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Miss Millie Millie Magnetism Electric Currents Electric currents produce magnetic fields. Magnets can exert a force on a current carrying wire. The direction of the force is always perpendicular to the direction of the current and also perpendicular to the direction of the magnetic field ! . = × ! = (magnitude) = × ! ! → () = !!" ! ! Force on an Electric Charge Moving in a Magnetic Field = × ! If ! ⊥ ! ! then !"# = = 0 if is parallel to Particle in a magnetic field with some velocity will move with centripetal force ; = !! ! ! = The time required for a particle of charge moving with constant speed to make on circular revolution in a uniform magnetic field ! (⊥ ) is = !!" ! where 2 is the circumference of its circular path. A Helical Path The velocity vector must be broken into its horizontal and vertical components. Lorentz Equation = ( ! + × ! ) Magnetic Fields 1. Do NO WORK = → = ∙ = ∙ = ! × ! ! ∙ () ∴ = 0 2. Do not change the kinetic energy of a moving charge ! 2 = 2 ! ! ! + ! ! + ! ! ! = 2 !2 ! ! + 2 ! + 2 ! ! ! = ∙ = = ∙ ! × ! ! The Hall Effect In the Hall effect, moving charges ( ! ) in a conductor placed in a magnetic field are forces to one side, producing an between the two sides of the conductor. ℰ ! = ! = ! ℰ ! = ! = = ℎ = ℎ = 1.60 ×10 !!" ! = Torque + Magnetic Dipole Moment Torque on a current loop in a magnetic field ! is = ! ! ! × ! Magnetic dipole moment of the loop is = = = = ℎ

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Miss Millie Millie

Magnetism �Electric Currents�

Electriccurrentsproducemagneticfields.Magnetscanexertaforceonacurrentcarryingwire.

Thedirectionoftheforceisalwaysperpendiculartothedirectionofthecurrentandalsoperpendiculartothedirectionofthemagneticfield𝐵!⃗ .

𝐹 = 𝐼𝑙 × 𝐵!⃗ 𝐹 = 𝐼𝑙𝐵 𝑠𝑖𝑛 𝜃 (magnitude)

𝑑�⃗� = 𝐼 𝑑𝑙 × 𝐵!⃗ 𝐵!⃗ → 𝑡𝑒𝑠𝑙𝑎(𝑇) = !!"

�Force on an Electric Charge Moving in a Magnetic Field�

�⃗� = 𝑞�⃗� × 𝐵!⃗

If!�⃗� ⊥ 𝐵!⃗ !then𝐹!"# = 𝑞𝑣𝐵

𝐹 = 0if𝑣isparallelto𝐵

Particleinamagneticfieldwithsomevelocitywillmovewith

centripetalforce;𝐹 = !!!

!= 𝑞𝑣𝐵

Thetime𝑇requiredforaparticleofcharge𝑞movingwithconstantspeed𝑣 tomakeoncircularrevolutioninauniformmagneticfield𝐵!⃗ (⊥ 𝑣) is𝑇 = !!"

!where2𝜋𝑟isthe

circumferenceofitscircularpath.

AHelicalPathThevelocityvectormustbebrokenintoitshorizontalandverticalcomponents.LorentzEquation

�⃗� = 𝑞(𝐸!⃗ + 𝑣 × 𝐵!⃗ )

Magnetic Fields 1. DoNOWORK

𝑑𝑙 = 𝑣𝑑𝑡 → 𝑠𝑚𝑎𝑙𝑙 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑚𝑒𝑛𝑡𝑊 = �⃗� ∙ 𝑑𝑙 = �⃗� ∙ �⃗� 𝑑𝑡 = 𝑞!𝑣×𝐵!⃗ ! ∙ �⃗�𝑑𝑡

(𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟) ∴ 𝑊 = 0

2. Donotchangethekineticenergyofamovingcharge𝑑𝑑𝑡𝑚𝑣!

2 =𝑑𝑑𝑡𝑚2 !𝑣!! + 𝑣!! + 𝑣!!!

=𝑚2 !2𝑣!

𝑑𝑣!𝑑𝑡 + 2𝑣! + 2𝑣!

𝑑𝑣!𝑑𝑡 !

= 𝑚𝑣 ∙𝑑�⃗�𝑑𝑡 = 𝐹 = 𝑞𝑣 ∙ !𝑣×𝐵!⃗ !

�The Hall Effect� IntheHalleffect,movingcharges(𝑒!)inaconductorplacedinamagneticfieldareforcestooneside,producingan𝑒𝑚𝑓betweenthetwosidesoftheconductor.

ℰ! = 𝐸!𝑑 = 𝑣!𝐵𝑑ℰ! = 𝐻𝑎𝑙𝑙 𝑒𝑚𝑓𝐸! = 𝐸𝑙𝑒𝑐𝑡𝑟𝑖𝑐 𝐹𝑖𝑒𝑙𝑑𝑑 = 𝑤𝑖𝑑𝑡ℎ 𝑜𝑓 𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑜𝑟𝑒 = 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛 𝑐ℎ𝑎𝑟𝑔𝑒 = 1.60 ×10!!"𝐶 𝑣! = 𝑑𝑟𝑖𝑓𝑡 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦

Torque + Magnetic Dipole Moment

Torqueonacurrentloopinamagneticfield𝐵!⃗ is

𝜏 = 𝜇 !!!⃗ × 𝐵!⃗ Magneticdipolemomentoftheloopis

𝜇 = 𝑁𝐼𝐴𝑁 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑙𝑜𝑜𝑝𝑠𝐼 = 𝑐𝑢𝑟𝑟𝑒𝑛𝑡𝐴 = 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑜𝑖𝑙

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