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~B Field CreationDetecting ~B fields
Magnetic Fields
PHYS 272 - David Blasing
Wednesday July 2nd, 2014
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
~B Field CreationDetecting ~B fields
Our “Road Map”
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
2/27
~B Field CreationDetecting ~B fields
Brief Review of Last Lecture
~Enet in the presence of a dielectric constant
~Ew/dielectric =~Ew/o dielectric
K
Capacitance of a Capacitor
Ccap = ε0As
Energy Density of ~E Fields
Electric fields locally carry energy per unit volume of 12ε0|~E |2
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
3/27
~B Field CreationDetecting ~B fields
Biot-Savart lawCreating ~B by a current
Magnetic, or “~B” Fields
This is a significant change, until now we have discussed just ~Efields. Now we are talking about a totally different field.
Everything that we have learned is still valid. Now we consideringan additional field that creates additional forces.
Magnetic fields (≡ ~B fields) are made by and exert forces on:
1 Moving charges (ex. can cause centripetal motion)
2 Permanent magnets (ex. can align compasses)
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
4/27
~B Field CreationDetecting ~B fields
Biot-Savart lawCreating ~B by a current
Magnetic, or “~B” Fields
This is a significant change, until now we have discussed just ~Efields. Now we are talking about a totally different field.
Everything that we have learned is still valid. Now we consideringan additional field that creates additional forces.
Magnetic fields (≡ ~B fields) are made by and exert forces on:
1 Moving charges (ex. can cause centripetal motion)
2 Permanent magnets (ex. can align compasses)
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
4/27
~B Field CreationDetecting ~B fields
Biot-Savart lawCreating ~B by a current
Magnetic, or “~B” Fields
This is a significant change, until now we have discussed just ~Efields. Now we are talking about a totally different field.
Everything that we have learned is still valid. Now we consideringan additional field that creates additional forces.
Magnetic fields (≡ ~B fields) are made by and exert forces on:
1 Moving charges (ex. can cause centripetal motion)
2 Permanent magnets (ex. can align compasses)
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
4/27
~B Field CreationDetecting ~B fields
Biot-Savart lawCreating ~B by a current
Biot-Savart law: single moving charge
Biot-Savart law: single moving charge
~B = µ04π
q~v×r̂r2 T
Units of Tesla (T) are kilogramsecond∗coulomb
This is how a single moving point charge makes a ~B field.
This is the magnetic analog of 14πε0
qr2 r̂ for a stationary
“point” charge creating an ~E field
q~v , (c ms ) is the “charge in motion” bit of the Biot-Savart law.
It is the magnetic analog of q, which created ~E fields
~v is the velocity of the point charge q in your reference frame
~r is still the position of a fixed observation location relative tothe qv source’s current location
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
5/27
~B Field CreationDetecting ~B fields
Biot-Savart lawCreating ~B by a current
Biot-Savart law: single moving charge
Biot-Savart law: single moving charge
~B = µ04π
q~v×r̂r2 T
Units of Tesla (T) are kilogramsecond∗coulomb
This is how a single moving point charge makes a ~B field.
This is the magnetic analog of 14πε0
qr2 r̂ for a stationary
“point” charge creating an ~E field
q~v , (c ms ) is the “charge in motion” bit of the Biot-Savart law.
It is the magnetic analog of q, which created ~E fields
~v is the velocity of the point charge q in your reference frame
~r is still the position of a fixed observation location relative tothe qv source’s current location
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
5/27
~B Field CreationDetecting ~B fields
Biot-Savart lawCreating ~B by a current
Biot-Savart law: single moving charge
Biot-Savart law: single moving charge
~B = µ04π
q~v×r̂r2 T
Units of Tesla (T) are kilogramsecond∗coulomb
This is how a single moving point charge makes a ~B field.
This is the magnetic analog of 14πε0
qr2 r̂ for a stationary
“point” charge creating an ~E field
q~v , (c ms ) is the “charge in motion” bit of the Biot-Savart law.
It is the magnetic analog of q, which created ~E fields
~v is the velocity of the point charge q in your reference frame
~r is still the position of a fixed observation location relative tothe qv source’s current location
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
5/27
~B Field CreationDetecting ~B fields
Biot-Savart lawCreating ~B by a current
Biot-Savart law: single moving charge
Biot-Savart law: single moving charge
~B = µ04π
q~v×r̂r2 T
Units of Tesla (T) are kilogramsecond∗coulomb
This is how a single moving point charge makes a ~B field.
This is the magnetic analog of 14πε0
qr2 r̂ for a stationary
“point” charge creating an ~E field
q~v , (c ms ) is the “charge in motion” bit of the Biot-Savart law.
It is the magnetic analog of q, which created ~E fields
~v is the velocity of the point charge q in your reference frame
~r is still the position of a fixed observation location relative tothe qv source’s current location
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
5/27
~B Field CreationDetecting ~B fields
Biot-Savart lawCreating ~B by a current
Permeability of Free Space
Definition: Permeability of Free Space, µ0
µ04π ≡ 10−7 tesla∗m2
coulomb∗m/s
µ04π is just a positive constant.
Recall: 14πε0≈ 109 but µ0
4π ≡ 10−7 so magnetic forces are usuallysmaller than electric forces.
|~Bearth| ≈ 2 ∗ 10−5 T
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
6/27
~B Field CreationDetecting ~B fields
Biot-Savart lawCreating ~B by a current
Permeability of Free Space
Definition: Permeability of Free Space, µ0
µ04π ≡ 10−7 tesla∗m2
coulomb∗m/s
µ04π is just a positive constant.
Recall: 14πε0≈ 109 but µ0
4π ≡ 10−7 so magnetic forces are usuallysmaller than electric forces.
|~Bearth| ≈ 2 ∗ 10−5 T
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
6/27
~B Field CreationDetecting ~B fields
Biot-Savart lawCreating ~B by a current
Biot-Savart law: single moving charge
Biot-Savart law: single moving charge
~B =µ0
4π
q~v × r̂
r2
Magnitude of the magnetic field:
|~B| =µ0
4π
|q||~v × r̂ |r2
|~B| =µ0
4π
|q||~v || sin(θ)|r2
Direction of the magnetic field depends on:
the sign of qthe direction of ~v × r̂
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
7/27
~B Field CreationDetecting ~B fields
Biot-Savart lawCreating ~B by a current
Biot-Savart law: single moving charge
Biot-Savart law: single moving charge
~B =µ0
4π
q~v × r̂
r2
Magnitude of the magnetic field:
|~B| =µ0
4π
|q||~v × r̂ |r2
|~B| =µ0
4π
|q||~v || sin(θ)|r2
Direction of the magnetic field depends on:
the sign of qthe direction of ~v × r̂
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
7/27
~B Field CreationDetecting ~B fields
Biot-Savart lawCreating ~B by a current
Biot-Savart law: single moving charge
Biot-Savart law: single moving charge
~B =µ0
4π
q~v × r̂
r2
Magnitude of the magnetic field:
|~B| =µ0
4π
|q||~v × r̂ |r2
|~B| =µ0
4π
|q||~v || sin(θ)|r2
Direction of the magnetic field depends on:
the sign of qthe direction of ~v × r̂
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
7/27
~B Field CreationDetecting ~B fields
Biot-Savart lawCreating ~B by a current
General Vector Cross Product of Any 2 Vectors
Definition: Vector Cross Product
~A× ~B = (AyBz − AzBy ,AzBx − AxBz ,AxBy − AyBx)
~A× ~B is perpendicular to both ~A and ~B and has magnitude|~A||~B||sin(θ)| where θ is the smallest angle between ~A and ~B
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
8/27
~B Field CreationDetecting ~B fields
Biot-Savart lawCreating ~B by a current
General Vector Cross Product of Any 2 Vectors
Definition: Vector Cross Product
~A× ~B = (AyBz − AzBy ,AzBx − AxBz ,AxBy − AyBx)
~A× ~B is perpendicular to both ~A and ~B and has magnitude|~A||~B||sin(θ)| where θ is the smallest angle between ~A and ~B
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
8/27
~B Field CreationDetecting ~B fields
Biot-Savart lawCreating ~B by a current
General Vector Cross Product of Any 2 Vectors
Alternatively, you can get it through a matrix definition:
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
9/27
~B Field CreationDetecting ~B fields
Biot-Savart lawCreating ~B by a current
Right Hand Rule
The direction of ~A× ~B can be gotten through a right hand rule:
1. 2.
Steps:
1 Point all four fingers in the direction of ~A
2 Flip hand towards the direction of ~B
3 Thumb points in the direction of ~A× ~B
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
10/27
~B Field CreationDetecting ~B fields
Biot-Savart lawCreating ~B by a current
Clicker Question 1
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
11/27
~B Field CreationDetecting ~B fields
Biot-Savart lawCreating ~B by a current
Biot-Savart law: single moving charge
~B =µ0
4π
q~v × r̂
r2
To find the direction of ~B created by a moving point charge:
1 Use the right hand rule to find the direction of ~v × r̂
2 +q =⇒ ~B is parallel to ~v × r̂
3 -q =⇒ ~B is anti-parallel to ~v × r̂
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
12/27
~B Field CreationDetecting ~B fields
Biot-Savart lawCreating ~B by a current
Biot-Savart law: single moving charge
~B =µ0
4π
q~v × r̂
r2
To find the direction of ~B created by a moving point charge:
1 Use the right hand rule to find the direction of ~v × r̂
2 +q =⇒ ~B is parallel to ~v × r̂
3 -q =⇒ ~B is anti-parallel to ~v × r̂
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
12/27
~B Field CreationDetecting ~B fields
Biot-Savart lawCreating ~B by a current
Biot-Savart law: single moving charge
Exercise: is the direction of ~B of correct?
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
13/27
~B Field CreationDetecting ~B fields
Biot-Savart lawCreating ~B by a current
Biot-Savart law: single moving charge
Biot-Savart law giving ~B at multiple locations:
Question: is this charge positive or negative?
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
14/27
~B Field CreationDetecting ~B fields
Biot-Savart lawCreating ~B by a current
Biot-Savart law: single moving charge
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
15/27
~B Field CreationDetecting ~B fields
Biot-Savart lawCreating ~B by a current
|~B | on the Surface of a Sphere
~B = µ04π
q~v×r̂r2 T
Consider every point a distance R away from a moving pointcharge. Is |~B| constant everywhere on the surface of that sphere(of radius R centered on the point charge)?...It is for the ~E fieldfrom a stationary point charge.
Even at a fixed distance, ~B fields depend on the relative anglebetween ~v and ~r by their cross product. So it is not constant asthat angle varies at different points on the surface of the sphere.
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
16/27
~B Field CreationDetecting ~B fields
Biot-Savart lawCreating ~B by a current
|~B | on the Surface of a Sphere
~B = µ04π
q~v×r̂r2 T
Consider every point a distance R away from a moving pointcharge. Is |~B| constant everywhere on the surface of that sphere(of radius R centered on the point charge)?...It is for the ~E fieldfrom a stationary point charge.
Even at a fixed distance, ~B fields depend on the relative anglebetween ~v and ~r by their cross product. So it is not constant asthat angle varies at different points on the surface of the sphere.
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
16/27
~B Field CreationDetecting ~B fields
Biot-Savart lawCreating ~B by a current
Biot-Savart law: single moving charge
Let’s say that you have a moving point charge, recall:
~B =µ0
4π
q~v × r̂
r2
~B =µ0
4π
qvsin(θ)
r2v̂ × r̂
|~B| =µ0
4π
|q||~v × r̂ |r2
|~B| =µ0
4π
|q||~v || sin(θ)|r2
Another question:
You have a positive charge moving with some velocity in the +x̂direction. Where is |~B| maximized and where is it minimum? Hint,one is a line and one is a plane...
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
17/27
~B Field CreationDetecting ~B fields
Biot-Savart lawCreating ~B by a current
Biot-Savart law: single moving charge
Let’s say that you have a moving point charge, recall:
~B =µ0
4π
q~v × r̂
r2
~B =µ0
4π
qvsin(θ)
r2v̂ × r̂
|~B| =µ0
4π
|q||~v × r̂ |r2
|~B| =µ0
4π
|q||~v || sin(θ)|r2
Another question:
You have a positive charge moving with some velocity in the +x̂direction. Where is |~B| maximized and where is it minimum? Hint,one is a line and one is a plane...
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
17/27
~B Field CreationDetecting ~B fields
Biot-Savart lawCreating ~B by a current
Clicker Question 2
Biot-Savart law: ~B = µ04π
q~v×r̂r2
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
18/27
~B Field CreationDetecting ~B fields
Biot-Savart lawCreating ~B by a current
Electron Current
1 Let many electrons move and make some current i
2 Each electron makes ~B according to the Biot-Savart law
3 Superposition holds for ~B fields just like it did for ~E fields
=⇒ ~Bnet =∑
charges
µ04π
q~v×r̂r2
Each charge has its own q, ~v , and ~r
4 Might be ~Bnet =∫
charges
µ04π
~dI×r̂r2 if the moving charges are well
approximated as a continuous current distribution. ~dI is aninfinetesimal chunk of current
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
19/27
~B Field CreationDetecting ~B fields
Biot-Savart lawCreating ~B by a current
Electron Current
1 Let many electrons move and make some current i
2 Each electron makes ~B according to the Biot-Savart law
3 Superposition holds for ~B fields just like it did for ~E fields
=⇒ ~Bnet =∑
charges
µ04π
q~v×r̂r2
Each charge has its own q, ~v , and ~r
4 Might be ~Bnet =∫
charges
µ04π
~dI×r̂r2 if the moving charges are well
approximated as a continuous current distribution. ~dI is aninfinetesimal chunk of current
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
19/27
~B Field CreationDetecting ~B fields
Biot-Savart lawCreating ~B by a current
Electron Current
1 Let many electrons move and make some current i
2 Each electron makes ~B according to the Biot-Savart law
3 Superposition holds for ~B fields just like it did for ~E fields
=⇒ ~Bnet =∑
charges
µ04π
q~v×r̂r2
Each charge has its own q, ~v , and ~r
4 Might be ~Bnet =∫
charges
µ04π
~dI×r̂r2 if the moving charges are well
approximated as a continuous current distribution. ~dI is aninfinetesimal chunk of current
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
19/27
~B Field CreationDetecting ~B fields
Biot-Savart lawCreating ~B by a current
Historical Note
Ørsted effect: discovered in 1820 byH. Ch. Ørsted
Conclusions about ~B field from a current in a wire:
|~B| is proportional to the amount of currentA wire with no current produces no ~B~B is perpendicular to the direction of currentThe direction of ~B underneath the wire is opposite to thatover it
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
20/27
~B Field CreationDetecting ~B fields
Biot-Savart lawCreating ~B by a current
Electron Current
Definition: Electron Current i
The number of electrons per second that traverse through a crosssection of a conductor
Electron current is the number of electrons per second passing thedashed line (the“cross-section”), so its units are just per second
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
21/27
~B Field CreationDetecting ~B fields
Biot-Savart lawCreating ~B by a current
Electron Current
Definition: Electron Current i
The number of electrons per second that traverse through a crosssection of a conductor
Electron current is the number of electrons per second passing thedashed line (the“cross-section”), so its units are just per second
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
21/27
~B Field CreationDetecting ~B fields
Biot-Savart lawCreating ~B by a current
Electron Current’s ~B Field
Negative (positive) charges flowing to the left (right) make a ~Bfield like:
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
22/27
~B Field CreationDetecting ~B fields
~B from Currents~B Fields and Compasses
Detecting ~B fields
Important: A compass needle points in the direction of the ~Bnet atits location, it does not respond to the local ~E field
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
23/27
~B Field CreationDetecting ~B fields
~B from Currents~B Fields and Compasses
Dipoles Interaction with ~B
The Dipole Interaction Energy with ~B
E = −~µ · ~B
Notes:
The magnet in a compass has a dipole moment described by ~µ
This interaction between ~µ and ~Bnet aligns a compass to ~Bnet
Energy is lowest when ~µ is aligned with ~B
Energy is highest when ~µ is anti-aligned with ~B
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
24/27
~B Field CreationDetecting ~B fields
~B from Currents~B Fields and Compasses
Detecting ~B fields
Say positive charges are moving North in the wire:
Is the direction of ~Bwire correct? (Use ~B = µ04π
q~v×r̂r2 )
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
25/27
~B Field CreationDetecting ~B fields
~B from Currents~B Fields and Compasses
Detecting ~B fields
Notes:
Compass needles point in the direction of~Bnet at their location
~Bnet = ~BEarth + ~Bwire
tan(θ) = |~Bwire ||~Bearth|
If |~Bwire | >> |~BEarth|, then θ ≈ π2 . If
|~Bwire | << |~BEarth|, θ ≈ 0
Doubling |~Bwire | does not double θ...tangentis not linear - it needs a (thankfullycountable) infininite number of powers of θin its definition!
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
26/27
~B Field CreationDetecting ~B fields
~B from Currents~B Fields and Compasses
Detecting ~B fields
Notes:
Compass needles point in the direction of~Bnet at their location~Bnet = ~BEarth + ~Bwire
tan(θ) = |~Bwire ||~Bearth|
If |~Bwire | >> |~BEarth|, then θ ≈ π2 . If
|~Bwire | << |~BEarth|, θ ≈ 0
Doubling |~Bwire | does not double θ...tangentis not linear - it needs a (thankfullycountable) infininite number of powers of θin its definition!
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
26/27
~B Field CreationDetecting ~B fields
~B from Currents~B Fields and Compasses
Detecting ~B fields
Notes:
Compass needles point in the direction of~Bnet at their location~Bnet = ~BEarth + ~Bwire
tan(θ) = |~Bwire ||~Bearth|
If |~Bwire | >> |~BEarth|, then θ ≈ π2 . If
|~Bwire | << |~BEarth|, θ ≈ 0
Doubling |~Bwire | does not double θ...tangentis not linear - it needs a (thankfullycountable) infininite number of powers of θin its definition!
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
26/27
~B Field CreationDetecting ~B fields
~B from Currents~B Fields and Compasses
Detecting ~B fields
Notes:
Compass needles point in the direction of~Bnet at their location~Bnet = ~BEarth + ~Bwire
tan(θ) = |~Bwire ||~Bearth|
If |~Bwire | >> |~BEarth|, then θ ≈ π2 . If
|~Bwire | << |~BEarth|, θ ≈ 0
Doubling |~Bwire | does not double θ...tangentis not linear - it needs a (thankfullycountable) infininite number of powers of θin its definition!
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
26/27
~B Field CreationDetecting ~B fields
~B from Currents~B Fields and Compasses
Detecting ~B fields
Notes:
Compass needles point in the direction of~Bnet at their location~Bnet = ~BEarth + ~Bwire
tan(θ) = |~Bwire ||~Bearth|
If |~Bwire | >> |~BEarth|, then θ ≈ π2 . If
|~Bwire | << |~BEarth|, θ ≈ 0
Doubling |~Bwire | does not double θ...tangentis not linear - it needs a (thankfullycountable) infininite number of powers of θin its definition!
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
26/27
~B Field CreationDetecting ~B fields
~B from Currents~B Fields and Compasses
Detecting ~B fields with a Compass Example
PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3
27/27