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1
TUTORIAL 8
1.
In lectures, the magnetic force on a charged particle was given by the empirical expression:
F
m
=
Q
v
!
B
( )
.
(i)
Use this expression to derive a general expression for the force on a
circuit
in a magnetic
field.
Consider the circuit shown in (a) below.
B
z
l
F
0
B
a
n
!
w
F
0
F
0
(a)
(b)
axis of rotation
(ii) Apply the formula derived in (i) to show that there is no net force on the circuit.
There is, however, a mechanical moment of force, i.e., a torque, about the axis of rotation
z
, on
the circuit in (a) given by
T
=
r
!
F
,
where
r
is the moment arm (of length
w
in the figure).
(iii) Referring to Figure (b), show that the torque is given by:
T
F
=
0
w
sin
.
(iv) Defining a quantity
m
a
=
IS
n
where
S
wl
=
, show that the torque can be expressed as
T
=
m
!
B
.
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2. HELMHOLTZ COILS
It is often desirable to make the magnetic field uniform over a region of space. Often, it is
not practical to use a solenoid, and a Helmholtz coil is used instead. A Helmholtz coil
consists of two circular coils of the same radius, with a common axis, separated by a
distance chosen to make the second derivative of
B
vanish at a point on the axis halfway
between the coils.
(i) Using the notation in the figure below, write an expression for the magnetic flux
density on the axis
B
z
z
( )
.
(ii) Determine the relationship between
b
and
a
in order that the second derivative
vanish.
(iii) By Taylor expanding
B
z
z
( )
evaluate the uniformity of the field over the region where
z
!
a
/ 2
"
a
/ 10
.
P
z
2
b
a
N
turns
N
turns
3.
(i)
Show that the magnitude of the magnetic flux density at point P due to a straight wire of
length
l
and carrying a current
I
is given by:
B
=
μ
0
I
4
!"
cos
#
1
$
cos
2
( )
.
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 Spring '07
 
 Electromagnet

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