B Field Creation Detecting B fields. Magnetic Fields. PHYS David Blasing. Wednesday June 26th 1 / 26

Transcription

1 Magnetic Fields PHYS David Blasing Wednesday June 26th 1 / 26

2 Magnetic ( B) Fields This is a significant change, until now we have discussed just E fields. Now we are talking about a totally different field. 2 / 26

3 Magnetic ( B) Fields This is a significant change, until now we have discussed just E fields. Now we are talking about a totally different field. Everything that we have learned is still valid. Now we considering an additional field that creates additional forces. 2 / 26

4 Magnetic ( B) Fields This is a significant change, until now we have discussed just E fields. Now we are talking about a totally different field. Everything that we have learned is still valid. Now we considering an 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) 2 / 26

5 : single moving charge : single moving charge B = µ 0 q v ˆr 4π r 2 T kilogram second coulomb Units of Tesla (T) are The is how a single moving point charge makes a B field. 3 / 26

6 : single moving charge : single moving charge B = µ 0 q v ˆr 4π r 2 T kilogram second coulomb Units of Tesla (T) are The is how a single moving point charge makes a B field. This is the magnetic analog of 1 4πɛ 0 q r 2 ˆr for a point charge creating an E field 3 / 26

7 : single moving charge : single moving charge B = µ 0 q v ˆr 4π r 2 T kilogram second coulomb Units of Tesla (T) are The is how a single moving point charge makes a B field. 1 q This is the magnetic analog of 4πɛ 0 ˆr for a point charge r 2 creating an E field The q v (units - coulomb m s ) is the charge in motion bit of the. It can be thought of as the analog of q, the thing that created E fields 3 / 26

8 : single moving charge : single moving charge B = µ 0 q v ˆr 4π r 2 T kilogram second coulomb Units of Tesla (T) are The is how a single moving point charge makes a B field. 1 q This is the magnetic analog of 4πɛ 0 ˆr for a point charge r 2 creating an E field The q v (units - coulomb m s ) is the charge in motion bit of the. It can be thought of as the analog of q, the thing that created E fields v is the velocity of the point charge q in your reference frame (more on this later) r is the same relative position vector as before - points from source location to the observation location 3 / 26

9 Permeability of Free Space µ 0 4π Definition: Permeability of Free Space, µ 0 is just a positive constant. µ 0 4π 10 7 tesla m2 coulomb m/s 4 / 26

10 Permeability of Free Space µ 0 4π Definition: Permeability of Free Space, µ 0 is just a positive constant. µ 0 4π 10 7 tesla m2 coulomb m/s 1 Recall: 4πɛ but µ 0 4π 10 7 so magnetic forces are usually smaller than electric forces. B earth T 4 / 26

11 : single moving charge : single moving charge B = µ 0 q v ˆr 4π r 2 Magnitude of the magnetic field: 5 / 26

12 : single moving charge : single moving charge B = µ 0 q v ˆr 4π r 2 Magnitude of the magnetic field: B = µ 0 q v ˆr 4π r 2 B = µ 0 4π q v sin(θ) r 2 Direction of the magnetic field depends on: 5 / 26

13 : single moving charge : single moving charge B = µ 0 q v ˆr 4π r 2 Magnitude of the magnetic field: B = µ 0 q v ˆr 4π r 2 B = µ 0 4π q v sin(θ) r 2 Direction of the magnetic field depends on: the sign of q the direction of v ˆr 5 / 26

14 Vector Cross Product of Any 2 Vectors in General Definition: Vector Cross Product A B = (A y B z A z B y, A z B x A x B z, A x B y A y B x ) 6 / 26

15 Vector Cross Product of Any 2 Vectors in General Definition: Vector Cross Product A B = (A y B z A z B y, A z B x A x B z, A x B y A y B x ) The resultant vector is perpendicular to both A and B and has magnitude A B sin(θ) where θ is the smallest angle between A and B. 6 / 26

16 Vector Cross Product of Any 2 Vectors in General Alternatively, you can get it through a matrix definition: 7 / 26

17 Right Hand Rule The direction can be found through a right hand rule: / 26

18 Right Hand Rule The direction can be found through a right hand rule: 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 8 / 26

19 Clicker Question 1 9 / 26

20 : single moving charge B = µ 0 q v ˆr 4π r 2 To find the direction of the B field created by a moving point charge: 1 First use the right hand rule to find the direction of v ˆr 10 / 26

21 : single moving charge B = µ 0 q v ˆr 4π r 2 To find the direction of the B field created by a moving point charge: 1 First use the right hand rule to find the direction of v ˆr 2 B is parallel to v ˆr if the moving charge is positive 3 B is anti-parallel to v ˆr if the moving charge is negative 10 / 26

22 : single moving charge Exercise, use the right hand rule to work out the cross product and verify that the direction of B is correct 11 / 26

23 : single moving charge giving B at multiple locations: Question, is this a positive or negative charge? 12 / 26

24 : single moving charge 13 / 26

25 B on the Surface of a Sphere B Field Creation B = µ 0 q v ˆr 4π r 2 T Suppose you fix the distance, R, away from a moving point charge. Is B field constant everywhere on the sphere that you just created (centered on the point charge, radius R)? 14 / 26

26 B on the Surface of a Sphere B Field Creation B = µ 0 q v ˆr 4π r 2 T Suppose you fix the distance, R, away from a moving point charge. Is B field constant everywhere on the sphere that you just created (centered on the point charge, radius R)? This is different than E fields, whose magnitude is constant when you fix the distance. Even at a fixed distance, B fields depend on the relative angle (θ) between v and r. So it is not constant even at a fixed distance. 14 / 26

27 : single moving charge Let s say that you have a moving point charge, recall: B = µ 0 q v ˆr 4π r 2 B = µ 0 4π B = µ 0 4π q v ˆr r 2 q v sin(θ) r 2 15 / 26

28 : single moving charge Let s say that you have a moving point charge, recall: Another question: B = µ 0 q v ˆr 4π r 2 B = µ 0 4π B = µ 0 4π q v ˆr r 2 q v sin(θ) r 2 You have a positive charge moving with some velocity in the +ˆx direction. Where is B maximized and where is it minimum? Hint, one is a line and one is a plane / 26

29 Clicker Question 2 : B = µ 0 q v ˆr 4π r 2 16 / 26

30 Electron Current 1 Lets the many electrons are moving and making some current i 17 / 26

31 Electron Current 1 Lets the many electrons are moving and making some current i 2 Each of the electrons make a B according to the Biot-Savart law 3 Superposition holds for B fields just like it did for E fields = B net = charges µ 0 q v ˆr 4π r 2 Each charge has its own q, v, and r 17 / 26

32 Electron Current 1 Lets the many electrons are moving and making some current i 2 Each of the electrons make a B according to the Biot-Savart law 3 Superposition holds for B fields just like it did for E fields = B net = charges µ 0 q v ˆr 4π r 2 Each charge has its own q, v, and r 4 Sum might switch to an integral if you consider the moving charges instead as a continuous current distribution 17 / 26

33 Historical Note Oersted effect: discovered in 1820 by H. Ch. Ørsted Conclusions: The magnitude of B depends on the amount of current A wire with no current produces no B B is perpendicular to the direction of current B under the wire is opposite to B over the wire 18 / 26

34 Electron Current Definition: Electron Current i The number of electrons per second that enter a certain section of a conductor (i.e. the rate of electrons passing through a cross section of the conductor) 19 / 26

35 Electron Current Definition: Electron Current i The number of electrons per second that enter a certain section of a conductor (i.e. the rate of electrons passing through a cross section of the conductor) Electron current is the number of electrons per second passing the dashed line (the cross-section ), so its units are just per second 19 / 26

36 Electron Current s B Field Negative charges flowing to the left make a B field like: 20 / 26

37 B from Currents B Fields and Compasses Important: A compass needle points in the direction of the B net at its location, it does not respond to the local E field 21 / 26

38 B from Currents B Fields and Compasses Dipoles Interaction with B Notes: The Dipole Interaction Energy with B E = µ B The magnet in a compass has a dipole moment described by µ This interaction between µ and B net is what aligns a compass to B net Energy is lowest when µ is aligned with B Energy is highest when µ is anti-aligned with B 22 / 26

39 B from Currents B Fields and Compasses Positive charges are moving up with some velocity. Use B = µ 0 4π to justify the direction of B wire q v ˆr r 2 23 / 26

40 B from Currents B Fields and Compasses A compass needle points in the direction of B net at its location B net = B Earth + B wire 24 / 26

41 B from Currents B Fields and Compasses Questions: 1 How can you calculate the angle θ if you know B wire and B Earth? 2 What would we observe if B wire were much larger or much smaller than B Earth? 3 If you double B wire does θ double? 25 / 26

42 B from Currents B Fields and Compasses with a Compass Example 26 / 26