Download B - Purdue Physics

Physics 272: Electricity and
Magnetism
Mark Palenik
Wednesday, June 27th
Midterm
• Reminder: Midterm 1 week from Thursday
Will cover up through tomorrow’s lecture
• There will be no lab.
• Midterm will start at 9:50 here (room 112)
• 2 hours long
• We will review solutions afterward
Not contradictory
• Two things may have sounded contradictory
1. Potential difference is path independent
2. You can’t know the potential from E at just two points
x
Need
Need
x
Don’t Need
• You need the electric field in enough points to do an
indefinite integral (i.e. a small region around both end
points)
Energy stored in a field
• Rather than potential energy, we can talk about the energy
of the EM field
• This is useful because radiation (e.g. light) carries energy,
and we may want to know how much
• Instead of a change in potential energy, we can say that
rearranging charges changes the field energy
• What is the energy density of a field (Energy/m3)?
iClicker question
• To find the field energy density, let’s start by thinking about
work
• Consider a capacitor with two plates. The charge on each
plate is +Q and –Q
• How much work does it take to move the second plate a
distance Ds (assume infinite plates)
a)
b)
c)
d)
0
Q/Ae0 Ds
Q2/Ae0 Ds
Q2/2Ae0 Ds
Field energy
• We could write
• DK + DU = W, since DK = 0, DU = W
• Or, instead of potential energy, we can think of it as
field energy
• DK + DUfield = W, DK = 0, DUfield = W
• This means DUfield = Q2/2Ae0 Ds = ½e0E2ADs = ½e0E2DV
Energy density
• So,
∆𝑈
∆𝑉
1
2
= ∈0 𝐸 2
• Not a rigorous proof, but the result is correct!
Volume, not
potential
Topics for today
• Magnetic fields:
– Why do we have magnetic fields
– Electron current
– Biot-Savart law
– If time, how do magnetic fields arise from
relativity
So, why magnetism?
• We all know a compass needle points North
• Force called magnetism pulls on it so it always
points that way
• What is magnetism? Why do we have it?
• Moving charges produce magnetism because of
relativity!
Electric fields and relativity
• Coulomb’s law applies to charges at rest
• Static charges produce fields that sit in space
• But relativity says nothing can move faster than light,
so moving a charge takes a finite time for its field to
propagate
• Also, relativity says things get squished in the direction
of motion.
Relativity gives rise to magnetism
• Relativity says: Changes in electric field get delayed,
electric fields and objects get squished when they
move, and also that time runs differently for a moving
observer.
• Because of relativistic effects on charges and fields, we
observe magnetic fields
– Moving charges produce magnetic fields!
Maxwell’s equations
Maxwell’s equations are fully relativistic, even though they were written
decades before Einstein came up with the theory of Relativity.
r
div( E ) = Ñ × E =
e0
Current running Radiation
out of the page
div( B) = Ñ × B = 0
¶B
¶B
curl(E) = Ñ ´ E = curl(E) = Ñ ´ E = ¶t
¶t
é
¶E ù curl( B ) = Ñ ´ B = m é J + e ¶E ù
curl( B ) = Ñ ´ B = m0 ê J + e 0
0ê
0
ú
ú
¶
t
¶t û
ë
û
ë
Current density
Currents are source of B
Electron current
• The “J” in Maxwell’s equations is “current density”. We
can start with the idea of electron current.
• Electron current is number of electrons per second that
pass through a section of wire. Labeled i (lower case).
• Conventional current is the amount of charge passing
through a section of wire per second. Labeled I (upper
case).
Conventional current
• Originally, it was thought that positive
charges that moved (Ben Franklin just
guessed)
• Conventional current points in the
direction that positive charges flow
• Opposite of direction that negative charges flow
• Current is a vector, so you can also think of it as
Charge*electron current
Current is a vector
• Electron current and conventional current are both
vectors.
• Because they have opposite sign, they point in
opposite directions
• 𝐼 = 𝑞𝑖 where q is charge of electron
Electron current
Conventional current
iClicker
• A wire has a circular cross section with an area of 10-6 m2.
• The space between electrons in each direction is 10-10 m (So each electron
occupies a “box” that is 10-30 m3)
• How many electrons are in a section of wire 1 m long?
a)
b)
c)
d)
e)
1030
1024
1016
1036
I have no idea
Circular surface
iClicker
• A wire has a circular cross section with an area of 10-6 m2.
• The space between electrons in each direction is 10-10 m (So each electron
occupies a “box” that is 10-30 m3)
• The electrons are moving at 2 m/s
• How many electrons per second pass through a surface in the wire
a)
b)
c)
d)
1024
2x1024
2x1030
1048
Circular surface
Electron current
• Electron current, i = nAv = electron density*Cross
sectional area of wire*velocity of electrons
• Because of friction-like losses, an electric field is
needed just to keep electrons moving at constant
speed
• v = mE where m is a constant of the metal (electron
mobility)
Biot-Savart Law
• Similar to Coulomb’s law, gives the magnetic field of a
current carrying wire
• 𝐵=
𝜇0 𝑞𝑣𝑥𝑟
4𝜋 𝑟 2
𝜇0
4𝜋
10−7
•
=
=
𝑞
𝜇0 𝑙 𝑣𝑥𝑟
4𝜋 𝑟 2
𝑡𝑒𝑠𝑙𝑎 𝑚2
𝑐𝑜𝑢𝑙𝑜𝑚𝑏 𝑚/𝑠
𝜇0 𝐼 𝑥𝑟
𝑑𝑙
4𝜋 𝑟 2
1
𝜇0 ∈ 0 = 2
𝑐
𝑑𝑙 =
also,
• Inside the integral charge*velocity bocomes charge/length
* velocity, which is simply current!
• dl means that the integration takes place over the length of
the wire.
• 𝐼 𝑥𝑟 is a “cross product”
Cross Products
• Two ways to think about cross products. First: purely
algebraic
Unit vectors in x,y,z directions
𝑖
𝑗
𝑘
• 𝐴𝑥𝐵 = 𝑑𝑒𝑡 𝐴𝑥 𝐴𝑦 𝐴𝑧
𝐵𝑥 𝐵𝑦 𝐵𝑧
= 𝐴𝑦 𝐵𝑧 − 𝐵𝑦 𝐴𝑧 , 𝐵𝑥 𝐴𝑧 − 𝐴𝑥 𝐵𝑧 , 𝐴𝑥 𝐵𝑦 − 𝐵𝑥 𝐴𝑦
• The magnitude of a cross product is|𝐴||𝐵|sin(𝜃)
• The cross product is perpendicular to both 𝐴 and 𝐵
• Since 𝐼 𝑥𝑟 appears in the cross product, magnetic field
is perpendicular to current and r
Right hand rule
• There are a few different ways to do the right hand rule
• The one I like for 𝐴𝑥𝐵
– Using your right hand, point thumb in the direction of A
– Point fingers in direction of B
– Hand will push in the direction of 𝐴𝑥𝐵
– Note, this only gives you direction, not magnitude
• Also, keep in mind, we use a “right handed” coordinate
system, so 𝑋𝑥𝑌 = 𝑍
• Can use |𝐴||𝐵|sin(𝜃) for magnitude
iClicker: cross products
• If our X and Y axes are labeled as below, which
way does the Z axis point?
y
x
a) Out of the page
b) Into the page
iClicker B field
• Remember, the Biot-Savart law is:
𝜇0 𝐼 𝑥𝑟
𝑑𝑙
4𝜋 𝑟 2
• Assuming the conventional current through the blue wire
points to the right, which way does the magnetic field point
at the x?
x
a)
b)
c)
d)
To the right
Up
Out of the page
Into the page
I (conventional current)
A second right hand rule for wires
• We don’t usually just want to know B at one point next to a
wire. We want to know field direction everywhere around
the wire.
• Field lines twist around wire, so:
• Thumb in the direction of current, fingers curl in direction
of field
The Magnetic Effects of Currents
Conclusions:
• The magnitude of B depends on the amount of current
• A wire with no current produces no B
• B is perpendicular to the direction of current
• B under the wire is opposite to B over the wire
Oersted effect:
discovered in 1820 by H. Ch. Ørsted
How does the field around a wire look?
Hans Christian Ørsted
(1777 - 1851)
Simple Circuits
Thinner
filament wire
Tungsten filament
Inert gas
Use socket
The Magnetic Effects of Currents
Make electric circuit:
Compass needle is a magnetic
dipole. Magnetic dipoles want to
align themselves along field lines
What is the effect on the compass needle?
What if we switch polarity?
What if we run wire under compass?
What if there is no current in the wire?
Use short bulb
Net field
In this picture, there are two
components to the field. Field from
wire is strongest, since the wire is
very close.
• 𝐵𝑛𝑒𝑡 = 𝐵𝑤𝑖𝑟𝑒 + 𝐵𝑒𝑎𝑟𝑡ℎ this is what the needle wants to
point along – you will figure out the specifics in lab and
recitation
The Magnetic Effects of Currents
Make electric circuit:
Needle deflection at different currents:
change light bulb (to long one)
short-circuit: two batteries, no light bulb
Relativity gives rise to magnetism
• Relativity says: Changes in electric field get delayed,
electric fields and objects get squished when they
move, and also that time runs differently for a moving
observer.
• Let’s do a simple example to see how this can create
magnetism.
• In reality, electric and magnetic fields are two parts of a
single relativistic object called the Faraday tensor
(don’t worry, we won’t talk about it!)
Electric field observed when moving
• Take two metal wires side by side. Electrons flow in
each of them (protons are fixed). Wires are neutral, so
electron density = Proton density.
Take a ride on an electron. The
electron sees the other electrons as
fixed and protons as moving
What the electron sees
Protons Electrons
Remember, relativity also says that
objects contract in the direction of
motion.
iClicker question
• The moving electrons in a neutral wire see protons as
moving. This means (because distances contract in
relativity) that the density of protons they see is
a) Greater than the electron density
b) Less than the electron density
c) The same as the electron density
Protons Electrons
Electrons see higher proton density
• Because electrons are moving, they see the distance
between protons as contracted (as well as the protons
themselves, since protons have a finite, but small size).
• The distance between protons is smaller, so the proton
density is higher. Electrons see this
iClicker question
• The fact that electrons see a higher proton
density means that the two wires will
a) Attract
b) Repel
c) Nothing
The protons see each wire as neutral. The electrons see each wire as positively
charged. Therefore, the electrons will be attracted to the other wire.
So, do we need a wire?
• We could make similar arguments if we had
just streams of flowing electrons (instead of
electrons and protons)
– In the moving frame, the electron density is still
lower, so there is less repulsive force
• Individual moving charges will also produce a
magnetic field