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