Chapter 26: Magnetism: Force and Field. Magnets. Magnetism. Magnetic forces. Magnetism. Magnetic field of Earth. Magnetism. S. N. S. N. S. N. Magnetic monopoles?. Perhaps there exist magnetic charges, just like electric charges.
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Magnetism
Magnetism
N
S
N
S
N
Perhaps there exist magnetic charges, just like electric charges.
Such an entity would be called a magnetic monopole (having + or magnetic charge).
How can you isolate this magnetic charge?
Try cutting a bar magnet in half:
Magnetism
Even an individual electron has a magnetic “dipole”!
Orbits of electrons about nuclei
Intrinsic “spin” of electrons (more important effect)
Magnetism
Magnetism
Magnetic force (Lorentz force)
Magnetism
B
B
x x x x x x
x x x x x x
x x x x x x
® ® ® ® ®
® ® ® ® ®
v
v
v
´
q
q
q
F
F = 0
F
Magnetic force
Magnetism
Magnetic force vs. electric force
Magnetism
Magnetic Field Lines and Flux
Electric Field Linesof an Electric Dipole
Magnetic Field Lines of a bar magnet
Magnetic Field Lines and Flux
Units:
A=C/s, T=N/[C(m/s)]
> Tm2=Nm/[C/s]=Nm/A
No magnetic monopole has been observed!
Motion of Charged Particles in a Magnetic Field
υ perpendicular to B
The particle moves at constant speed υ in a circle in the plane perpendicular to B.
F/m = a provides the acceleration to the center, so
v
R
F
B
x
Motion of Charged Particles in a Magnetic Field
Motion of Charged Particles in a Magnetic Field
Velocity selector
Motion of Charged Particles in a Magnetic Field
Mass spectrometer
Motion of Charged Particles in a Magnetic Field
Mass spectrometer
Motion of Charged Particles in a Magnetic Field
Mass spectrometer
Motion of Charged Particles in a Magnetic Field
Mass spectrometer
Motion of Charged Particles in a Magnetic Field
Motion of Charged Particles in a Magnetic Field
Since the magnetic field does not exert
force on a charge that travels in its direction,
the component of velocity in the magnetic
field direction does not change.
Magnetic Force on a CurrentCarrying Conductor
Magnetic Force on a CurrentCarrying Conductor
Magnetic Force on a CurrentCarrying Conductor
Magnetic Force on a CurrentCarrying Conductor
Magnetic Force on a CurrentCarrying Conductor
Magnetic Force on a CurrentCarrying Conductor
Magnetic Force on a CurrentCarrying Conductor
Magnetic Force on a CurrentCarrying Conductor
Magnetic Force on a CurrentCarrying Conductor
Magnetic Force on a CurrentCarrying Conductor
Magnetic Force on a CurrentCarrying Conductor
Force and Torque on a Current Loop
Force and Torque on a Current Loop
Force and Torque on a Current Loop
The same magnetic dipole moment formulae work for any shape of planar loop.
Any such loop can be filled by a rectangular mesh as in the sketch. Each subloop is made to carry the current NI. You will now see that all the interior wires have zero current and are of no consequence. Nevertheless, each subloop contributes to μ in proportion to its area.
Force and Torque on a Current Loop
Force and Torque on a Current Loop
Work done by the torque when the magnetic
moment is rotated by df :
In analogy to the case of an electric dipole
in Chapter 22, we define a potential energy:
Potential energy of a magnetic
dipole at angle f to a magnetic
field
Current decreased
Applications
We have seen that a magnet can exert a torque on a loop of current – aligns the loop’s “dipole moment” with the field.
Slightly tip the loop
Restoring force from the magnetic
torque
Oscillations
Now turn the current off, just as the loop’s μ is aligned with B
Loop “coasts” around until itsμ is ~antialigned withB
Turn current back on
Magnetic torque gives another kick to the loop
Continuous rotation in steady state
t
Applications
charges accumulate
(in this case electrons)



+
+
+
Measuring
Hall voltage
(Hall emf)
In a steady state
qEH =qvdB
Charges move sideways until the Hall field EH grows to balance the force due to the magnetic field:
n can be measured
1
x
x
r
x
x
2
Exercises
If a proton moves in a circle of radius 21 cm perpendicular to a B field of 0.4 T, what is the speed of the proton and the frequency of motion?
v
N
B
Exercises
Example of the force on a fast moving proton due to the earth’s magnetic field. (Already we know we can neglect gravity, but can we neglect magnetism?)
Let v = 107 m/s moving North.
What is the direction and magnitude of F?
Take B = 0.5x104 T and v B to get maximum effect.
(a very fastmoving proton)
vxB is into the paper (west). Check with globe
Magnetic field produced by a moving charge
Magnetic Field of a Moving Charge
Note the factor of μ0 /4π the constant of proportionality needed just as 1/(4πε0) is needed in electrostatics.
Magnetic Field of a Current Element
ds
For an element ds of a conductor carrying a current I there are n A ds charges with drift velocity υd (using priciple of superposition).
number of charge q
ds
ds
Magnetic Field of a Current Element
ds
Note: ds is dL in the textbook.
ds
Note that ds is in the direction of I, but has a magnitude which is ds the length of wire considered.
Deduced by Biot and Savart c. 1825 from experiments with coils
The magnitude of the field dB is:
Total magnetic field at P is found by summing over all the current elements ds in the wire.
Magnetic Field of a Current Element
P
r
I
ds
P
r
R
x
ds
I
x
So the magnitude of dB is given by:
Magnetic Field of a Straight Current Carrying Conductor
dB
dB
P
r
R
x
ds
I
x
Magnetic Field of a Straight Current Carrying Conductor
dB
P
r
R
x
ds
I
x
Magnetic Field of a Straight Current Carrying Conductor
In the limit (L/R) →∞
Magnetic field by a long straight wire
The magnetic field due to segments A´A and CC´ is zero because ds is parallel to along these paths.
Along path AC, dsand are perpendicular.
Magnetic Field of a Current Element
Calculate the magnetic field at point O due to the wire segment shown. The wire carries uniform current I, and consists of two straight segments and a circular arc of radius R that subtends angle .
A´
A
ds
C´
I
C
R
O
Note: B field at the centre of a loop, =2
Force Between Parallel Conductors
At a distance afrom the wire with current I1the magnetic field due to the wire is given by
Force Between Parallel Conductors
Parallel conductors carrying current in the same direction attract each other. Parallel conductors carrying currents in opposite directions repel each other.
Force Between Parallel Conductors
The chosen definition is that for a = L = 1m, The ampere is made to be such that F2 = 2×10−7 N when I1=I2=1 ampere
This choice does two things (1) it makes the ampere (and also the volt) have very convenient magnitudes for every day life and (2) it fixes the size of μ0 = 4π×10−7. Note ε0 = 1/(μ0c2). All the other units follow almost automatically.
Use to find B field at the center of a loop of wire.
I
R
First find
is a vector coming out of the paper at the same angle anywhere on the circle. The angle is constant.
R
i
Magnetic Field of a Circular Current Loop
Loop of wire lying in a plane. It has radius R and total current I flowing in it.
Magnitude of B field at center of loop. Direction is out of paper.
Magnetic Field of a Circular Current Loop
Loop of wire of radius R = 5 cm and current I = 10 A. What is B at the center? Magnitude and direction
Direction is out of the page.
Total length of arc is S.
0
R
where S is the arc length S =R0
0 is in radians (not degrees)
Magnetic Field of a Circular Current Loop
What is the B field at the center of a segment or circular arc of wire?
P
Why is the contribution to the B field at P equal to zero from
the straight section of wire?
Previously from the BiotSavart’s law we had
Ampere’s Law
On substitution for B
Ampere’s Law
^
k
^
y
r
^
q
q
x
Let us look at the integral along any shape of closed path in 3D. The most general ds is
Where unit vectors are used for the radial r and the tangential directions q and for z along the wire k. In this system we have
^
^
^
tangential component of ds
For any path which encloses the wire
For any path which does not enclose the wire
A long straight wire of radius R carries a steady current I that is uniformly distributed through the crosssection of the wire. Outside R.
In region where r< Rchoose a circle of radius r centered on the wire as a path of integration. Along this path, B is again constantin magnitude and is always parallel to the path.
Consider circular current carrying loop. Calculate B field at point P, a dist x from the centre of the loop on the axis of the loop.
Ids
ds
Again in this case vector I dsis tangent to loopand perp to vector r from current element to point P. dB is in direction shown, perp to vectors r and I ds. Magnitude dB is:
ds
ds
ds
ds
Ids
ds
Field due to entire loop obtained by integrating:
But I, R and x are constant
ds
ds
Ids
B on the axis of a current loop
x >>R
Limits: x 0
Compare case of electric field on axis of electric dipole far from dipole
vs.
When the coils of the solenoid are closely spaced, each turn can be regarded as a circular loop, and the net magnetic field is the vector sum of the magnetic field for each loop. This produces a magnetic field that is approximately constant inside the solenoid, and nearly zero outside the solenoid.
I
The ideal solenoid is approached when the coils are very close together and the length of the solenoid is much greater than its radius. Then we can approximate the magnetic field as constant inside and zero outside the solenoid.
I
Use Ampère’s Law to find B inside an ideal solenoid.
N is the number of loops in the toroid, and I is the current in each loop
Applications of Ampere’s Law
A toroid can be considered as a solenoid “bent” into a circle as shown. We can apply Ampère’s law along the circular path inside the toroid.
The wire semicircles shown in Fig. have radii a and b. Calculate the net
magnetic field that the current in the wires produces at point P.
I
Since point P is located at a symmetric position
with respect to the two straight sections where the
current I moves (anti)parallel to the radial direction.
So there is no contributions from these segments.
The contribution from the semicircle of radius a is
a half of that from a complete circle of the same radius:
Similarly the contribution from the semicircle of radius b is:
From principle of superposition, the net magnetic field at point P is:
I
P
Long, straight conductors with square cross sections and
each carrying current I are laid sidebyside to form an
infinite current sheet. The conductors lie in the xyplane,
are parallel to the yaxis and carry current in the +y
direction. There are n conductors per unit length measured
along the xaxis. (a) What are the magnitude and direction
of the magnetic a distance a below the current sheets?
(b) What are the magnitude and direction of the magnetic
field a distance a above the current sheet?
x
z
y
B
B
L