}\space$

• Field Lines

Coil

• a coil can be represented as $N$ stacked current loops such that on the axis of the loop:

$B_x=\frac{\mu_0NIa^2}{2(x^2+a^2)^{3/2}}$

• the magnetic moment for 1 loop is $\mu =I\pi a^2$, so for $N$ loops → $\mu =NI\pi a^2$
• this implies the field can also be written as

$B_x=\frac{{\mu }_0\mu }{2\pi (x^2+a^2{)}^{3/2}}$

Ampere’s Law

• Ampere’s law works on highly symmetric situations using the line integral in place of Gauss’ flux on a closed path ($l$) enclosing the current element

$\oint \overrightarrow{B}\cdot d\overrightarrow{l}={\mu }_0{I}_{\text{enc}}$

• it is intended to be used on an enclosed path such that $d\overrightarrow{l}\parallel \overrightarrow{B}$

• if $\oint \overrightarrow{B}\cdot d\overrightarrow{l}=0⟹I_{\text{enc}}=0$ NOT that the magnetic field is zero

due to Cylindrical Conductor

• on a thick wire of outer radius $R$ with total current $I$ distributed evenly across the cross-sectional area of the conductor
• the enclosed charge can be expressed as

$I_{\text{enc}}=\frac{\pi r^{2}I}{\pi R^2}$

• magnetic field inside the conductor using Amperian loop of radius $r<R$

$\oint \overrightarrow{B}\cdot d\overrightarrow{l}⟹B=\frac{{\mu }_{0}I}{2\pi }\frac{r}{R^2}$

• field outside the conductor with Amperian loop of $r>R$

$\oint \overrightarrow{B}\cdot d\overrightarrow{l}⟹B=\frac{{\mu }_{0}I}{2\pi r}$

due to a straight Solenoid

• magnetic field at the center of a solenoid of $n$ turns per unit length and current $I$passing through each turn using an Amperian loop through a length $L$
• assume the solenoid is infinitely long as to define field lines as straight and contained within the solenoid (no leakage outside)

$I_{\text{enc}}=\frac{NIL}{L}=nLI$

$\oint vecB\cdot d\overrightarrow{l}=BL⟹B=\frac{{\mu }_{0}NI}{L}={\mu }_{0}nI$

• Field Lines

due to a Toroidal Solenoid

• has N turns around a doughnut shaped ferrous metal
• the field lines are circular around the inside of the metal
• the central hollow region has $B=0$
• the outside region has $B\approx 0$ (due to helical windings instead of circular)
• inside the solenoid

$I_{\text{enc}}=NI\space$

• Field Lines

Bohr Magneton

• in atomic physics: magnetism begins with the motion of electrons

• we can model a quantum mechanically incorrect model of an electron with orbit of radius $r$ and tangential velocity $v$
• using the charge of the electron $e$ and period of motion $T$

$I=\frac{e}{T}=\frac{ev}{2\pi r}$

• resulting magnetic moment

$\mu =IA=\frac{ev}{2\pi r}(\pi r^2)=\frac{evr}{2}$

• given the angular moment $L=mvr$

$\mu =\frac{e}{2m}L$

Quantized Angular Momentum

• atomic angular moment is quantized: its component of ang. mom. in any particular direction is an integer multiple of the reduced Planck’s constant: $\hslash =\frac{h}{2\pi }$ where $h=6.626\times {10}^{-34}$ is Planck’s Constant
• thus for $L=\frac{h}{2\pi }$ the Bohr Magneton

${\mu }_B=\frac{e}{2m}\frac{h}{2\pi }=\frac{eh}{4\pi m}=9.274\times {10}^{-24}[A\cdot m^2]$

• the spin of an electron is very close to the Bohr magneton

Paramagnetism and Diamagnetism

• most orbital and spin magnetic moments are cancelled out, but in some materials, they can have a net magnetic moment on the order of Bohr Magneton
• if such materials are place in an external magnetic field ${\overrightarrow{B}}_0$, the individual magnetic moments tend to align with the field → produce another magnetic field proportional to total magnetic moment ${\overrightarrow{\mu }}_{\text{total}}$ of the material
• magnetization of a material is its magnetic moment per unit Volume

$\overrightarrow{M}=\frac{{\overrightarrow{\mu }}_{\text{total}}}{V}={\chi }_m\frac{{\overrightarrow{B}}_0}{{\mu }_0}$

• therefore the total magnetic field is

$\overrightarrow{B}={\overrightarrow{B}}_0+{\mu }_0\overrightarrow{M}={\overrightarrow{B}}_0{K}_m$

• materials that exhibit this behavior are paramagnetic: the magnetic field inside the material is stronger than outside

Diamagnetism

• paramagnetism implies the internal field is greater than an equivalent field in the vacuum around by a factor of $K_m$ known as the relative permeability of the material (i.e. depends on the material)
• this can give us the permeability ($\mu$) of the material (NOT MAGNETIC MOMENT)

$\mu =K_m{\mu }_0$

• magnetic susceptibility is the amount the relative permeability differs from 1

${\chi }_m=K_m-1$

• diamagnetism is the property that the internal fields of the material oppose the external field in direction such that $K_m<1$

Magnetism Table at 20 ºC

Ferromagnetism

• ferromagnetic materials (iron, nickel, cobalt, etc.) have strong interactions between magnetic moments that cause magnetic moments to align parallel to each other in magnetic domains regardless of external magnetic field
• when there is no external fields, the magnetic fields are randomly oriented → presence of magnetic field causes domains to orient in the net direction of the field → most materials return to normal after being removed from the field → ferromagnetic materials do not
• such materials are magnetized so their domains don’t revert (bar magnet, other magnets) and have relative permeabilities $K_m$ on the order of ${10}^3,{10}^5$

Hysteresis

• the behavior of the relationship of magnetization $M$ to the external magnetic field $B_0$ strength is hysteresis

• this occurs until the material reaches saturation $M_{\text{sat}}$

Resources

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