Current Balance

R. Thakrar Department of Physics and Astronomy University College London

16th March 2009

Abstract: The relationship between magnetic force on a conductor (F), the current running through the conductor (I), the amount of conductor within a magnetic field (L), and the magnetic flux density (B) was verified by using two different tests: one test varied F and I while keeping B constant; another varied F and L while also keeping B constant. The value of the magnetic force was determined by using the principle of moments between this force and small counterweight called a Rider, which was used to balance the conductor when the magnetic force displaced it. The force was found to be directly proportional to both I and L, hence verifying the relationship F=BIL for a current carrying object in a magnetic field. For both experiments, a value of B was calculated. For the first experiment, it was calculated to be B1= (0.059±0.0038) T, while for the second experiment, it was calculated as B2= (0.055±0.0009) T. These values are comparable to a value determined by a Gaussmeter, but the Gaussmeter value of B=0.065T lies outside the range of both values, suggesting that sources of additional experimental errors may affected the accuracy of results.

[1] Introduction The aim of this experiment was to measure the force on a current-carrying conductor inside a magnetic field. The results obtained from the experiments would be used to verify the relationship between force, current, the length of the conductor in a magnetic field, and the magnetic field strength. The experiment involves a copper wire balancing on a pivot, which is placed in a constant magnetic field. When a current is passed through the copper, the copper feels a vertical force, and hence is deflected upwards or downwards depending on the direction of current flow. In this experiment the force experienced will be vertically upwards. This deflection can be counteracted by moving a small weight along the copper wire called a “rider” (shown in figure 1). This may be moved to make the conductor vertical again.

Figure 1 - Diagram of experimental setup (the scales are not shown on the diagram but they sit next to the length of the conductor, and parallel to the rail) The magnitude of force can be measured using the current, length of conductor in the field and the magnetic field strength, such that:

(1) Where F is the force, B is the magnetic field strength, I is the Current, and L is the length of the conductor in the field.

The amount the rider has moved along the scale is directly proportional to the force felt by the conductor in the field. When the conductor is balanced by the rider, it is because it has produced a turning moment that counteracts the moment produced by the magnetic force on the copper wire. This leads to the equation: (2)

Where x1 is the distance between the pivot and the magnetic force (the middle of the magnet) and x2 is the perpendicular distance of the rider from the pivot.

The experiment tries to verify equation 1 by performing two tests:  Varying the current, I, with the force while keeping B and L constant;

[2]  Varying the length, L, with the force while keeping B and I constant. This is achieved by moving the magnet laterally along the equipment, and hence changing the length of magnet that the conductor is in.

Method

Figure 2 - Schematic Diagram of Apparatus The first experiment was set up to verify the relationship between the current and the rider position (and hence the force). The apparatus was set up as shown in figure 1 and 2, with the conductor fully inside the magnet (feeling maximum strength of B). The position of the magnet was kept constant throughout this experiment. The counterweight was then moved so that the conductor was horizontal when the rider was in the zero position on the scale and no current was flowing. A current was then passed through the conductor, between zero and two Amperes at 0.2A intervals. At each increment, the rider was moved so that the conductor is horizontal again. The rider position was recorded at each increment.

The second experiment was set up to verify the relationship between lengths of conductor within a magnetic field and the rider position (and hence the force). The experiment was set up like in figure 1 and 2, and the conductor balanced so that it is horizontal when the rider was in the zero position. The rider was then moved to near full scale, and a current was passed through the magnet to rebalance the conductor. The magnet was then slid along the rail until its leading edge was near the scale zero position, so the conductor was almost entirely outside the magnetic poles. The rider was moved to zero and the counterweight was used to rebalance the conductor. The magnet assembly was then slid up in increments. At each increment, the rider was used to rebalance the conductor, and its position was recorded.

Results & Analysis Theory suggests there is proportionality between the magnetic force with the current, and the force and length of conductor within the magnetic field. Hence there must also be proportionality between the rider position and the current and length L respectively.

[3] From tables one and two we can see straight away that there is a correlation between the x2 and I and L respectively: when the current or the length of conductor in the field is increased, a larger

value of x2 is needed to balance the conductor. Hence a larger force is needed for balance to occur, meaning that a larger magnetic force was exerted when I or L was increased. These trends are emphasised in table three and four.

I (A) Error in I Average Error in x2 ±

(A) ± x2 (mm) (mm) 0.0 0.01 0 0.5

0.2 0.01 4 0.5

0.4 0.01 7 0.5

0.6 0.01 10 0.5

0.8 0.01 13 0.5

1.0 0.01 17 0.5

1.2 0.01 19 0.5

1.4 0.01 22 0.5

1.6 0.01 26 0.5

1.8 0.01 28 0.5

2.0 0.01 31 0.5

[4] L Error in L Average Error in x2 ±

(mm) (mm) ± x2 (mm) (mm) 0 1 0.0 0.5 5 1 4.0 0.5 10 1 7.5 0.5 15 1 10.0 0.5 20 1 14.0 0.5 Table 1 - Table showing the variation of rider position (x2) with steadily increasing current. 25 1 17.0 0.5 30 1 20.0 0.5 Table 2 - Table showing the variation in rider position (x ) 2 35 1 24.0 0.5 with increasing amounts of conductor inside the magnetic field (L). 40 1 27.0 0.5 45 1 29.0 0.5 2 An R value can be found for trends of data using 50 1 31.0 0.5 Microsoft Excel©. The value is between 0 and 1, and shows how well the data correlates: 1 being a perfect correlation and 0 being no correlation. For both graphs, the R2 value is over 0.99 and so this shows that the data correlates very well with the trend line. The graphs show that there is a direct proportionality between the current and L, and the rider position and hence a proportionality to the force. This therefore verifies a relationship between F, I and L for a constant value of B.

Table 3 - Graph showing the variation or rider position (x2) with current (I).

[5] Table 4 - Graph showing the variation of rider position (x2) with the length of conductor in the magnetic field (L) The value of B in both experiments can be found using equations 1 and 2 to give: (3) (4) (5)

Where x2/I is the gradient in table 3, and x2/L is the gradient in table 4. These values were used to get values of: B1= (0.059±0.0038) T B2= (0.055±0.0009) T When a Gaussmeter was used to measure the magnetic field strength, it was seen to be B=0.065T. Compared to the experimental uncertainty, this value is far from the value calculated in the experiment as it lies outside the bounds of this uncertainty. This suggests that there were possibly sources of systematic error.

Conclusion

In conclusion, the experiment was successful in showing that force was indeed directly proportional to the current and the amount of conductor in a magnetic field. Thus the experiment verifies the relationship between the force, current, magnetic flux density, and the length of conductor in the magnet to follow equation 1 (F=BIL). The two values of B found using these relationships were found to be B1= (0.059±0.0038) T, and B2= (0.055±0.0009) T. A value of B determined using a Gaussmeter was found to be outside the range of both B1 and B2, suggesting that there were additional sources of systematic error that would have made the error larger than the predicted standard uncertainty, such as a human error when judging the conductors horizontality.

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