Physics 272
March 11
Spring 2014
http://www.phys.hawaii.edu/~philipvd/pvd_14_spring_272_uhm.html
Prof. Philip von Doetinchem
philipvd@hawaii.edu
Phys272 - Spring 14 - von Doetinchem - 32
Summary
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Magnetic field of a current element (vector version):
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Law of Biot and Savart:
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Important law to find the total magnetic field of at
any point in space due to currents
Phys272 - Spring 14 - von Doetinchem - 33
Summary
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Two protons moving in the same direction.
Magnetic force on
the upper proton
from the magnetic
field of the lower
proton: attractive
Magnetic force on
the lower proton
from the magnetic
field of the upper
proton: attractive
Phys272 - Spring 14 - von Doetinchem - 37
Force between moving charges
http://www.youtube.com/watch?v=43AeuDvWc0k
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Moving in the same direction for likewise signed
charges creates an attractive force and moving in
the opposite direction creates a repulsive force
Phys272 - Spring 14 - von Doetinchem - 38
Force between parallel conductors
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Interaction force between
conductors
Important when wires are
close to each other
Force on wire 1 due to
magnetic field from wire 2:
Phys272 - Spring 14 - von Doetinchem - 39
Magnetic forces and defining the Ampere
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Definition of the SI unit for current “Ampere”
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One ampere is that unvarying current that, if present
in each of two parallel conductors of infinite length
and one meter apart in empty space, causes each
conductor to experience a force of exactly 2x10 -7N
per meter of length.
Definition of 1 Coulomb based on the same
approach: 1C=1A/s
Phys272 - Spring 14 - von Doetinchem - 41
Magnetic field of a circular current loop
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Coils of wire with a large number of turns spaced
closely
Each turn is nearly
planar
Current in such a coil
causes a magnetic field
http://www.youtube.com/watch?v=7wmZrU6VQzE
Look at one loop first
→ then study the case of multiple turns
Phys272 - Spring 14 - von Doetinchem - 42
Magnetic field on the axis of a coil
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Magnetic field on axis
of the loop along the
x axis at a certain
distance x using
Biot-Savart
Phys272 - Spring 14 - von Doetinchem - 43
Magnetic field on the axis of a coil
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All magnetic field
contributions in the
plane of the loop
cancel out and only
a component along
the loop axis remains
Phys272 - Spring 14 - von Doetinchem - 44
Magnetic field on the axis of a coil
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Integration along the circular loop → B field along
the axis of the loop:
Phys272 - Spring 14 - von Doetinchem - 45
Magnetic field on the axis of a coil
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Stack a number of N loops on top of each other
Assume that each loop is planar and that the loops are
very closely spaced
Also assume that the distance from the center of each
loop is the same to the point along the coil axis under
study
Superposition principle at work:
Many loops are able to produce very strong fields
→ easier than having high currents
Phys272 - Spring 14 - von Doetinchem - 46
Magnetic field on the axis of a coil
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Express with the help of magnetic dipole moment:
Magnetic dipole moment does not only react to
magnetic field, but also acts a source of magnetic
field in this sense
Be careful: equations on last 4 slides are only
valid along the symmetry axis of the coil
Phys272 - Spring 14 - von Doetinchem - 47
Magnetic field of a coil
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A coil with 100 circular loops with radius 60cm
carries a current of 5.0A:
Phys272 - Spring 14 - von Doetinchem - 48
Magnetic field of a coil
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A coil with 100 circular loops with radius 60cm
carries a current of 5.0A:
Phys272 - Spring 14 - von Doetinchem - 49
Ampere's law
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Magnetic field calculation so far based on adding
contribution of small line segments
→ analogous to electric field calculation with adding
up electric fields of individual charges
Gauss's law was a different way to look at the
problem from the outside in symmetrical cases
Gauss's law for magnetism does not allow to make
conclusions on current
→ magnetic flux through enclosed surface is always
zero
Phys272 - Spring 14 - von Doetinchem - 50
Ampere's law
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Integrate along a closed path around a wire:
Phys272 - Spring 14 - von Doetinchem - 51
Ampere's law
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Integrate along a closed path around a wire:
Phys272 - Spring 14 - von Doetinchem - 52
Ampere's law
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Integrate along a closed path, but not totally
enclosing the wire:
If the current is not enclosed the line integral around
a closed path is zero
Phys272 - Spring 14 - von Doetinchem - 53
Ampere's law: general statement
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The scalar product always takes the parallel projection for
the integration
→ as long as the closed path integration encloses the
current we get:
This also true if multiple conductors are present within the
closed integration path
(magnetic field is the sum of the individual magnetic field of
the different conductors)
This law is true for steady currents
→ time variation next lectures
Important:
does not mean that the
magnetic field is zero
Phys272 - Spring 14 - von Doetinchem - 54
Important differences electric field vs. magnetic field
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Electric field:
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Line integral along a closed path is zero
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Electric force is conservative
→ force does zero net work on charge that returns to
start point
Magnetic field:
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Magnetic force on a moving charge is always
perpendicular
→ line integral over magnetic field is not related to the
work done by magnetic force
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Magnetic force is not conservative
→ force does not only depend on position, but also on
velocity
Phys272 - Spring 14 - von Doetinchem - 55
Application of Ampere's law
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Ampere's law useful for problems with symmetries
to determine the magnetic field
To calculate magnetic field at a certain point the
integration path has to go through this point
Look for symmetries to make the analytical solution
of the integral possible
Integration path should be tangent in regions of
interest and perpendicular in other regions (makes
the scalar product easy)
No net current enclosed: line integral is zero
Make sanity checks: how does the calculated field
behaves in different limits
Phys272 - Spring 14 - von Doetinchem - 56
Field of a long, straight, current-carrying conductor
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Similar to the example before, but reversed
situation: magnetic field unknown, current known
Identify symmetry:
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magnetic field is tangent to a circle around the conductor
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Magnetic field magnitude is the same everywhere on the
circle
Phys272 - Spring 14 - von Doetinchem - 57
Field of a long cylindrical conductor
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Cylindrical conductor with radius R carries a current
I. Current is uniformly distributed over the crosssectional area of the conductor.
conductor
Phys272 - Spring 14 - von Doetinchem - 58
Field of a long cylindrical conductor
Phys272 - Spring 14 - von Doetinchem - 59
Field of a solenoid
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Helical winding of
wire on a cylinder
Each winding can
be treated as circular
loop
All turns carry the
same current
Field is very uniform
in the middle
External field near the middle is very small
→ Assumption: magnetic field zero outside and uniform
inside
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Field is the strongest in the center
Phys272 - Spring 14 - von Doetinchem - 60
Field of a solenoid
Phys272 - Spring 14 - von Doetinchem - 61
Field of a solenoid
Phys272 - Spring 14 - von Doetinchem - 62