Download - Lecture 21: FRI 06 MAR
Lecture 21: FRI 06 MAR Lecture 21: FRI 06 MAR Magnetic fields Magnetic fields
Ch.28.8-10
Physics 2102
Jonathan Dowling
28.8 Force on Current in Wire
28.9 Torque on Current Loop
28.10 Magnetic Dipole Moment
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Magnetic Force on a Wire.Magnetic Force on a Wire.
L
BLiFrrr
×= BLiFrrr
×= BLdiFdrrr
×= BLdiFdrrr
×=
. dLr
Br
dFr
i
φ
If we assume the more general case for which the
magnetic field forms an angle with the wire
the magnetic force equation can be written
B φ
Magnetic Force on a Straight Wire in a Uniform Magnetic Field
r
in vector
form as . Here is a vector whose
magnitude is equal to the wire length and
has a direction that coincides with that of the current.
The magnetic force magnitude is s
B
B
F iL B L
L
F iLB
= ×
=
r r r r
in .
In this case we divide the wire into elements of
length , which can be considered as straight.
The magnetic force
dL
φ
Magnetic Force on a Wire of Arbitrary Shape
Placed in a Nonuniform Magnetic Field
on each element is
. The net magnetic force on the
wire is given by the integral .
B
B
dF idL B
F i dL B
×
= ×∫=
r r r
r r r
BF iL B= ×r r r
BdF idL B×=r r r
BF i dL B= ×∫r r r
(28-12)
Torque on a Current Loop: Principle behind electric motors.
Net force on current loop = 0
iaBFF == 31
)sin(1 θFF =⊥
)sin(θτ iabBbFTorque === ⊥
For a coil with N turns,τ = N I A B sinθ, where A is the area of coil
Rectangular coil: A=ab, current = i
But: Net torque is NOT zero!
nNiA ˆ)(=μr n̂,μr
Magnetic Dipole MomentMagnetic Dipole Moment
N = number of turns in coilA=area of coil.
We just showed: τ = NiABsinθRight hand rule:
curl fingers in direction of current;
thumb points along μDefine: magnetic dipole moment μ
Brrr
×=μτ
As in the case of electric dipoles, magnetic dipoles tend to align with the magnetic field.
Electric vs. Magnetic DipolesElectric vs. Magnetic Dipoles
€
UE = −r p ⋅
r E
-Q
θQE
QE
+Q
p=Qa
€
rτ B =r μ ×
r B
nNiA ˆ)(=μr
€
rτ E =r p ×
r E
€
UB = −r μ ⋅
r B