Magnetic Fields - physics.purdue.edu

46
~ B Field Creation Detecting ~ B fields Magnetic Fields PHYS 272 - David Blasing Wednesday July 2nd, 2014 PHYS 272 - David Blasing Matter and Interactions: 18.1 - 18.3

Transcript of Magnetic Fields - physics.purdue.edu

Page 1: Magnetic Fields - physics.purdue.edu

~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

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~B Field CreationDetecting ~B fields

Our “Road Map”

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~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

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~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)

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Page 5: Magnetic Fields - physics.purdue.edu

~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

Page 6: Magnetic Fields - physics.purdue.edu

~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

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~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

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~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

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Page 9: Magnetic Fields - physics.purdue.edu

~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

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Page 10: Magnetic Fields - physics.purdue.edu

~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

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~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

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~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

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~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

q~v × r̂

r2

Magnitude of the magnetic field:

|~B| =µ0

|q||~v × r̂ |r2

|~B| =µ0

|q||~v || sin(θ)|r2

Direction of the magnetic field depends on:

the sign of qthe direction of ~v × r̂

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~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

q~v × r̂

r2

Magnitude of the magnetic field:

|~B| =µ0

|q||~v × r̂ |r2

|~B| =µ0

|q||~v || sin(θ)|r2

Direction of the magnetic field depends on:

the sign of qthe direction of ~v × r̂

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~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

q~v × r̂

r2

Magnitude of the magnetic field:

|~B| =µ0

|q||~v × r̂ |r2

|~B| =µ0

|q||~v || sin(θ)|r2

Direction of the magnetic field depends on:

the sign of qthe direction of ~v × r̂

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~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

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~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

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~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:

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~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

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~B Field CreationDetecting ~B fields

Biot-Savart lawCreating ~B by a current

Clicker Question 1

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~B Field CreationDetecting ~B fields

Biot-Savart lawCreating ~B by a current

Biot-Savart law: single moving charge

~B =µ0

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

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~B Field CreationDetecting ~B fields

Biot-Savart lawCreating ~B by a current

Biot-Savart law: single moving charge

~B =µ0

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

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~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?

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~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?

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~B Field CreationDetecting ~B fields

Biot-Savart lawCreating ~B by a current

Biot-Savart law: single moving charge

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~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.

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~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.

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~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

q~v × r̂

r2

~B =µ0

qvsin(θ)

r2v̂ × r̂

|~B| =µ0

|q||~v × r̂ |r2

|~B| =µ0

|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...

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~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

q~v × r̂

r2

~B =µ0

qvsin(θ)

r2v̂ × r̂

|~B| =µ0

|q||~v × r̂ |r2

|~B| =µ0

|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...

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~B Field CreationDetecting ~B fields

Biot-Savart lawCreating ~B by a current

Clicker Question 2

Biot-Savart law: ~B = µ04π

q~v×r̂r2

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~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

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Page 32: Magnetic Fields - physics.purdue.edu

~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

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Page 33: Magnetic Fields - physics.purdue.edu

~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

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~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

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~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

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~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

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~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:

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~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

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~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

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~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 )

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~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!

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~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!

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~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!

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Page 44: Magnetic Fields - physics.purdue.edu

~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!

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~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!

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~B Field CreationDetecting ~B fields

~B from Currents~B Fields and Compasses

Detecting ~B fields with a Compass Example

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