Magnetism

The interaction of moving charges

Charged Particles

influence is described in terms of

The influence

and interactions

of

Story so far was about

Electric

Fields

3-D Electric Field Caused by two identical Point Charges

interaction is described in terms of

Electric

Forces

Magnetism

Charged Particles

The influence

and interactions

of

Today's story...

moving

3-D Electric Field Caused by two identical Point Charges

Magnetism

influence is described in terms of

Magnetic

Fields

interaction is described in terms of

Magnetic

Forces

Magnetism

Introduction

Intrinsic Magnetism

Magnetism

I. Sources of Magnetic Fields

Magnetism

Sources of Magnetic Fields

Intrinsic Magnetism

Electric charges in motion

Magnetism

I. Sources of Magnetic Fields

\vec{B}=\frac{\mu_0}{4\pi}\frac{q\ \vec{v}\ \times \ \hat{r}}{r^2}

Direction is given by

Magnitude is given by

{B}=\frac{\mu_0}{4\pi}\frac{q\ v\ \text{sin}\theta }{r^2}

\mu_0=4\pi\times 10^{-7} T\ m/A

[B]=\text{T for Tesla}

Magnetism

Sources of Magnetic Fields

Biot-Savart Law

Right-Hand-Rule

Direction is given by

Magnitude is given by

{B}=\frac{\mu_0}{4\pi}\frac{q\ v\ \text{sin}\theta }{r^2}

\mu_0=4\pi\times 10^{-7} T\ m/A

[B]=\text{T for Tesla}

Magnetism

Sources of Magnetic Fields

Biot-Savart Law

Right-Hand-Rule

\vec{B}=\frac{\mu_0}{4\pi}\frac{q\ \vec{v}\ \times \ \hat{r}}{r^2}

Direction is given by

Magnitude is given by

{B}=\frac{\mu_0}{4\pi}\frac{q\ v\ \text{sin}\theta }{r^2}

\mu_0=4\pi\times 10^{-7} T\ m/A

[B]=\text{T for Tesla}

Magnetism

Sources of Magnetic Fields

Biot-Savart Law

Right-Hand-Rule

\vec{B}=\frac{\mu_0}{4\pi}\frac{q\ \vec{v}\ \times \ \hat{r}}{r^2}

Sources are moving charges–if the velocity is zero, the magnetic field is zero!

–

Superposition principle applies–just like electric fields, magnetic field vectors add linearly!

–

Magnetic field strength has inverse square dependence on the distance from the moving charge

Magnetism

Sources of Magnetic Fields

Biot-Savart Law -- example

\vec{B}=\frac{\mu_0}{4\pi}\frac{q\ \vec{v}\ \times \ \hat{r}}{r^2}

Magnitude is given by

{B}=\frac{\mu_0}{4\pi}\frac{q\ v\ \text{sin}\theta }{r^2}

\mu_0=4\pi\times 10^{-7} T\ m/A

[B]=\text{T for Tesla}

Magnetism

Sources of Magnetic Fields

Biot-Savart Law -- example

\vec{B}=\frac{\mu_0}{4\pi}\frac{q\ \vec{v}\ \times \ \hat{r}}{r^2}

Magnitude is given by

{B}=\frac{\mu_0}{4\pi}\frac{q\ v\ \text{sin}\theta }{r^2}

\mu_0=4\pi\times 10^{-7} T\ m/A

[B]=\text{T for Tesla}

Magnetic Field Lines

Magnetism

I. Sources of Magnetic Fields

Magnetism

Sources of Magnetic Fields

Magnetic Field Lines

Read about the properties of magnetic field lines

Magnetism

Sources of Magnetic Fields

Magnetic Field Lines

animation from National MagLab

Electric Currents

Magnetism

I. Sources of Magnetic Fields

Magnetism

Sources of Magnetic Fields

Biot-Savart Law

d\vec{B}=\frac{\mu_0}{4\pi}\frac{I\ d\vec{l}\ \times \ \hat{r}}{r^2}

\vec{B}=\frac{\mu_0}{4\pi}\frac{q\ \vec{v}\ \times \ \hat{r}}{r^2}

q \vec{v} = q \frac{d\vec{l}}{dt} = I\ d\vec{l}

Biot-Savart Law

d\vec{B}=\frac{\mu_0}{4\pi}\frac{I\ d\vec{l}\ \times \ \hat{r}}{r^2}

Magnetic Field due to a Current Segment

Magnetism

Sources of Magnetic Fields

Biot-Savart Law

Magnetism

Sources of Magnetic Fields

Biot-Savart Law

Magnetic Field due to a thin straight wire

Magnetism

Sources of Magnetic Fields

Biot-Savart Law

Magnetic Field on the axis of a circular loop

Magnetism

Sources of Magnetic Fields

Biot-Savart Law

Magnetic Field on the axis of a solenoid

Biot-Savart Law

d\vec{B}=\frac{\mu_0}{4\pi}\frac{I\ d\vec{l}\ \times \ \hat{r}}{r^2}

Magnetism

Sources of Magnetic Fields

Biot-Savart Law -- TL;DR

a distance d away from a Long straight segment

B=\frac{\mu_0 I}{2 \pi d}

@ center of Circular Loop

B=\frac{\mu_0 I}{2 R}

On the axis of a circular Loop

B=\frac{\mu_0 I R^2}{2 (y^2+R^2)^\tfrac{3}{2}}

On the axis of a solenoid

B=\frac{\mu_0 N I}{l}

Magnetic Force on a moving charge -- The Rules

Magnetism

II. Magnetic Forces

Magnetism

Magnetic Forces

Magnetic Force on a moving charge

\text{experiences a Magnetic Force}

\text{a charge}

\text{moving with velocity}

\text{in a magnetic field}

\text{N}

\text{S}

q

\vec{v}

\vec{B}

q

\vec{v}

\vec{B}

\vec{F}_B=q\ \vec{v}\ \times \vec{B}

Magnetism

Magnetic Forces

Magnetic Force on a moving charge

\vec{F}_B=q\ \vec{v}\ \times \vec{B}

\text{The Magnetic Force}

\text{on a charge}

\text{moving with velocity}

\text{in a magnetic field}

\text{N}

\text{S}

q

\vec{v}

\vec{B}

Magnetism

Magnetic Forces

Magnetic Force on a moving charge

\text{The Magnetic Force}

\text{on a charge}

\text{moving with velocity}

\text{in a magnetic field}

\text{N}

\text{S}

q

\vec{v}

\vec{B}

\vec{F}_B=q\ \vec{v}\ \times \vec{B}

Magnetism

Magnetic Forces

Magnetic Force on a moving charge

\vec{F}_B=q\ \vec{v}\ \times \vec{B}

\text{N}

\text{S}

q

\vec{v}

\vec{B}

\text{Magnitude:}

{F}_B=q\ v\ B \ \sin\theta

\theta

q=0 \quad \text{neutral charge }

v=0 \quad \text{stationary charge }

B=0 \quad \text{region w/ no field}

\theta=0 \quad \text{(or } \pi\text{)} \quad \vec{v}\ //\ \vec{B}

F_B=0 \quad \text{for: }

Magnetism

Magnetic Forces

Magnetic Force on a moving charge

\text{N}

\text{S}

q

\vec{v}

\vec{B}

\theta

\text{RHR-Force:}

\vec{v}

\vec{B}

\vec{F}_B

on (+) charges

\vec{F}_B

on (-) charges

\theta

\vec{F}_B=q\ \vec{v}\ \times \vec{B}

Magnetism

Magnetic Forces

Magnetic Force on a moving charge

An electron travelling at 20% the speed of light in a direction 30 degrees North of East, passes through a region where Earth’s magnetic field is uniform, pointing North, with a magnitude of 50µT. What is (the magnitude and direction of) the magnetic force experienced by the electron?

The trajectory of charged particles in uniform magnetic fields

Magnetism

II. Magnetic Forces

Magnetism

Magnetic Forces

Particle trajectory

+

Consider a charged particle traveling perpendicular to a uniform magnetic field.

The particle experiences a magnetic force in a direction given by the right-hand-rule.

The trajectory of the particle follows a circular arc at a constant speed.

+

An acceleration perpendicular to the direction of motion will only change the direction of motion.

The change in the direction of motion induces a change in the direction of the force (& acceleration)

Magnetism

Magnetic Forces

Particle trajectory

+

+

F_c=m\ a_c

q\ v_{\tiny \perp} {B}=m\ \frac{v_{\tiny \perp} ^2}{R}

R= \frac{m}{q}\frac{\ v_{\tiny \perp}}{B}

The magnetic force acts as a centripetal force, keeping the charge in a circular arc of radius R

The radius, R, depends on:

m and q (mass and net charge) are intrinsic quantities related to the particle itself,

v is the speed of the particle (which is related to its kinetic energy)

B is the strength of the magnetic field (aka magnetic flux density)

Magnetism

Magnetic Forces

Particle trajectory

+

+

F_c=m\ a_c

q\ v_{\tiny \perp} {B}=m\ \frac{v_{\tiny \perp} ^2}{R}

The magnetic force acts as a centripetal force, keeping the charge in a circular arc of radius R

The radius of the circular trajectory

net electric charge

magnetic field strength

\perp \vec B

component of velocity

mass

R= \frac{m}{q}\frac{\ v_{\tiny \perp}}{B}

Magnetism

Magnetic Forces

Particle trajectory

In general, the particle's motion can be decomposed into components:

The component parallel to the magnetic field is unchanged by its interaction with the field

The resulting motion is helical about the direction of the magnetic field

The component perpendicular to the magnetic field undergoes uniform circular motion.

Magnetism

Magnetic Forces

Particle trajectory

The basic cause of Aurora involves the interaction of the solar wind with Earth's magnetosphere

Magnetism

Magnetic Forces

Magnetic Force on a moving charge

The electric force

is parallel to

the electric field

The magnetic force

is perpendicular to

the magnetic field

\vec{F}=q\ \vec{v}\ \times\ \vec{B}

\vec{F}=q\ \vec{E}

Magnetism

Magnetic Forces

Magnetic Force on a moving charge

The electric force

is not perpendicular to

the motion

The magnetic force

is perpendicular to

the motion

W_\text{magnetic force} = 0

W_\text{electric force}\neq 0

particle speeds up or slows down

particle changes direction of motion

Magnetic Force on a moving charge -- Applications

Magnetism

II. Magnetic Forces

Magnetism

Magnetic Forces

Particle trajectory

Application: Velocity Selector

A velocity selector is a device that allows charged particles with a particular velocity to pass through, while deflecting all other charged particles.

Magnetism

Magnetic Forces

Particle trajectory

The device operates by applying electric and magnetic forces to the particle in such a way that these

forces balance.

For the situation shown in the figure, how should an electric field be applied so that the force it applies to the particle can balance the magnetic force?

For the situation shown in the figure, what is the selected velocity?

Magnetism

Magnetic Forces

Particle trajectory - Hall Effect

Application: Hall Effect Sensors

Magnetism

Magnetic Forces

Particle trajectory - pair production

Charged

Particles

in

Cloud/Bubble

Chambers

Magnetism

Magnetic Forces

Particle trajectory - pair production

If the magnetic field is directed out of the "page"-- What are the signs of the charges of the three particles?

All 3 particles have the same mass and (magnitude of) charge-- which particle is initially moving most rapidly?

The particles follow a spiraling path -- Are we able to explain why?

The tracks going counter clockwise are left by negatively charged particles.

The bigger the speed, the bigger the radius (for same q, m and B)

Loss of energy (signified by loss of speed, leads to ever decreasing radius)

Magnetism

Magnetic Forces

Particle trajectory - the mass spectrometer

r= \frac{m\ v}{q\ B}

For fixed v, q, and B

r \propto m

Detector

particle injection

Magnetism

Magnetic Forces

Particle trajectory - the mass spectrometer

\tfrac{1}{2}mv^2=qV

r= \frac{m\ v}{q\ B}

\}

m=\left(\frac{q\ r^2}{2\ V}\right)\ B^2

Magnetic Force on a current-carrying conductor

Magnetism

II. Magnetic Forces

Magnetism

Magnetic Forces

Magnetic Force on a current-carrying conductor

Consider a conductor carrying a an electric current in a region where there is a magnetic field

charges flowing inside the conductor will experience a magnetic force

\vec{F}_B=q\ \vec{v}\ \times \ \vec{B}

q \vec{v} = q \frac{d\vec{l}}{dt} = I\ d\vec{l}

\vec{F}_B=\int I\ d\vec{l}\ \times \ \vec{B}

d\vec{F}_B= I\ d\vec{l}\ \times \ \vec{B}

Magnetism

Magnetic Forces

Magnetic Force on a current-carrying conductor

For a straight segment in a uniform magnetic field:

The Magnetic Force

electric current

magnetic field strength

angle between current & field

length of segment

F= I\ L\ B \sin\theta

Torque on a current-carrying loop

Magnetism

II. Magnetic Forces

Magnetism

Magnetic Forces

Torque on a current-carrying loop

q \vec{v} = q \frac{d\vec{l}}{dt} = I\ d\vec{l}

Consider a rectangular loop, L x W, carrying a current I, in a uniform magnetic field of strength B

F= I\ L\ B \sin\theta

The magnitude of the force on each arm of the loop is given by

These forces are in opposite directions, as given by the RHR

\implies \Sigma\vec{F} =0\quad \&\quad \Sigma\vec{\tau} \ne0

Magnetism

Magnetic Forces

Torque on a current-carrying loop

\Sigma\vec{\tau}=2\times F\times (\frac{W}{2}\sin\phi)

=(ILB\sin\theta) (W\sin\phi)

=IAB\sin\phi

\vec{\tau}=\vec{\mu}\times\vec{B}

\vec{\mu}=IA\ \hat{n}

To calculate the net torque about the axis of rotation, consider the situation where the normal to the loop makes an angle with the magnetic field:

\phi

where

\implies