Sources of Magnetic Fields
Sources of Magnetic Fields
Level 5 Physics
January 2013
Material adapted from MIT 8.02 course notes
Sources of Magnetic Fields
Introduction
Observation
Magnets
Where do you think magnetic fields originate from?
Sources of Magnetic Fields
Introduction
Observation
Magnets
Where do you think magnetic fields originate from?
A magnet is perhaps the most commonly known source of magnetic
fields, but the actual cause of the magnetic field is not obvious.
Sources of Magnetic Fields
Introduction
Observation
Magnets
Where do you think magnetic fields originate from?
A magnet is perhaps the most commonly known source of magnetic
fields, but the actual cause of the magnetic field is not obvious.
An interesting feature of magnetism is that the magnetic poles
always come in a pair. The north pole and south pole of a magnet
cannot be separated to create magnetic “monopoles”.
Sources of Magnetic Fields
Introduction
Observation
Magnetic Field Observations
Some observations we have already made and formulas we have
derived give us clues for the source of magnetic fields:
Sources of Magnetic Fields
Introduction
Observation
Magnetic Field Observations
Some observations we have already made and formulas we have
derived give us clues for the source of magnetic fields:
Unlike poles attract while like poles repel
~
Magnetic force on a moving point charge: F~B = q~v × B
~
Magnetic force on a current-carrying wire: F~B = I ~` × B
Sources of Magnetic Fields
Introduction
Observation
Magnetic Field Observations
Some observations we have already made and formulas we have
derived give us clues for the source of magnetic fields:
Unlike poles attract while like poles repel
~
Magnetic force on a moving point charge: F~B = q~v × B
~
Magnetic force on a current-carrying wire: F~B = I ~` × B
Any ideas?
Sources of Magnetic Fields
Introduction
Observation
Magnetic Field Observations
Some observations we have already made and formulas we have
derived give us clues for the source of magnetic fields:
Unlike poles attract while like poles repel
~
Magnetic force on a moving point charge: F~B = q~v × B
~
Magnetic force on a current-carrying wire: F~B = I ~` × B
Any ideas?
The motion of charges causes magnetic fields.
Sources of Magnetic Fields
Introduction
Ørsted
Ørsted’s Discovery
In 1820, Ørsted noticed that a wire carrying an electric current
caused a compass needle to deflect.
He was one of the first to discover a
connection between electricity and
magnetism, and he eventually
determined that an electric current
produces a circular magnetic field.
Sources of Magnetic Fields
Introduction
Magnetic Field Lines
Straight Wire Magnetic Field
Remember that iron filings align towards the magnetic field lines.
Sources of Magnetic Fields
Introduction
Magnetic Field Lines
Straight Wire Magnetic Field
Remember that iron filings align towards the magnetic field lines.
If we apply a current along a straight wire, the alignment of the
iron filings indicates a circular magnetic field.
Sources of Magnetic Fields
Introduction
Magnetic Field Lines
Semicircle Wire Magnetic Field
The direction of the magnetic field in the center of a semicircle
wire consists of parallel lines.
This suggests that the direction of the current affects the direction
of the magnetic field (otherwise we might expect the magnetic
field to have zero magnitude in the center).
Sources of Magnetic Fields
Introduction
Magnetic Field Lines
Another Right Hand Rule
The magnetic field produced by a current-carrying wire is circular,
with the direction given by another right hand rule.
Right Hand Rule
If your thumb points along the direction of the current, your
fingers curl in the direction of the magnetic field.
Sources of Magnetic Fields
Introduction
Magnetic Field Lines
Solenoid
A solenoid is a piece of wire wrapped into a coil. An ideal solenoid
has infinite length and the coil is tightly packed.
Sources of Magnetic Fields
Introduction
Magnetic Field Lines
Solenoid
A solenoid is a piece of wire wrapped into a coil. An ideal solenoid
has infinite length and the coil is tightly packed.
What do you think the magnetic field for a solenoid looks like,
considering the magnetic field for a straight wire and the right
hand rule?
Sources of Magnetic Fields
Introduction
Magnetic Field Lines
Solenoid Magnetic Field
The magnetic field is strong inside the solenoid and consists of
parallel lines with direction determined by the direction of the
current.
The magnetic field is weak outside the solenoid and for an ideal
solenoid is zero.
Sources of Magnetic Fields
Introduction
Magnetic Field Lines
Toroid
A toroid is a solenoid wrapped around into a circle.
Sources of Magnetic Fields
Introduction
Magnetic Field Lines
Toroid
A toroid is a solenoid wrapped around into a circle.
What do you think the magnetic field for a toroid looks like,
considering the magnetic field for a solenoid?
Sources of Magnetic Fields
Introduction
Magnetic Field Lines
Toroid Magnetic Field
The magnetic field lines for a toroid are circular in the region inside
the loops.
Outside the toroid, the magnetic field is zero.
Sources of Magnetic Fields
Theory
Biot-Savart
Biot-Savart Law
Biot-Savart Law
~
The magnetic field contribution, d B,
from a current source, Id~s, at a field
point P a distance r in the direction of
r̂ is given by
~ =
dB
µ0 Id~s × r̂
4π r 2
where µ0 is the permeability of free
space constant with value
µ0 = 4π × 10−7 T · m/A
Sources of Magnetic Fields
Theory
Biot-Savart
Summing the Contributions
If the differential contribution is given by
~ =
dB
µ0 Id~s × r̂
4π r 2
then taking the integral over the length of the wire gives an
equation for calculating the magnetic field:
Z
Z
d~s × r̂
~ =
~ = µ0 I
B
dB
4π
r2
wire
wire
Remember that an integral represents a sum where the d~s are very
small.
Sources of Magnetic Fields
Theory
Biot-Savart
Finite Length Straight Wire
The magnetic field caused at point P by a finite length straight
wire can be calculated using Biot-Savart’s Law.
Sources of Magnetic Fields
Theory
Biot-Savart
Finite Length Straight Wire
The magnetic field caused at point P by a finite length straight
wire can be calculated using Biot-Savart’s Law.
Performing the calculation gives for the magnitude
B=
µ0 I
(cos θ2 + cos θ1 )
4πa
Sources of Magnetic Fields
Theory
Biot-Savart
Infinite Length Straight Wire
What is the magnitude of the magnetic field at P if the wire has
infinite length rather than finite length?
Sources of Magnetic Fields
Theory
Biot-Savart
Infinite Length Straight Wire
What is the magnitude of the magnetic field at P if the wire has
infinite length rather than finite length?
Infinite length will require θ1 → 0 and θ2 → 0.
Sources of Magnetic Fields
Theory
Biot-Savart
Infinite Length Straight Wire
What is the magnitude of the magnetic field at P if the wire has
infinite length rather than finite length?
Infinite length will require θ1 → 0 and θ2 → 0.
B=
=
lim
θ1 ,θ2 →0
µ0 I
2πa
µ0 I
(cos θ2 + cos θ1 )
4πa
Sources of Magnetic Fields
Theory
Biot-Savart
Magnetic Field for Current-Carrying Wire
Infinite Length Wire
The magnitude of the magnetic field a distance a away from a wire
with current I is
B=
µ0 I
2πa
The magnetic field line direction is given by a right hand rule.
Sources of Magnetic Fields
Questions
Concept Questions
One Current Loop
A current loop lies in the xy-plane and has counterclockwise
current. The positive z-axis is coming out of the page. What is the
direction of the magnetic field along the entire z-axis?
1
Into the page
2
Out of the page
3
It switches direction
Sources of Magnetic Fields
Questions
Concept Questions
One Current Loop
A current loop lies in the xy-plane and has counterclockwise
current. The positive z-axis is coming out of the page. What is the
direction of the magnetic field along the entire z-axis?
1
Into the page
2
Out of the page
3
It switches direction
Correct Answer: 2
(Use the right hand rule and remember that magnetic field lines
cannot cross.)
Sources of Magnetic Fields
Questions
Concept Questions
Three Current Loops
Three loops with equal radii and equal current magnitude lie on
planes parallel to the yz-plane. What is the direction of the
magnetic field at the origin?
1
Positive x direction
2
Negative x direction
3
Zero magnitude (no
direction)
Sources of Magnetic Fields
Questions
Concept Questions
Three Current Loops
Three loops with equal radii and equal current magnitude lie on
planes parallel to the yz-plane. What is the direction of the
magnetic field at the origin?
1
Positive x direction
2
Negative x direction
3
Zero magnitude (no
direction)
Correct Answer: 1
(The effects of the two outer current loops cancel out, leaving the
magnetic field generated from the middle loop.)