082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

250 Chapter 7 Vectors
F
q
B
v
Figure7.31
Force,F,exerted on aparticle with charge,q,when moving with
velocity,v,in a magnetic field,B.
where θ is the angle between v and B, v is the modulus of v andBis the modulus
of B.
Note thatif θ = 90 ◦ , sin θ = 1,thenB = F qv .
Theseformulaeareusefulbecausetheycanbeusedtocalculatetheforcesexerted
on a conductor in an electric motor. They are also used to analyse electricity generators
in which the motion of a conductor in a magnetic field leads to movement of
charges within the conductor, thus generating electricity.
Arelatedquantityisthemagneticfieldstrength,orthemagneticfieldintensity,
H, defined from
B=µH
µ is called the permeability of a material and has units of webers per ampere per
metre (Wb A −1 m −1 ). The units of H are amperes per metre (A m −1 ). Confirm for
yourself thatthe units match on both sides of the equation.
Engineeringapplication7.10
Magneticfieldduetoamovingcharge
A chargeqmoving with velocity v gives rise to a magnetic field with magnetic flux
densityBinitsvicinity.Asaresultofthis,anothermovingchargeplacedinthisfield
will experienceamagnetic force.Themagnetic fluxdensityis given by
B = qµ 0
4πr 2 (v׈r)
where r is a vector from the charge to the point at which B is measured, and µ 0
is a
constant called the permeability of free space.
Thisequationcanbeusedtofindthemagneticfieldduetoacurrentinawire.Supposeasmallportionofwirehaslength
δsandcontainsacurrentI.Bywriting δsasa
vectoroflength δsinthedirectionofthewire,itcanbeshownthatthecorresponding
contribution tothe magnetic fluxdensityis given by
δB = µ 0 I (δs × ˆr)
4πr2 ThisistheBiot--Savartlaw.Techniquesofintegrationarerequiredinordertocompletethecalculation,
butusingthisitispossibletoshow,forexample,thatthemagnetic
flux density a distance r from a long straight wire has magnitude B = µ 0 I
2πr .