Dipole in Uniform Magnetic field
Most people are familiar with magnets primarily as toys or as decorative items on a refrigerator door
Magnetism has a much broader range of applications
Considering two opposite poles (i.e., North and South poles) as a system is known as a magnetic dipole
Therefore, based on the definition we can say that a bar magnet forms a magnetic dipole
Let us see what happens when we place a Dipole in Magnetic field
When a bar magnet or magnetic dipole is placed in a uniform Magnetic field, each pole will experience a force
Due to this Force, the Dipole is bound to experience a Torque
Let us see the Force and Torque acting on a Magnetic Pole in a field
When an electric charge is placed in an Electric field, Force acts on the charge due to the Electric Field Intensity
Similarly, a Magnetic pole experiences a Force due to a Magnetic Field
The Force acting on the Magnetic Pole is given by
F
→
=
m
B
→
Where,
m
=The Magnetic pole strength
B
→
=Magnetic induction
The pole strength acting on the North Pole is
+
m
, so the Force acting on it will be
F
→
=
m
B
→
The direction of the Force acting on the North magnetic Pole will be in the direction of the Magnetic Field
The pole strength of the South magnetic Pole will be
−
m
, so the Force acting on it will be
F
→
=
−
m
B
→
The direction of the Force acting on the South magnetic Pole will be opposite to the direction of Magnetic Field
Now, let us consider a Dipole placed in a Uniform Magnetic Field
B
→
at an angle
θ
The Force acting on the dipole is zero since the forces acting on the individual poles are equal and opposite
The Torque acting on a dipole can be mathematically expressed as,
τ
=
F
×
perpendicular distance between North Pole N and South Pole P
The vertical distance between the North Pole and the South Pole is
N
P
In
△
S
P
N
,
sin
θ
=
S
N
N
P
∴
N
P
=
S
N
sin
θ
=
2
l
sin
θ
Replacing the value of
N
P
=
2
l
sin
θ
,
τ
=
m
B
×
2
l
sin
θ
Rearranging the terms,
τ
=
(
m
)
(
2
l
)
B
sin
θ
∴
τ
=
M
B
sin
θ
Where,
M
=
m
(
2
l
)
is the Magnetic Moment on the Dipole
The equation of Torque can be written in vector form as,
τ
→
=
M
→
×
B
→
In the equation
τ
=
M
B
sin
θ
, if
θ
=
0
o
or
θ
=
1
8
0
o
, the Dipole is lying Parallel or Anti-Parallel to the Magnetic Field
In this case, the Torque acting on the Dipole will be Zero
This is because the Torques acting on the North Pole
m
B
and South Pole
−
m
B
will be balanced
Now, if
θ
=
9
0
o
,
sin
9
0
o
=
1
In this case, the Torque acting on the Dipole will be
τ
=
M
B
Additionally, if
B
=
1
T
, the Torque acting on the Dipole will be equal to Magnetic Moment
M
The Torque acting on a Dipole placed in a Unit Magnetic field perpendicular to the direction of Magnetic field is known as the Magnetic Dipole Moment
Let us calculate the Work done on the Dipole
Work is done whenever a Torque is used to rotate a Dipole by an angle
Work done can be mathematically expressed as
Work done = Torque x Angular Displacement in the direction of the Force
The Work done to rotate a bar magnet or Dipole by a small angle
θ
is given by,
d
W
=
τ
d
θ
Now, further Work done is required to move the Dipole from an initial angle
θ
0
to final angle
θ
This Work done is calculated by,
W
=
∫
d
W
∫
d
W
=
∫
θ
0
θ
M
B
sin
θ
d
θ
⇒
W
=
M
B
∫
θ
0
θ
sin
θ
d
θ
⇒
W
=
M
B
θ
0
θ
−
c
o
s
θ
⇒
W
=
M
B
[
cos
θ
0
−
cos
θ
]
If
θ
0
=
0
,
W
=
M
B
[
1
−
cos
θ
]
If
θ
0
=
9
0
o
,
W
=
−
M
B
cos
θ
∴
W
=
−
M
→
.
B
→
Since,
A
→
.
B
→
=
A
B
cos
θ
0
This Work done will be stored in the form of Potential Energy. Therefore,
P
.
E
=
−
M
B
cos
θ
Revision
A magnet in which opposite poles (i.e., North and South poles) are on opposite sides of the magnet is called a Dipole magnet
When a bar magnet or magnetic dipole is placed in a uniform Magnetic field, it will experience a torque
The Force acting on a Dipole multiplied by the perpendicular distance between the Poles is the Torque acting on the Dipole
The work done on the dipole if
θ
0
=
9
0
o
, will be
W
=
−
M
→
.
B
→
The End