Ch.8 Magnetic Field

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Overview

• created by moving charges
• alwaysd dipoles with a north and south pole
• practical magnetic fields created using currents, collections of moving charges
• electron spin gives certain elements magnetic properties

Magnetism

• magnet has a north pole pointing north and a south pole which attracts the north pole and repels other north poles (magnetic dipoles)
• cutting a magnet gives two weaker magnets each with a north and south pole
• magnetic materials are attracted to both poles
• if the right thumb is the direction of current in a wire, the direction of the curled other fingers is the direction of the north pole tangent to the circle around the wire (right hand rule)

Earth

• currents in the core create a magnetic field
  • slightly offset poles from geographic poles
  • magnetic south pole at geographic north pole

Magnetic Field

• magnetic field$\vec{B}$'s properties
  • magnetic field created around all points in space around current-carrying wire
  • is a vector field
  • exerts forces in magnetic poles; force of north pole is parallel to $\vec{B}$, south is opposite
  • obeys principle of superposition
• compass needle can be used to probe
• magnetic field lines are circles around a point charge
• Biot-Savart Law
  $\vec{B}_\text{point charge}=\frac{\mu_0}{4\pi}\frac{qv\sin\theta}{r^2} (\text{direction}\perp \hat{r})=\frac{\mu_0}{4\pi}\frac{q\vec{v}\times\vec{r}}{|\vec{r}|^3}\\d\vec{B}=\frac{\mu_0}{4\pi}\frac{dq(d\vec{l}/dt)\times\vec{r}}{|\vec{r}|^3}=\frac{\mu_0}{4\pi}\frac{Id\vec{l}\times\vec{r}}{|\vec{r}|^3}\\\vec{B}(\vec{r})=\int\frac{\mu_0}{4\pi}\frac{Id\vec{l}\times\hat{r}}{|\vec{r}|^2}$
  where $\mu_0=1.26\cdot10^{-6}\text{ T m/A}$ is the permeability constant
• SI Unit tesla, where $1\text{ T}=1\text{ N/A m}$
  • Earth's magnetic field is $5\cdot10^{-5}\text{ T}$, a refrigerator magnet is $0.01\text{ T}$, an industrial electromagnet is $0.1\text{ T}$, and a superconducting magnet is $10\text{ T}$

Magnetic Fields of Currents

Straight Wire

constant current $I$, distance $d$ away
$B=\int\frac{\mu_0I}{4\pi}\frac{ds\sin\theta}{r^2}=\frac{\mu_0I}{4\pi}\int\frac{ds}{r^2}\frac{d}{\sqrt{x^2+d^2}}\\=\frac{\mu_0Id}{4\pi}\int_{-L/2}^{L/2}(x^2+d^2)^{-3/2}dx=\frac{\mu_0Id}{4\pi}\left(\frac{x}{d^2\sqrt{x^2+d^2}}\right)\Bigg|_{-L/2}^{L/2}=\frac{\mu_0IL}{2\pi d\sqrt{L^2+4d^2}}$
As $L\to\infty$, this approaches
$B=\frac{\mu_0I}{2\pi r}(\text{tangent to the circle around the wire in the right-hand direction})$

Flat Coils (Current Loop)

constant current $I$, $N$ turns of coil
find field at the center of a single ideal loop
$B=\frac{\mu_0I}{4\pi}\int\frac{ds\sin\theta}{r^2}=\frac{\mu_0I}{4\pi R^2}\int_0^{2\pi}Rd\theta=\frac{\mu_0I}{2R}$
if the $N$ coils are tightly coiled, then its simply this times $N$
$B_\text{center}=\frac{\mu_0}{2}\frac{NI}{R}$

current loops create a magnetic dipole acting like a permanent magnet, also known as an electromagnet

If the axis of the dipole was the $z$-axis, then the magnetic field is
$B_\text{loop}=\frac{\mu_0}{2}\frac{IR^2}{(z^2+R^2)^{3/2}}$
which at $z\gg R$, approaches
$B_\text{loop}=\frac{\mu_0}{2}\frac{IR^2}{z^3}=\frac{\mu_0}{4\pi}\frac{2(\pi R^2)I}{z^3}=\frac{\mu_0}{4\pi}\frac{2AI}{z^3}$
Define the magnetic dipole moment$\vec{m}=AI$ where $A$ is the area of the loop and $I$ is the current. At large $z$, it does not have to be a circular loop. This makes the magnetic field
$\vec{B}_\text{dipole}=\frac{\mu_0}{2\pi}\frac{2\vec{m}}{z^3}(\text{on the axis of the dipole})$
This closely resembles the field of an electric dipole
$\vec{E}_\text{dipole}=\frac{1}{4\pi\epsilon_0}\frac{2\vec{p}}{z^3}(\text{on the axis of the dipole})$

Helical Coils (Solenoid)

constant current $I$, $n=N/L$
$B=\mu_0nI$

Ampere's Law

the magnetism counterpart to Gauss's Law
uses a line integral, an integral taken over a curve
$\int_i^f\vec{B}\cdot d\vec{s}$
If the magnetic field is perpendicular at every point on the line, the integral is $0$. If the field is tangent and has the same magnitude as $\vec{B}$ at every point, then the integral is $Bl$ where $l$ is the length.
Note for a closed loop, this is denoted $\oint$, though the meaning is the same.

Consider a closed loop. The field at the center is $\mu_0I/2\pi r$, and the field is everywhere tangent with constant magnitude, so we can easily compute
$\oint\vec{B}\cdot d\vec{s}=Bl=B2\pi r=\mu_0I$
Similar to Gauss's Law, it can be shown that this relationship

• is independent of shape around the current
• is independent of where the current passes through the curve
• depends only on total current through the area enclosed

Thus, if $I_\text{through}$ is the total current passing through an area bounded by a closed curve, the following relatioship holds
$\oint\vec{B}\cdot d\vec{s}=\mu_0I_\text{through}$
This is Ampere's Law

Solenoid

A uniform magnetic field may be produced with a solenoid, a helical coil of wire of length $L$ formed with $N$ loops or turns
An ideal solenoid has uniform magnetic field parallel to to the axis inside and field of zero outside.

Consider a rectangular cross section of length $l$ encompassing $N$ turns. The top part should be outside, the bottom should be inside, and the sides should be perpendicualr to the field.
The total current is $I_\text{through}=NI$
To compute $\oint\vec{B}\cdot d\vec{s}$, note that the sides contribute $0$ because the field is perpendicular to the path, the bottom is simply $Bl$ due to constant parallel field, and outside is $0$ because the field is zero. Therefore,
$\oint\vec{B}\cdot d\vec{s}=Bl=\mu_0NI\implies B_\text{solenoid}=\mu_0I\frac{N}{l}=\mu_0In$
where $n$ is the number of turns per unit length

With a limited number of turns, there is some field outside, and resembles a bar magnet. Therefore, a solenoid is an electromagnet.

Magnetic Force on Moving Charge

Magnetic fields exert forces on currents, or moving charges
$\vec{F}_{\text{on }q}=q\vec{v}\times\vec{B}$
Properties:

• only moving charges $\vec{v}\ne 0$ experience magnetic force
• no force on charges moving parallel/antiparallel to field
• the force is perpendicular to the plane containing $\vec{v}$ and $\vec{B}$
• magnetic is force is always perpendicular to direction of motion, so it does no work

Magnetic fields are created by and forces exerted on moving charges -> magnetism is the interaction of moving charges

In a uniform magnetic field, a charged particle moving perpendicular to the field experiences a force that causes it to move in uniform ciruclar motion. This is the cyclotron motion of a charged particle.
If there is a parallel component, then it is not affected, creating a helical movement.

The radius of cyclotron orbit is given by $F=mv^2/r$, giving $r_\text{cyc}=mv/qB$
The cyclotron frequency is given by $f=v/2\pi r$, giving $f=qB/2\pi m$

Cyclotron

The cyclotron was the first practical particle accelerator invented in the 1930s

A potential with voltage $\Delta V$ that oscillates changes the sign of the potential across the gap. When a proton starts moving, it is accelerated across the gap due to the potential difference, then when it comes back, it is accelerated again, since the potential changed sign, looping over and over again until it exists after being accelerated with kinetic energy $K=2Ne\Delta V$

Hall Effect

The Hall Effect states charges moving through a conductor as a current is also deflected by a magnetic field
If a magnetic field perpendicular to a charge carrier is created, there is a force pushing charge to the edge, leaving excess charge, like capacitor plates. This creates a potential difference and an electric field inside, exerting an opposite electric force.
The Hall voltage is the steady-state potential difference where the magnetic force and electric force cancel out.
$F_B=ev_dB=F_E=eE=e\frac{\Delta V}{w}\implies\Delta V_H=wv_dB$
where $w$ is the separation distance (width of the charge carrier) and $v_d$ is the electron drift speed
Using $J=nev_d$ and $A=wt$ (the cross-sectional area of the conductor), we have
$v_d=\frac{J}{ne}=\frac{I}{wtne}\implies\Delta V_H=\frac{IB}{tne}$

Magnetic Force on Current-Carrying Wires

For a small segment of wire with width $\Delta x$ and charge $\Delta q$ moving at velocity $v_d$ and in a magnetic field $\vec{B}$ perpendicular to current $I$, the magnitude of force felt is
$F=\Delta qv_dB=\frac{\Delta q}{\Delta t}(v_d\Delta t)B=I\Delta x B$
In general, the force on length $l$ of wire is
$\vec{F}_\text{wire}=I\vec{l}\times\vec{B}$

Now suppose there are two wires of length $l$, distance $d$ apart with currents $I_1$ and $I_2$ in the same direction. Assume they are long enough to use the infinite wire approximation
$B=\frac{\mu_0I}{2\pi r}$
current $I_1$ is affected by $\vec{B}_2$ created by $I_2$, and current $I_2$ is affected by $\vec{B}_1$ created by $I_1$, which in turn creates a magnetic force attracting each other.
If going in opposite directions, they repel.

From the formula, set distance $\Delta x=d$ to get
$F_\text{parallel wires}=I_1\vec{l}\times\vec{B}_2=I_1l\frac{\mu_0I_2}{2\pi d}=\frac{\mu_0lI_1I_2}{2\pi d}$

A current loop in a magnetic field has the top and bottom with currents in opposite directions. This creates no net force, but a net torque
$\tau=\vec{m}\times\vec{B}=I\vec{A}\times\vec{B}$
where $\vec{m}$ is the magnetic dipole moment

Electric Motor

current is passed through a coil inside a magnetic field to create a torque
every $180^\circ$, a commutator reverses the current so the motor doesn't reach equilibrium

Explanations of Magnetic Properties

Atomic Magnet

orbiting electrons act as a moving current that creates a magnetic field; atoms with more electrons and molecules tend to cancel out the field, so this does not explain strength of things like iron