M3-1
THE CURRENT BALANCE
Introduction
Ampere’s law tells us that the magnetic field around a current carrying wire is proportional to the
current flowing in the wire. Parallel, current-carrying wires impose a magnetic force on one
another through their magnetic fields, and that force 𝐹 acts perpendicular to the direction of
current flow, as in the figure below.
Figure 1. Force at a point on wire 2 due to B field of wire 1.
This essential physical law is used to define one of the seven base units of the International
System of Units (SI). The unit of current, the ampere (A), is defined as follows:
“The ampere is that constant current which, if maintained in two straight parallel conductors of
infinite length, of negligible circular cross-section, and placed 1 meter apart in vacuum, would
produce between these conductors a force equal to 2 × 10−7 newton per meter of length.”
NIST Special Publication 330, 2008 Edition. International System of Units (SI).
From this definition, the permeability of vacuum, 𝜇! = 4𝜋×10!! 𝐻/𝑚 in the SI system of units,
is set. We can see that this definition can be used primarily in two different ways to
experimentally validate Ampere’s law. The first is to accept the value of the permeability, and
validate the measuring tools at our disposal which measure amperes. The second is to accept the
value of the ampere, and measure the permeability of vacuum.
M3-2
Objective
The objective of this experiment is to study the magnetic forces produced by electric currents in
straight, parallel wires at a fixed separation. You will use a null method of determining the force
between two current carrying wires to measure the permeability of air, 𝜇! .
Theory
Based on Figure 1, the relationship between force 𝐹, in newtons, between two long current
carrying conductors can be shown to depend on current according to:
𝐹=
!!" !
𝐼
!
(1)
where 𝐿 is the overlapping length of current carrying conductors in metres, 𝑑 is the centre-tocentre separation of the conductors in metres, 𝐼 is the current in each wire in amperes, and 𝑘 is
𝜇! /4𝜋.
When 𝐹 is plotted against 𝐼 ! , a straight line should be obtained with a slope equal to 2𝑘𝐿 𝑑.
The value for 𝑘 may be determined from this slope. With the wires surrounded by air, k should
have a value near 1.00×10!! 𝐻/𝑚 (or equivalently, in units of 𝑁/𝐴! ), the value for vacuum.
To realize this experiment, we will be using a current balance apparatus depicted schematically
in Figure 2 and Figure 3. A steel bar which pivots on a pair of knife edges lies above a fixed
steel bar. By changing the mass on the top bar, and correspondingly increasing the current
passing through the bars and thus the repulsive force between the bars, maintaining constant
separation, we will use the null method to determine the force between the wire. With the above
equation, we can then determine the permeability of air.
M3-3
3
b
1
2
0
telescope
mirror
1
free loop
2
3
end view of parallel
current-carrying bars
scale
counterpoise
a
knife edge supports
Figure 2. Schematic of current balance apparatus (side view).
Equipment:
•
•
•
•
•
•
•
•
Current Balance Apparatus
Telescope mounted with scale
DC power supply
Ammeter
Cables with banana plug terminations
Set of tweezers, 5x20 mg masses, and a coin
Vernier calipers
Metre stick
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DC power
supply
ammeter
A
lower bar
top bar
support bar
L
knife edge
support
Figure 3. Circuit diagram for the current balance.
Procedure
This experiment incorporates very high currents, which can be extremely harmful if the proper
precautions aren’t taken. Always turn power supply off before adjusting connections, and when
not in use.
1. Set up the apparatus as shown in the diagrams on the previous page. The current balance
mirror should be at least two metres from the scale by the telescope. It is important that
the current carrying bars run in a North-South direction to minimize the effect of Earth’s
magnetic field.
2. Observe the beam lift mechanism designed to lift the beam so that the knife edges come
clear of their supports and then to return them to a particular position. This mechanism
allows the knife edges to be repositioned to the previous position should they move a
little during the experiment. Use the lift mechanism to gently lift the knife edges clear of
the supports and then lower them back down. The frame should oscillate freely. The
metal plate between the poles of the damping magnets should not touch the magnets and
the poles of the magnets should be approximately 2.0 mm apart.
3. Adjust the levelling screws to make sure the base of the apparatus is firmly situated on
the bench. If necessary, adjust the counterpoise behind the mirror until the frame comes
to rest with the front horizontal bar a few millimetres above the stationary bar. If
necessary, adjust the counterpoise below the mirror until the period of the oscillations of
the frame is 1-2 seconds.
4. Note that thumb screws on each front post permit either end of the lower bar to be raised
M3-5
5.
6.
7.
8.
or lowered. Similar thumb screws near the rear of the frame permit either end of the
upper bar to be adjusted forward or backward. These two sets of thumb screws allow the
bars to be adjusted to be parallel to each other. Gently use the lift mechanism to lift the
knife edges and to replace them in the proper spot. If necessary, adjust the bars to be
parallel without moving the position of the knife edges on their supports. When viewed
from the front, and with a white paper between the bars, the two bars may appear to be
slightly lacking in straightness. If this is very serious, ask your lab instructor to have
them straightened.
Aim the telescope at the mirror on the current balance but do not focus it on the mirror.
With one student looking into the telescope, another student can slowly rotate the current
balance base on the table until a reflection of the scale can be seen in the telescope. Note
that the adjustment screw behind the mirror permits the angle of tilt of the mirror to be
adjusted so that the scale can be seen in the telescope with the telescope at a convenient
height for the observer. Engage the beam lift gently and release it to insure that the knife
edges are still in their proper positions.
Focus the crosshairs of the telescope, then adjust the main focusing of the telescope until
the scale can be seen clearly. The telescope is properly focused only when there is no
parallax between the crosshairs and the scale as seen through the telescope. Read the
scale as seen through the telescope with the beam of the current balance at rest in its
equilibrium position. The separation of the two bars at equilibrium may need additional
adjusting. Place five 20 mg weights on the pan on the upper bar, then increase the
current gradually until the scale reading indicates that the beam has returned to its
equilibrium position. Do not exceed 15 amps. Adjust the separation so that the required
current is about 10 amps, but not more than 15 amps. Do not adjust the separation while
the current is on.
Record the current required to restore the beam to its equilibrium position for weights on
the pan ranging from 20 mg to 100 mg in 20 mg increments.
Measure the length 𝐿 of the upper front bar from the centre-to-centre of its supporting
bars (see Figure 3)
M3-6
TABLE 1: DETERMINING SLOPE OF F = f(I2) GRAPH AND μ0
Run
No
Mass on
Pan on
the
Upper
Bar
Y-Axis
Gravitational
Force on Upper
Bar
I (A)
F =mg (N)
m (mg)
1.
20
2.
40
3.
60
4.
80
5.
100
Current
through
the bars
X-Axis
Square of
Current
through
the bars
I2 (A2)
Slope
of the
Graph
2
(N/A )
Overlapping
Length of
Bars
L (m)
d –from Table 2
Permeability of air
µ0 =
2π d
⋅ slope
L
(N/A2)
9. The separation of the two bars at equilibrium will be needed for the calculations and can
be determined in the following manner. The scale reading at equilibrium is noted. The
upper bar is then depressed by placing a coin or similar-sized object on the pan, until it is
in contact with the lower bar. The new scale reading is then noted. Simple geometry can
be used to show that the open space separation, 𝑑! , is given by the following equation:
𝑑! = 𝐷𝑎/2𝑏
where D is the difference in the scale readings, a is the mean distance from knife edge to
bar, and b is the distance from the mirror to the scale (see Figure 3). The required centreto-centre separation, 𝑑, is obtained by adding the diameter of one bar to 𝑑! (see Figure 4).
Take necessary measurements to determine 𝑑. Make a rough check on this value for 𝑑
by measuring directly with a millimetre scale.
M3-7
Figure 4. Separation of current carrying bars.
TABLE 2: DETERMINING CENTRE-TO-CENTRE DISTANCE BETWEEN BARS
Diameter
of the two
bars
Φ (m)
Initial
Mark on
the Ruler
without
the penny
Di (cm)
Final
Mark on
the Ruler
with the
penny
Df (cm)
D = Di -Df
(m)
Distance
from
knife
edge to
upper bar
a (m)
Distance
from
mirror to
scale
Side to Side
Separation
of bars
b (m)
d0 =
(m)
D⋅a
2b
Centre to
Centre
Separation
of bars
d = d0 + φ
(m)
Analysis
Plot a graph of 𝐹 vs. 𝐼 ! . The values of 𝐹 are the weights of the masses placed on the pan. Find
the slope of the straight line graph and use it to determine the permeability of air. As always,
incorporate the uncertainty of your measurements into your calculations. Compare to the value
of the permeability found for the known value for vacuum.
References:
Barry N. Naylor and Ambler Thompson, NIST Special Publication 330, The International
System of Units. NIST, 2008.
Bernhard Kurrelmeyer and Walter H. Mais, Electricity and Magnetism, Van Nostrand, 1967.