Download electromagnetism

Electric Motor
z
What happens if we put
a loop of wire carrying a
current in a B field ?
z
FB on opposite sides of
the loop produce a
torque on the loop
causing it to rotate.
Electric motor – a commutator reverses the
direction of the current every half turn to that
the torque is always in the same direction.
Electric Motor
z
z
z
Define normal n to plane
using right-hand rule
Torque tends to rotate loop
to align n with B field
Torque for single loop
τ = iAB sin θ
where A is the area of the loop
and θ is between n and B
z Replace single loop with coil
of N loops or turns
τ = ( NiA) B sin θ
Magnetic Dipole
• Define magnetic dipole moment
μ = NiA
τ = ( NiA) B sin θ = μB sin θ
z
The direction of the magnetic
dipole moment is the same as
the normal vector to the plane.
r
r
μ =n
z
The torque becomes
r
τ = μ×B
r
r
Magnetic Dipole
z
z
z
A magnetic dipole in a
magnetic field has a
magnetic potential
energy, U
Lowest energy when
dipole moment lined
up with B field
Highest energy when
dipole moment directed
opposite B field
r
U = −μ • B
r
Magnetic Dipole
• Magnetic dipole moment μ has
Torque
r
τ = μ×B
r
z
r
Potential Energy
r
U = −μ • B
r
Remember electric dipole moment p
Torque
r r
τ = p× E
r
Potential Energy
r r
U = −p•E
B Fields from Currents
z
Calculate B field produced
by distribution of currents
z
Similar to finding E from
distribution of charges
1 dq
dE =
4πε 0 r 2
z
B fields, like E fields, can
be superimposed to find
net field
Biot-Savart Law
μ 0 i ds sin θ
dB =
2
4π
r
z
z
z
Current-length element,
i ds, is product of a
scalar and a vector
Find net B field by
integrating
BUT remember it is z Permeability constant, μ0
a vector sum
−7
μ 0 = 4 π × 10 T ⋅ m A
Biot-Savart Law
μ0 ids sin θ
dB =
2
4π
r
z
Rewrite in vector form
r
r
r μ0 ids × r
dB =
3
4π r
z
Known as Biot-Savart Law