Magnetic Fields, Magnetic Forces, and Sources of Magnetic Fields

W07D2

Announcements

Problem Set 6 Due Thursday at 9 pm .This problem set is an introduction to Experiment 1 and the two problems from W06D2.

Week 8 LS1 due Mon at 8:30 am

Week 8 LS2 due Mon at 8:30 am

Week 8 LS3 due Wed at 8:30 am

Week 8 LS4 due Wed at 8:30 am

Outline

Magnetic Field

Lorentz Force Law

Magnetic Force on Current Carrying Wire

Sources of Magnetic Fields

Biot-Savart Law 3 Magnetic Field of the Earth

North magnetic pole located in southern hemisphere

http://www.youtube.com/watch?v=AtDAOxaJ4Ms 4 Magnetic Force on Moving Charges Force on ! B ! ! positive charge F = qv × B q q

B B q vq F F q q B q + B Force on negative charge Magnetic force is perpendicular to both velocity of the charge and magnetic field 5 CQ: Cross Product and Magnetic Force

An electron is traveling up in a magnetic field that points to the right. What is the direction of the force on the electron? v 1. Up. q 2. Down. B 3. Left. q = e 4. Right. 5. Into plane of figure. 6. Out of plane of figure.

6 Group Problem: Cyclotron Motion

A positively charged particle

vq with charge +q and mass m is B moving with speed v in a + uniform magnetic field of + q magnitude B directed into the plane of figure will undergo circular motion. Find

(1) the radius R of the orbit, (2) the period T of the motion, (3) the angular frequency ω. (4) Sketch the motion of the

particle. 7

Lorentz Force Law

Force on charged particles in electric and magnetic fields ! ! ! B ! ! E F qv B F = qE q = q × q Electric Force Magnetic Force

Lorentz Force Law ! ! ! ! F = q(E + v × B) q q

8 CQ: Velocity Selector +

+ + + + + + + + + + + + + + + + C v B E e A q = e

+ Electrons with charge – e and mass m are emitted from the cathode C and accelerated toward slit A with different speeds in the direction shown. The electrons enter a region with a downward pointing electric field (magnitude E) and a magnetic field (magnitude B) that points into the plane of the figure. The electrons that travel on a straight trajectory through the plates have speed 1. v = B/E 2. v = (1/2)(eE/m)t2 3. v = 1/eB 4. v = E/B 9 Magnetic Force on Current-Carrying Wire Current source infinitesimal element ! d s dq dF ! ! ! mag vdq dqv dq = dq = d s = I d s dt dt ! dq B d s ext Direction of is the direction of I. Force on source element in external magnetic field ! ! ! ! ! dF = dqv × B = Id s × B mag dq ext ext I d s dF Force on a current carrying wire in an mag external magnetic field ! ! ! B F = I d s × B ext mag ∫ ext wire 10 Magnetic Force on Current-Carrying Wire If the wire is in a uniform magnetic field then ! ⎛ !⎞ ! F = I d s × B mag ⎜ ∫ ⎟ ext ⎝ wire ⎠ F I where the direction of the vector mag ! d s is the direction of the current

If the wire is also straight then L ! ! ! Bext Fmag = I(L × Bext )

where! the direction of the vector L is the direction of the current

11 Magnetic Force on Current-Carrying Wire

Current is moving charges, and we know that moving charges feel a force in a magnetic field with direction given by    F = I(L × B) mag ! where the direction of the vector L is the direction of the current 12 Group Problem: Current Loop

A conducting rod of uniform + current source mass per length λ and length l is suspended by two flexible wires in a uniform magnetic field of magnitude B which B points out of the page and a uniform gravitational field of l g magnitude g pointing down. If the tension in the wires is zero, what is the direction and magnitude of the current?

13 Moving Charges are Sources of Magnetic Fields

http://youtu.be/JmqX1GrMYnU 14 Recall: Electric Field of Point Charge An electric charge produces an electric field:

 1 q E = rˆ 4πε r 2 o

rˆ : unit vector directed from the charged object

to the field point P 15 Magnetic Field of Moving Charge  Moving charge with velocity v produces magnetic field at the point P:  rˆ  µ q v ×rˆ P B 0 . = 2 B 4π r r v ˆr: unit vector directed q 0 from charged object > to P

−7 −1 permeability of free space µ0 = 4π × 10 T ⋅ m ⋅ A 16 Current is a Source of Magnetic Field

17 Biot-Savart Law  Current element d s of length ds pointing in direction of current I produces a magnetic field at point P: rˆ ! dq ! ! dqv d s I d s dB dq = = P dt I d s dq   µ0 I ds × rˆ v dq I dB = 2 4π r

http://web.mit.edu/viz/EM/visualizations/magnetostatics/MagneticFieldConfigurations/CurrentElement3d/CurrentElement.htm 18 The Right-Hand Rule #2 rˆ  ˆ  µ I ds × rˆ dB 0 kˆ = 2 B 4π r P

I current directed out of plane of figure ! dir d s = kˆ ! dir B(P) kˆ rˆ ˆ = × = θ 19 CQ: Force between Two Parallel Wires

Consider two parallel current carrying wires with the currents running in opposite directions.

1. attracted (opposites attract?)

2. repelled (opposites repel?) I1 I2 3. pushed another direction

4. not pushed – no net force wire 1 wire 2

20 Can we understand why? Whether they attract or repel can be seen in the shape of the created B field

http://youtu.be/nQX-BM3GCv4 http://youtu.be/5nKQjKgS9z0

You tube videos of field line animations showing why showing parallel wires attract: or repel: 21 Group Ring of Radius R

Consider a ring with radius R and carrying a current I What is a vector expression for the magnetic field at point P? P z ˆ kˆ R rˆ