4. Electric & Magnetic Forces
Electromagnetic (Lorentz) force
Magnetic force
magnetic flux density B at a point in space in
terms of the magnetic force Fm that acts on a
charged test particle moving with velocity u
through that point.
If a charged particle resides in the presence of
both an electric field and a magnetic field then
the total electro-magnetic force acting on it is
5. Magnetic Force on a Current Element
Differential force dFm on a differential
current Idl:
8. Torque
d = moment arm
F = force
T = torque
When a force F is applied on a rigid body that can
pivot about a fixed axis, the body will rotate about
that axis. The angular acceleration depends on the
cross product of the applied force vector F and the
distance vector d, and it is called torque:
9. Magnetic Torque on Current Loop
No forces on arms 2 and 4 (because B is
parallel to the direction of the current flowing
in those arms.)
Magnetic torque:
Area of Loop
The moment arm is a/2 for both forces, but d1
and d3 are in opposite directions
10. For a loop with N turns and whose surface
normal is at angle theta relative to B direction:
Magnetic Field Perpendicular to the Axis of
a Rectangular Loop
11.
12. Biot-Savart Law
The Biot–Savart law states that the
differential magnetic field dH generated
by a steady current I flowing
through a differential length vector dl is
For the entire length:
13. 5-2-1 Magnetic Field due to Current Densities
We can express Bio-Savart Law in terms of
surface current density Js and volume current
density J.
19. Magnetic Field of a Loop
Cont.
dH is in the r–z plane , and therefore it has
components dHr and dHz
z-components of the magnetic fields due to dl and
dl’add because they are in the same direction,
but their r-components cancel
Hence for element dl:
Magnitude of field due to dl is
23. 5- Maxwell’s Magnetostatics Equations
Gauss Law for Magnetism
Recall that in electrostatics, Gauss law says that flux through a
surface, equals the charge enclosed by that surface:
Magnetic flux
31. Example
The magnetic vector potential of a current distribution in free
space is given by: Ԧ
𝐴 = 15𝑒−𝑟
sin ф Ƹ
𝑧 (Wb/m). Find 𝐻 at (3,
π/4, -10).
32. 5-5 Magnetic Properties of Materials
Where M is called the
magnetization vector
Where is called the
magnetic susceptibility
33. Diamagnetic
materials have a weak, negative susceptibility to magnetic
fields. Diamagnetic materials are slightly repelled by a magnetic field.
do not retain the magnetic properties when the external field is
removed.
Paramagnetic
materials have a small, positive susceptibility to magnetic fields. These
materials are slightly attracted by a magnetic field.
do not retain the magnetic properties when the external field is
removed.
Ferromagnetic
materials have a large, positive susceptibility to an external magnetic field
They exhibit a strong attraction to magnetic fields and are able to retain their
magnetic properties after the external field has been removed.
5-5 Magnetic Properties of Materials
34.
35. Magnetic Hysteresis
B–H magnetization curve, where B and H refer
to the amplitudes of the B flux and H field in
the material
residual flux density Br
38. The magnetic field in the region S between
the two conductors is approximately
Example 5-7: Inductance of Coaxial Cable
Total magnetic flux through S:
Inductance per unit length:
