while the electric force acting on Qis
Fe=QE (coul )
(N
coul
)
(9 .146)The total force acting on the moving charge Qis
F=Fm+Fe=Q(E+v ×B) (9 .147)The magnetic forces can be summed over lines, surfaces or volumes. This gives
rise to a one-dimensional, two-dimensional and three-dimensional representation for
the magnetic force. In one-dimension the element of force acting on an element of
wire is dFm= (v ×B)ρds where ρis the charge density per unit length (coul/m)
and ds is an element of arc length. In two-dimensions the element of force acting on
a surface is dFm= (v ×B)ρSdσ where ρSis the charge density per unit area (coul/m^2 )
and dσ is an element of area. In three-dimensions the element of force acting within
a volume is dFm = (v ×B)ρVdV where ρV is the charge density per unit volume
(coul/m^3 )and dV is an element of volume. An integration gives the total magnetic
force as
Fm=∫(v ×B)ρds one-dimension
Fm=∫∫(v ×B)ρSdσ two-dimension
Fm=∫∫∫(v ×B)ρVdV three-dimension
(9 .148)Example 9-11. The Maxwell-Faraday Equation
Faraday’s law^10 of induction investigates the magnetic
flux
∫∫SB ·dS across a surface^11 S determined by a sim-
ple closed curve C. Think of a simple closed curve in space
drawn on a sheet of rubber and then hold the simple closed
curve fixed but deform the rubber surface into any kind of
continuous surface Shaving Cfor its boundary. The direc-
tion of the unit normal ˆen to the surface Sis determined
by the right-hand rule of moving the fingers of the right
(^10) Michael Faraday (1791-1867) English physicist who studied electricity and magnetism.
(^11) Think of a rubber sheet across Cand then deform the sheet to form the surface S.