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ECT1026 Field Theory
Chapter 3
Magnetostatics
3.2 Magnetic Force
By Dr Mardeni Roslee
[email protected]
0383125481
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ECT1026 Field Theory
Lecture 3-2
2007/2008
Charge Particles
3.2 Magnetic Force on
Current Carrying Conductor
3.3 The Biot-Savart Law
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ECT1026 Field Theory
3.2 Magnetic Force
I. Force on a Moving Charge in B
Fm = qu u  B (N)
-q
q
+
u
A test charge q is in a
magnetic field B experiences a
magnetic force, Fm.
This force is:
 proportional to q;
 the direction of Fm at any
point is at right angle to the
velocity vector, u and
magnetic filed B
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ECT1026 Field Theory
The magnetic force acting on the particle of charge “q” =
Fm = q u × B
(N)
Magnitude of Fm
 is the angle
between
u and B
= 90o  Fm is maximum
 = 0o or 180o  Fm =0
Direction of Fm = direction of u  B
Right Hand Rule
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ECT1026 Field Theory
Right Hand Rule - Direction of Fm
Fm = q u × B
Thumb: direction of Fm
Fingers: direction of velocity u
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ECT1026 Field Theory
3.2 Magnetic Force
2007/2008
If a charged particle is present in both an
electrical field, E and magnetic field, B.
According to Lorentz Force law,
the total electromagnetic force acting on the particle =
F = Fe + Fm
=qE+quB
= q (E + u  B)
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ECT1026 Field Theory
3.2 Magnetic Force
Differences between Electric Force and Magnetic Force

Electric force is always in the direction of the electric field.
The magnetic force is always perpendicular to the magnetic field
Fe

The electric force acts on a charged particle whether or not it is moving,
the magnetic force acts on it only when it is in motion
Fe

Fm
The electric force expends energy in displacing a charged particle, the
magnetic force does no work when a particle is displaced
V1
Fe
V2
V1
Fm
V2
(V1=V2)
Work done
(V1=V2)
No Work done
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ECT1026 Field Theory

The electric force expends energy in displacing a charged
particle, the magnetic force does no work when a particle
is displaced
Since Fm is always perpendicular to u
 Fm . u = 0
(Fm.U.Cos 90 = 0)
The work performed when a charged particle is displaced by a
(S=ut)
differential distance dl = u dt is
dW = Fm . dl = (Fm . u ) dt = 0
(W=Fs)
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ECT1026 Field Theory
Since no work is done, a magnetic field
cannot change the kinetic energy
of a charged particle
Magnetic field
can change the direction of motion
of a charged particle, but
it
cannot change its speed
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ECT1026 Field Theory
3.2 Magnetic Force
II. Force on a Current Carrying Conductor
A slightly flexible vertical wire is oriented along the z-direction,
is placed in a magnetic field B directed into the page
a) No current flowing in
the wire, Fm= 0. The
wire maintains its
vertical orientation
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ECT1026 Field Theory
3.2 Magnetic Force
Force on a Current Carrying Conductor
dFm= I dl  B
Fm
Fm
Current is flowing upward
in the wire
Current is flowing downward
in the wire
Wire deflects to the left
(-y–direction)
Wire deflects to the right
(+y–direction)
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ECT1026 Field Theory
3.2 Magnetic Force
Force on a Current Carrying Conductor in B
For a closed contour C carrying a current I,
the total magnetic force is:
(N)
The equation is applied to two special cases:
1) Closed circuit in an uniform B field
2) Curved wire in an uniform B field
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ECT1026 Field Theory
3.2 Magnetic Force
Force on a Current Carrying Conductor in B
Case 1) Closed Circuit in a Uniform B Field
Consider a closed wire carrying a
current I and placed in a uniform
external magnetic field B.
B is constant, thus it is taken
outside the integral.

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ECT1026 Field Theory
3.2 Magnetic Force
Force on a Current Carrying Conductor in B
Case 1) Closed Circuit in a Uniform B Field
a
The integral of vectors dl
over a closed path is equal to
zero.
Therefore, the total magnetic
force on any closed current
loop in a uniform magnetic
field is zero.
(a=a)
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ECT1026 Field Theory
b
=
a
b
3.2 Magnetic Force
2007/2008
Force on a Current Carrying Conductor in B
Case 2) Curved Wire in a Uniform B Field
Consider a curved wire carrying a
current I and placed in a uniform
external magnetic field B.
B is constant, thus it is taken outside
the integral.
a
Fm = I
(a = b)
(∫adl)  B = I l  B
b
l is the vector directed from a to b,
integral of dl from a to b has the
same value irrespective of the path
from point a to b
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ECT1026 Field Theory
3.2 Magnetic Force
Force on a Current Carrying Conductor in B
Case 2) Curved Wire in a Uniform B Field
The net magnetic force
on a line segment is
proportional to the
vector between the end
points (point a and b),
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Coulomb’s Law
dq=vΔv
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ECT1026 Field Theory
3.3 The Biot-Savart Law
Point R
B
Rr
I

dl
The magnetic flux density at a point R from an element dl of a
wire that is carrying a current I.
“dB is generated by I”
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ECT1026 Field Theory
3.3 The Biot-Savart Law
The magnetic flux density at a point R from an element dl of a
wire that is carrying a current I. This expression is known as the
Biot-Savart law, and it is expressed as:
 ^
 o0 IIddl×R
  Rˆ
dB 
dB
2
2
4
4 RR
(Tesla)
It is also defined as the differential magnetic flux density
dB generated by a steady current I flowing through a
differential length dl
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ECT1026 Field Theory
3.3 The Biot-Savart Law
(Observation point)
Difference between E and B
Electric field vector E, whose
direction is along the distance
vector joining the ‘charge’ and
‘observation point’.
q
^
E=R
4eR2
The magnetic field or
magnetic
flux density is orthogonal to
the
plane containing dl and R. (the
direction is dictated by dl  R
Point R
B
B
R
Rr
dl
E
I

(Charge)
dl
B
U
Fm
Hence, B is directing “out
of the page” by using right
hand rule.
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ECT1026 Field Theory
3.3 The Biot-Savart Law
Surface Current Distribution
o
B  s
4
Js × Rˆ
R2
Volume Current Distribution
(I dl = Js ds = Jdv)
ds
Js = Surface Current Density
o
B 
v 4
J × Rˆ
R2
J = Volume Current Density
dv
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