Lab Partner’s Name(s)
TA’s Name
Abstract
The permeability of free space is an important physical constant when dealing with electromagnetic theory in physics. By carrying out this lab experiment knowledge about procedure to find this physical constant was gained. During this experiment several safety issues as well as various careful measures were taken as group members were dealing with current and sensitive balancing device.
Objectives
Purpose of this experiment was to calculate an experimental value for μ0, the permeability of free space after balancing repulsive magnetic force between two wires with antiparallel currents by utilizing a known gravitational force.
Procedure
The procedure of the lab experiment was described with help of following sketch of the current balance apparatus.
At first lower bar was fixed in place. In the experiment the upper bar was movable and could be brought to within a few millimeters of the fixed bar by means of an adjustable counterweight (not shown). During the experiment a series of small masses was placed in the pan attached to the upper bar and to compensate for this added weight, the current I was slowly increased until the movable bar returned to its initial position and the repulsive magnetic force was equal to the weight of the added mass. The measured magnetic force was then be equated with the calculated magnetic force FB using
FB = μ0LI2/2πr . (1)
where L was the length over which the currents were antiparallel and r is the separation distance between the two currents. Finally, the distance r between currents; i.e. the center-to- center distance between the two bars, was determined with an optical lever. The geometry of this device was shown in the following figure.
One group member looked through the scope at the mirror, and thus, by reflection, at some point on the ruler. By recording two observed ruler locations, xt when the two bars were touching and xeq when the upper bar was in its equilibrium position, the distance r was calculated from following equation.
r = gap + 3.18 mm = a/2b | xeq – xt | + 3.18 mm .. (2)
where 3.18 mm was the diameter of each bar. Equation (2) was explained by following figure.
The mirror position and a reflected light ray when the bars were touching were shown by left-hand figure. ϕ was the angle of incidence and/or reflection for the reflected light ray. When the bars were touching, the center-to-center distance between the bars was 3.18 mm. But at the equilibrium position, the center-to-center distance was r. the mirror position and a reflected light ray when the upper bar is in its equilibrium position was shown by right-hand figure. The mirror was tilted by an angle θ with respect to its initial position and for this reason the angles of incidence and reflection have both been reduced by θ. The original light ray from xt was also drawn in the right-hand figure; the angle between the two light rays, one coming from xt and the other from xeq, was 2θ. As 2θ was a small angle the difference between a triangle containing the angle 2θ (or θ) and a piece of a circle containing the same small angle were neglected. Therefore, for θ in radians,
θ = gap/a = (r-3.18 mm)/a and 2θ = |xeq- xt|/b
Eliminating θ and solving for r yields equation (2).
A few preliminary measurements like a (length of the upper arm of optical lever, b (the distance from mirror to the ruler attached to the telescope), L (effective length of parallel bars) were made. In the next step, it was needed to adjust the telescope for use. Then sharp crosshairs was adjusted by moving the part of the eyepiece nearest the eye. After focusing the crosshairs the length of the scope was adjusted by pulling and pushing on the tube that emerged from the body of the scope to get a sharp image of the ruler. Then xeq, the scale reading when the upper bar was at its equilibrium position was recorded. Then, a few weights were piled on the pan until the bars touched and xt was recorded. Finally, the weights were removed to insure that the upper bar returned to its equilibrium position r was calculated to insure that a reasonable value was obtained.
Then the formal experiment was started. Masses were added to the pan starting with 20 mg and going to 120 mg in units of 20 mg. For each mass, I was adjusted so that the upper bar returned to its equilibrium position. Measurement of m and I were recorded. After this, the masses in units of 20 mg were removed as well as values of m and I were recorded for equilibrium. When all masses were removed, it was checked that the equilibrium position had not changed by the action of adding and removing masses. To reduce the effect of magnetic field of the Earth, the two-way knife-edge switch was used to reverse the current direction and the entire experiment was repeated. A value of μ0 was computed for each experiment and average of the two values was taken.
Experimental Data
Following data were obtained by carrying out this experiment.
D = 0.318
a = 0.2175 m
L = 0.265 m
b = 1.315 m
Kp = 0.407 m
xt = 0.423 m
r = 4.503 mm
Gap = 1.325 mm
θ = 0.00608 radians
Following table shows the experimental data measured.
Following graphs were obtained from experimental data –
Result
Value of μ0 based on experimental data is
μ0 = 8.14 x 10^7 N/A2
The defined value of μ0 = 4π × 10-7 N/A2 = 12.56 x 10 -7 N/A2
Percentage error = [(12.56-8.14)/12.56] x100% = 35%
Discussion and Analysis
FB = μ0LI2/2πr
I2 = 2πr FB/μ0L
So, slope of the graph for I2 is 2πr/μ0L = 1.542 x 10^5
or, μ0 = 2πr/(L x 1.542 x 10^5)
or, μ0 =( 2x3.14x 4.503)/( 265 x 1.542 x 10^5) = 6.92 x 10^7 N/A2
slope of the graph for I1 is 2πr/μ0L = 1.123 x 10^5
or, μ0 = 2πr/(L x 1.123 x 10^5)
or, μ0 =( 2x3.14x 4.503)/( 265 x 1.123 x 10^5) = 9.37x 10^7 N/A2
Average value of μ0 = (6.92 x 10^7 + 9.37x 10^7)/2 = 8.14 x 10^7 N/A2
Conclusion
The experiment was carried out safely. But the percentage error for averaged μ0 was quite high like 35 %. This may be due to not maintaining careful measures mentioned in the lab manual or error in taking reading. However, by performing this experiment knowledge about procedure to find this physical constant was gained and here lies the success of the experiment.
