The Current Balance


Introduction and Theory:

When a current I passes through a long, straight conductor, a magnetic field B is 
created.  According to AmpereÕs law, the magnitude of the field at a perpendicular 
distance d from the conductor is given by the relation.

	B = µoI/2¹d.		 (1)

In SI or MKS units the magnetic field is measured in tesla when I is in amperes and d is 
in meters.  The quantity  µo  is a constant known as the permeability of free space and 
has the assigned value of  4¹ x10-7 tesla.meter/amp. It is assigned a value in order to 
maintain consistent units when the permittivity of free space has the value associated 
with CoulombÕs law for the electrostatic force between charges, with the additional 
requirement that the square of the speed of light in a vacuum be given by 
 
	c = ( eo mo)-1/2.			 	(2)

If a long, straight conductor carrying current IÕ is placed in a magnetic field of strength 
B, the conductor will experience a force whose magnitude on a length L of the 
conductor is given by

	F = IÕLB sin q,				(3)

where q is the angle between the current and field directions.  The force F has the units 
of  newtons if B is measured in teslas, IÕ in amperes, and L in meters.

	Now,  if two long, parallel conductors carrying currents I and IÕ are separated 
by a distance d, each will experience a force due to the magnetic field set up by the 
other.  Combining Eqs. 1 and 3 for the case of parallel wires (q = 90û), the following 
result is obtained:

					F = µoI IÕL/2pd. 				(4)

The forces exerted on the two conductors are equal in magnitude and oppositely 
directed, as required by NewtonÕs third law.  If the currents are in the same direction 
the conductors experience an attractive force, while oppositely directed currents will 
produce a repulsive force.  This equation is valid only for infinitely long conductors.  
However, if the separation (d) between the conductors is very much less than the length 
of either conductor, then the error in the equation is negligible.  If the currents in the 
two conductors are equal (I = IÕ), then  Eq. 4  becomes

					F = 2x10-7 I2L/d,				(5)
				




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in which we have incorporated the assigned value for  µo.  This last result is used to 
define the unit of current, the ampere.  Formally,

Òone ampere is that current which, if present in each of two parallel conductors of 
infinite length and separated by a distance of one meter in a vacuum, causes each 
conductor to experience a force of exactly 2x10-7 newton per meter of length.Ó 

At the National Institute of Science and Technology, primary measurements of current 
are made using a current balance.  In a current balance, the conductors form part of an 
arm of a sensitive balance and the force between them, when carrying a current, is 
counterbalanced by weights added to the other arm.  Current balance measurements are 
used to calibrate secondary standards (ammeters) which are more convenient to use for 
current measurements.  When at static equilibrium, the gravitational force (mg) on the 
moveable conductor is equal to the repulsive magnetic force.  If these quantities are  
equated, the result is that 

			
					m = CI 2,					(6)

with the constant C given by 

					C= µoL/2¹gd.				(7)

It is clear that the plot of m vs. I2 should be a straight  line passing through the origin, 
having a slope equal to C. 

Apparatus:

DO NOT HANDLE THE CURRENT BALANCE UNTIL YOU READ THESE 
INSTRUCTIONS.
 
 	The current balance is shown in Fig. 1.  The balance has four terminal posts, one 
near each corner.  In setting up the circuit, the two posts  at one side should be joined 
by a fairly long wire.  The other two posts are used to connect the balance in the circuit 
(Fig. 2).  It is important that the lead wires connected to the binding posts of the 
balance leave at right angles to the side of conductors which are part of the frame.  
Voltage control is achieved by use of a  step-down 6-volt transformer.  The control 
knob on the carbon block resistor used for current variations should be loosened to 
increase the resistance to a maximum before power is applied.
	The apparatus is leveled by removing the frame from the balance and adjusting the 
leveling screw under the base.  The frame should be replaced with the knife edges (K) 
positioned on the bearing posts so that the pins on the beam-lift engage easily.  
Operation of the beam-lift will place the beam  in the same position each time the lift is 
raised. During this experiment, it is preferable to disturb the balance beam as little as  
possible.
	 The alignment of the conductors is done by placing a coin or other mass on the pan 
(P) and adjusting the appropriate thumbscrews;  however, the beam lift must be 
operated after each adjustment to be sure that the knife edges are always in the same 
place.  After the balance has been leveled and aligned, the telescope is adjusted so that 
the scale is visible in the mirror.

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	Figure 1.  Schematic Diagram of Current Balance




                                                              


	Figure 2.     Diagram of Current Balance for AC Operation


Safety Precautions: 

The conductors in the current balance carry  AC currents as large as 15 amperes (do not 
exceed this limit).  The carbon block rheostat will become quite hot through the course 
of the experiment. Use care when handling this device even after the experiment is 
completed.





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Procedure:   

1.     The equilibrium separation of the two conductors do should initially be set at a 
few millimeters by moving the counterpoise (C).  This separation is measured 
indirectly, using the telescope to observe the image of the scale in the mirror on the 
current balance. With a coin or other mass on the pan, the scale reading for the 
conductors in contact is recorded.  Then, the coin or mass is removed and the scale 
reading with the conductors at the equilibrium separation is read and recorded.  The 
value for do is given by the relation

					do =  Da/2b,					(8)

where D is the difference in scale readings, a is the distance from knife edge to the 
center of the upper conductor, and b is the distance from the back of the mirror to the 
scale (for a derivation, see the Appendix.   When the apparatus is  set up, the value of 
b should be at least 150 cm, and preferably 175-180 cm.  The quantity d discussed in 
the Theory section is the center-to-center distance between the conductors.  A 
micrometer caliper is used to determine the diameter of each conductor,  and this 
average diameter is added to do to obtain a value for d.  Also measure L, the  
(conducting) length of the moveable wire.  Once the equilibrium separation, do, and the 
values for d and L have been found, no further mechanical adjustments should be made 
to the current balance.

                                                      


	Figure 3.  End-on Detail of Parallel Current-Carrying Bars


2.  	Data is acquired by placing a small mass on the pan (initially 50 milligrams or 
less), making sure that the rheostat is loosened (to give maximum resistance), and 
closing the switch.  The current in the circuit is slowly increased until the scale reading 
as seen through the telescope returns to the same reading it had for the equilibrium 
separation do.  When this occurs, the force on the upper conductor due to the mass in 
the pan (given by mg) is balanced by the force of repulsion due to the magnetic field 
interaction of the two currents.  The current indicated on the ammeter for the 
equilibrium separation should be recorded with the value of the mass in the pan. To 
insure consistency in the data, the current should be measured at least three times for 
each value of mass added to the pan.  This procedure should be repeated  for at least 5 
well-spaced values of mass.  It is preferable to gently remove and place masses on the 
pan without touching or disturbing the balance beam in any way.  If the beam is jarred 
inadvertently, the beam lift may be used to reposition the beam.  If this is done, be sure 
to check (with the current turned off) the equilibrium reading on the telescope, making 
sure it agrees with the previous equilibrium value.  At no time during the 
experiment should the current be allowed to exceed 15 amps.   

3.   	From your data, prepare a table of added mass, average current, and average 
current squared for each of the masses.  

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Analysis: 

1.  	Using the table of data from step (3) of the Procedure section, make a plot of 
added mass versus average current squared,  I2.
 
2.   	Using your graph, calculate the slope of the resulting line.
 
3. 	Using the slope of this line, the value of g for Rochester (g = 9.80453 m/s2), 
and your values for the length of and the equilibrium separation of the conductors, 
determine an experimental value for  µo. 


Uncertainty in Results: 

Using the measured and calculated uncertainties in m, I2, and do, determine the 
uncertainty in the experimental value of µo.  Using this uncertainty, comment on the 
comparison between your experimental value of µo and its accepted value.  


Appendix: Magnification Using the Optical Lever

When the upper bar of the current balance (in Fig. 4) is at position P1, one then reads 
S1 on the scale  in the telescope.  The angle of incidence of the ray from S1 is f1, as is 
the angle of reflection.   When the bar moves through an angle q to position P2 (a 
distance do above P1), the point  S2 on the scale is seen in the telescope. The ray from 
S2 is incident on the mirror at an angle of f2, and again the angle of incidence is equal 
to the angle of reflection.  Since the normal to the mirror has moved through an angle 
q, then f1+ f2 = q, and the total angle defined by the rays from S1 and S2 is 2q.  For 
large b and small q,  the bisector of the base bisects this angle.  From the two similar 
triangles of apex angle q, one then has

					(D/2)/b = do/a.				(9)




















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				Figure 4.  Geometry of the Optical Lever







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