A Current Carrying Conductor
Lets recap how we connected moving charge in a conductor to the current.
The electric current is:
I=lim t 0
Q dQ
=
t
dt
We connected the “macroscopic” measurable quantity
– the electric current I to a microscopic quantity – the density of charge carriers n
1
Relating Macroscopic (electrical current) to
Microscopic (density of charge carriers)
The total charge in the section of interest:
Q=number of charge carriers x charge per carrier = n A x q
Q=n A v d t q
or
I=
Q
=n A v d q
t
x
+
+
+
-
-
-
+
+
-
-
+
-
vd
The current is proportional to the cross-sectional area of the wire.
All the other quantities in the equation are microscopic quantities.
2
The Magnetic Force on a Current Carrying
Conductor
For a single charge moving with a drift speed vd in a conductor
we have learned that there is a force on the charge when it is placed
in a magnetic field:
FB =q vd x B
Now, lets say there are many charge carriers each of charge q.
the density of charge carriers is n
the conductor cross-sectional area is A
the length of the conductor is l
The total charge on the conductor is q nAl
And the total force on the conductor is
FB =q vd x B nAl
3
The Magnetic Force on a Current Carrying
Conductor
Collecting two pieces of information:
The current can be written in terms of n,A,q and vd
I =n A v d q
1
The total magnetic force on a length of current-carrying conductor is:
FB =q vd x B nAl
2
Plugging (1) into (2) we get:
FB = I l x B
Since current (and vd direction), is along the length, we can put the vector sign on
the length instead.
4
The Magnetic Force on a Current Carrying
Conductor
How do we determine the force on a conductor that is not straight ?
ds
ds
I
ds
ds
ds
b
a
We tend to use the same tricks over and over. “Use the differential form”.
Divide the length segment into several infinitesimal (very small) segments.
Calculate the force due to each segment.
x
d FB =I ds
B
To add up all the forces, you integrate:
b
b
xB
FB =a d FB=a I ds
5
Question: Magnetic force on a stroke of
lightning.
What is lightning ?
6
Question: Magnetic force on a stroke of
lightning.
When lightning strikes, negative charge moves from the cloud to ground.
Therefore, a positive current flows from the ground to the cloud.
(Geographic) North Pole
N
S
I
Earth
B
N
S
(Geographic) South Pole
7
Magnetic force: a semicircular current loop
Section 1 of Loop
FB = I l x B=I 2R B sin 90 o =2 I R B
ds
2
R
1
Direction: out of the page
d
Section 2 of Loop
Here we must “use the differential
form”:
x
d FB =I ds
B
8
Magnetic force: a semicircular current loop
Section 2 of Loop
x
d FB = I ds
B
ds
2
R
1
Geometry: The angle between the
length element and the magnetic field is
d
d FB = I ds Bsin = I R Bsin d F B =0 d FB = I R B0 sin d =2 I R B
Direction: into of the page
9
Magnetic force: a semicircular current loop
Section 1 of Loop
FB =2 I R B out of the page
ds
2
R
1
Section 2 of Loop
d
FB =2 I R B into the page
The net force on the loop is zero.
But the torque on the loop is not.
10
Magnetic torque: a rectangular current loop
We pick a simple geometry to
consider torque on a current loo.
Section 1 of Loop
Section 2 of Loop
Section 3 of Loop
Section 4 of Loop
What are the forces on each
section ?
What is the net force ?
11
Magnetic torque: a rectangular current loop
We pick a simpler geometry to
consider torque on a current loo.
Section 1 of Loop
FB =0
=180o
Section 2 of Loop
FB = I a B
Out of the page
Section 3 of Loop
FB =0
=0 o
Section 4 of Loop
FB = I a B
Into the page
Again net force is zero.
But net torque is not.
12
Magnetic torque: a rectangular current loop
The torque is:
=r x F
B
The most relevant pivot point is the
centre of the line joining section 2
and 4
O
b
b
= F2 F4
2
2
b
b
= I a B I a B
2
2
max =2 Ia b /2 B=I A B
Bottom
view
Side view
We label this torque “max” because
this is the value when the loop is
perpendicular to the field:
sin =1
13
Magnetic torque: a rectangular current loop
The torque is:
=r x F
The most relevant pivot point is the
centre of the line joining section 2
and 4
F2
b
b
= F2 sin F 4 sin
2
2
= I A Bsin F4
14
Magnetic torque on a current loop
Lets recap:
Moving charge experiences a force
A current is a continuuum description of moving charge: a current
carrying conductor feels a force
Even though th net force on a current loop is zero, it experiences a
torque
Lets generalize:
= I A Bsin = I Ax
B
We introduce an new physical quantity:
= I A
“Mu” is the magnetic moment and it thr product of the
current through a loop and the area inside the loop.
=
xB
15
Magnetic Moment on a current “coil”
A coil is a loop with more than one turn
In this case, each turn feels the torque for one loop
If there are N turns, the torque will multiply N fold.
= N I Ax
B
If we still write
=
xB
Then the magnetic
moment of the coil is:
= N I A
16
Example 22.5 : Torque on a current “coil”
Dimensions 5.4cm x 8.5 cm
N = 25 turns
I = 15 mA
B =0.35 T parallel to the plane of the loop.
17