Physics 227: Lecture 13
More Magnetic Fun
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Lecture 12 review:
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RC circuits: voltage across capacitor does not jump. Voltages
and currents vary with time constant τ = RC like e-t/τ when
switch opens or closes to charge or discharge capacitor.
Magnets have North and South poles in pairs - no known
magnetic monopoles. Opposites attract, like poles repel. Field lines
for a bar magnet like those for an electric dipole - but since
there are no magnetic charges the field lines don’t start / stop.
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Currents generate field lines and cause forces on magnets, so
magnetic fields cause forces on currents or moving charges:
F = qE + qvxB - the magnetic force is perpendicular to the
motion and magnetic field.
Thursday, October 20, 2011
Examples of Magnetic Fields - Dipoles
PHET of coil
Thursday, October 20, 2011
Examples of Magnetic Fields
Near constant field between
poles of magnet. Note that
with a ``highly magnetic
material’’, the magnetic
field lines largely follow
along through the material.
Thursday, October 20, 2011
Circular field around a long wire
carrying a current.
Flux iClicker
We have already learned about electric
fields and electric flux, and now magnetic
fields.
What is the magnetic flux through a closed
surface (like the outside of your iClicker)?
A. 0.
B. q/ε0.
See next slide for reasoning.
C. μ/μ0.
D. ∫B.dA.
E. Magnetic flux is not like electric flux -
there is no simple answer to this question.
Thursday, October 20, 2011
Magnetic Field Flux
The magnetic flux is defined as a surface integral, in a similar
way to the electric flux: φE = ∫E.dA = qenclosed/ε0 ➮ φB = ∫B.dA =
0, since there are no magnetic charges.
Think about a box with field lines entering it. Since there are
no charges in the box, the field lines cannot start or stop in
the box. Thus every field line that enters the box must leave
it. So the total flux through the surface of the box (# field
lines leaving - # field lines entering the box) must be 0.
The units of field are Tesla (T). The units of magnetic flux are
Tesla x meters2 = Webers (Wb).
Thursday, October 20, 2011
Motion of a Charged Particle in a Uniform
Magnetic Field
The magnetic force is perpendicular to the direction of
motion of a charged particle.
What shape will the motion of a charged particle be in a
uniform magnetic field?
Assume this is out in space where you can neglect gravity.
A. Straight line.
B. Parabola.
C. Hyperbola.
D. Circle.
E. Helix.
Thursday, October 20, 2011
If the initial velocity is
parallel to the B field,
there is no magnetic
force on the particle
and it travels in a
straight line.
See next slides for reasoning
for circle and helix.
Two-dimensional Circular Orbits
Consider a particle moving in a circle of radius
r in the xy plane, centered on the origin. If its
velocity is v, then there is a force directed to
the origin, perpendicular to the velocity, with
magnitude Fc = mv2/r. (Last years physics.)
Assume a particle is moving in the xy plane,
with a magnetic field in the z direction. There
is a force FM = qvB that is always
perpendicular to the velocity.
The motion is the same, circular, whether we
have a string or a rod or the magnetic field
causing the particle to go around in a circle.
Thus Fc = mv2/r = FM = qvB ➮ r = mv/qB = p/qB.
Also, we call ω = v/r = qB/m the ``cyclotron
frequency’’. (Recall circular / simple harmonic
motion.)
Thursday, October 20, 2011
Three-dimensional Helical Orbits
If the particle has a z
component of velocity,
this component is
constant: there is no
force in the z direction.
The component vz also
does not affect the force
- it does not change the
cross product.
The path is a helix similar to a screw thread.
� = q(vx x̂ + vy ŷ + vz ẑ) × Bz ẑ = q(−vx ŷ + vy x̂)Bz = qBz (vy x̂ − vx ŷ)
F�M = q�v × B
DEMO - electron beam
Thursday, October 20, 2011
Three-dimensional Circular Orbits
Bubble chamber
image of particles
moving in a
magnetic field:
γe- → e- + e+e-
The particles gradually
lose energy, and the
orbits spiral in.
Thursday, October 20, 2011
A magnetic bottle: the charged particle
spirals around the field lines. At each end,
near the current loops where the field is
strongest, the transverse B field reflects
the particle back to the other end. (Q>0
particles spiral around one way, Q<0
particles would spiral around the other
way.)
Three-dimensional Circular Orbits
The earth’s magnetic field also acts like a magnetic bottle. Near
the poles, the particles make it deepest into the atmosphere,
strike and ionize gas atoms, which flouresce, leading to the
aurora borealis and australis - northern and southern lights.
Thursday, October 20, 2011
A Velocity Filter
Consider a particle of charge q moving in the
z direction (v = vz) at some point in time, in a
region in space where the electric field is in
the -y direction (E = -Ey) and the magnetic
field is in the x direction (B = Bx).
The electric force has magnitude FC = qE = qEy
in the -y direction.
The magnetic force is FM = qvxB = qvzBx, in the
+y direction.
If qE = qvB, or E = vB, then the net force on
the particle is 0, independent of the charge.
Thus crossed electric and magnetic fields act
as a ``velocity filter’’, allowing particles with v
= E/B to pass undeflected. Particles with other
velocities will be deflected to ±y.
Thursday, October 20, 2011
Mass Spectrometer
A velocity filter selects out charged particles
of a given speed, v = E/B.
The particles then move into a region of
constant magnetic (but no electric) field.
They move in circular orbits of radius
r = mv/qB.
For fixed v, B, q (source prepares particles to
have charge +e), we have m = r (qB/v). The
measured position gives the radius gives the
mass.
Thursday, October 20, 2011
Measuring e/m of the electron
Accelerate the electron to some kinetic energy T = qV = ½mv2.
Solve for velocity v: v = (2qV/m)1/2.
Put it through a velocity filter v = E/B.
Setting these equal: q/m = E2/(2VB2).
Everything on the r.h.s. is measured, determining q/m.
Thursday, October 20, 2011
Stereo Speaker
As current varies with time, force on wire loop varies.
The loop and the speaker cone move back and forth at ``same’’
frequency as electrical signal, causing air to move at the same
frequency.
Pressure wave in air = sound you hear.
Thursday, October 20, 2011
Magnetism iClicker
I have a wire going through a horseshoe magnet.
I will put a current through the wire. What will the wire do?
A.
B.
C.
D.
The wire will move from the
magnetic force on the current
Nothing will happen. The forces are
- the electrons do not get
on the current, not on the wire.
pulled out of the wire. The
orientation was such that the
The wire will jump up or down.
magnet jumped up and down,
It will move forward or backward.
but we also could turn the
magnet sideways and make
We will not see anything, but
electrons are pulled out of the wire. the wire move forward/
backward.
E. No current can flow due to the
magnetic field, so nothing happens.
Thursday, October 20, 2011
DEMO - rod with current
DEMO - wire
Thank you.
See you Monday.
Thursday, October 20, 2011