Lab V, Problem 6 : Measuring the Magnetic Field Between Two Parallel Coils

Emily YoungApril 6, 2015

Physics 1202W, Professor: Thomas Walsh, TA: Sergey Monin

AbstractA current was applied to a set of identical, parallel Helmholtz coils to produce a

magnetic field. The magnetic field between the coils was predicted using the Biot-Savart law, and confirmed to within experimental error with an average percent

error of ±4.53%.

Introduction

As part of a research group researching magnetotactic bacteria, the behavior of the bacteria in uniform magnetic fields was studied. In an attempt to develop a suitable configuration with uniform magnetic field, the magnetic field between two identical coils of wire separated by a distance equal to their radii was determined. To ensure the desired parameters for the study, it is necessary to reliably predict the magnetic field between the coils. The experiment studied the magnetic field between two coils as a function of the applied current and radii of the coils.

Prediction

The magnetic field between two coils can be predicted using the Biot-Savart law.

The magnetic field B at a distance r from a segment dl of the current I causing it is given by the Biot-Savart law.

dB=k ' Idl⋅ r

r 3 (1)

Where k’ is a proportionality constant equal to 10-7 N/A2.

The dot product of dl and r is dlrsinθ, where θ is the angle between the segment of length dl and the vector r to a point P, so equation 1 can be rewritten as:

dB=k ' Idlsinθ

r 2 (2)

For a single coil, the total resulting magnetic field is the sum of the fields produced by small increments of the coil. The direction of the resulting magnetic field is determined by the right hand rule.

The total magnetic field resulting from a single coil can be written as the total x component and y components of the contributing segments of the coil.

∆Btot = ∆Bx + ∆By (3)

Bx can be rewritten asBx= Bcosα (4)

Where α is the angle between the plane of the loop and the vector r from the line segment dl to the point P along axis x where the magnetic field is being derived.

By can be rewritten asBy= Bsinα (5)

Where α is the angle between the plane of the loop and the vector r from the line segment dl to the point P along axis x where the magnetic field is being derived.

Over the entire coil, the magnitude of the vertical components are equal and point in opposite directions such that they cancel, resulting in a net force along the x axis (positive or negative, depending on the direction of current). Since By= Bsinα=0, the total magnetic field can be expressed as

∆Btot = ∆Bx =∆Bcosα (6)

Substituting in equation 2 for this result gives

dB=k ' Idlsinθ

r 2cosα (7)

dl is always perpendicular to r, so θis 90 ° and sinθ= 1. The cosine of is R/r. By the pythagorean theorem, the distance r from the coil to a point P on the x axis can be expressed in terms of the

horizontal distance x from the center of the coil and the radius of the circle R as r= (R2 + x2)1/2 and cosα can be expressed as

cosα=R

(R 2+x2)½ (8)

Substituting equations 8, 9, and 10 into equation 7 gives

∆Btot = ∫❑

❑

❑k ' Idl

(R 2+x2)⋅

R(R 2+x2)½

(9)

Simplifying this for a single coil with N turns and radius R, the magnetic field B at a distance x from the center of the coil is:

∆Btot =∫❑

❑

❑Nk ' IdlR

(R 2+x2)3/2 (10)

The integral of dl in this case is the circumference of a circle, which is 2πR. Then The magnetic field can be expressed as

∆Btot =2 πk ' INR 2

(R 2+x2)3/2 (11)

With two coils, the total magnetic field at a point x is the sum of the fields produced by each coil at that point:

∆Btot= ∆B1 + ∆B2 (12)

If D is the distance between the two coils and x is the distance from the midpoint between the two coils, the total magnetic field at a point x is the sum of the magnetic field resulting from each coil:

Then the distance from the center of the coil to a point P is x1 for one ring and x2 for the other, and the total magnetic field can be expressed as

∆Btot=Nk ' I 2π Rx

((x1)2+R 2)3 /2 + Nk ' I 2π Rx

((x2)2+R 2)3 /2 (13)

If x is the distance from the midpoint between the two rings R/2, then x1 in equation 13 can be rewritten as (D/2 - x) and x2 can be rewritten as (D/2 + x):

∆Btot=Nk ' I 2 π Rx

((D /2−x )2+R 2)3 /2 + Nk ' I 2 π Rx

((D /2+x)2+R 2)3/2 (14)

With Helmholtz coils, the separation D is equal to the radius R. Solving for the magnetic field at any point x along the x axis gives:

∆Btot=Nk ' I 2 π R 2

((R/2−x )2+R 2)3 /2 + Nk ' I 2 π R 2

((R/2+x)2+R 2)3/2 (15)

Simplifying this result gives:∆Btot= 2πk’INR2[(R2 + (R/2 - x)2)-3/2) + (R2 + (R/2 + x)2)-3/2)] (16)

Equation 16 gives the magnetic field at any point between two identical parallel coils with radii and separation R and applied current I.

For the midpoint between the coils, x=0 and B can be expressed as

∆Btot=32 π k '∋ ¿R(125)1/2

¿ (17)

Equation 17 gives the magnetic field for the midpoint between two coils.

For any point along the x axis that is not between the two coils, the horizontal distance is given by (x- R/2) for the the coil the point is closest to, and (x + R/2) for the other coil, where x is the distance from the midpoint between the two coils to point P. Then, for any point along the x axis not between the coils, the magnetic field is given by

∆Btot= 2πk’INR2[(R2 + (x- R/2)2)-3/2) + (R2 + (x + R/2)2)-3/2)] (18)Equation 18 gives the magnetic field at any point along the x axis that is not between the two coils.

Procedure

Figure 1: The mechanism used to create a magnetic field

Figure 1 depicts the mechanism used to create a magnetic field in this experiment. Two Helmholtz coils with N turns and radii R separated by R both have a current I running through them. A DC power supply connected directly across the coils provides the current as well as measures it. A Hall Probe (magnetic field sensor) placed in the center of the coils (lying horizontally) is attached to logger pro, recording the magnetic field between the coils.

Data

Radius R= 0.1m Number of coils N= 200 Proportionality constant k’= 10-7 N/A2

Applied Current, Measured and Theoretical Magnetic Fields:

Trial Applied Current (A) Experimental Magnetic Field (T)

Theoretical Magnetic Field (T)

1 0 0 0

2 0.01 1.65x10-5 1.7x10-5

3 0.02 2.5x10-5 3.4x10-5

4 0.03 3.4x10-5 5.1x10-5

5 0.04 4.2x10-5 6.8x10-5

6 0.05 5.1x10-5 8.5x10-5

7 0.06 5.9x10-4 1.02x10-4

8 0.07 6.8x10-5 1.19x10-4

9 0.08 7.6x10-5 1.36x10-4

10 0.09 8.5x10-5 1.53x10-4

11 0.10 9.3x10-5 1.7x10-4

12 0.11 1.01x10-4 1.87x10-4

13 0.12 1.1x10-4 2.2x10-4

14 0.13 1.2x10-4 2.3x10-4

15 0.14 1.35x10-4 2.5x10-4

Uncertainty of Magnetic Field (Theor.): ±1x10-4T Uncertainty of Magnetic Field (Exp.): ±1x10-6TExpected Range of Error of Magnetic Field: ±1.01x10-4TStandard Deviation of Magnetic Field: ±1.1x10-4TAverage Percentage Error of Magnetic Field: ±24.4%

Experimental versus Theoretical Magnetic Field at Midpoint with Varying Currents

Graph 1: the experimental vs. theoretical magnetic fields for varying currents at the midpointGraph 1 shows the magnetic field (in Gauss) at the midpoint between the Helmholtz coils for varying applied currents. The green line depicts the experimental results, while the purple line shows the results expected from the prediction in equation 18.

Applied current I= 0.05A Placement of coil 1 along x axis= 0.0m Placement of coil 2 along x axis= 0.10m Point along x axis, measured and theoretical magnetic fields (constant current):

Trial Point along x axis (m)

value of x Experimental Magnetic Field (T)

Theoretical Magnetic Field (T)

1 0.00 0.05 7.0x10-5 8.5x10-5

2 0.01 0.04 8.0x10-5 8.77x10-5

3 0.02 0.03 8.4x10-5 8.91x10-5

4 0.03 0.02 8.7x10-5 8.97x10-5

5 0.04 0.01 8.7x10-5 8.98x10-5

6 0.05 0.00 8.7x10-5 8.99x10-5

7 0.06 0.01 8.5x10-5 8.98x10-5

8 0.07 0.02 8.3x10-5 8.97x10-5

9 0.08 0.03 8.0x10-5 8.91x10-5

10 0.09 0.04 8.0x10-5 8.77x10-5

11 0.10 0.05 7.5x10-5 8.5x10-5

12 0.11 0.06 7.0x10-5 8.1x10-5

13 0.12 0.07 5.5x10-5 7.57x10-5

14 0.13 0.08 4.0x10-5 6.94x10-5

15 0.14 0.09 2.0x10-5 6.26x10-5

16 0.15 0.10 1.0x10-5 5.75x10-5

Uncertainty of Magnetic Field (Theor.): ±1x10-4TExpected Range of Error of Magnetic Field: ±1.01x10-4TStandard Deviation of Magnetic Field: ±3.5x10-6TAverage Percentage Error of Magnetic Field: ±4.53%

Experimental vs. Theoretical Magnetic Field at Different Points Along X Axis

Graph 2: the experimental versus theoretical magnetic fields at various points along the x axisGraph 2 compares the curve as expected based on the results as predicted in the equations section (in red) versus the experimental results (in blue).

Analysis

The theoretical and experimental uncertainties for the magnetic field in both the experiment with varying currents and varying points along the x axis were determined by calculating the maximum possible variation in either direction by accounting for any uncertainty in the measurements used to calculate the figures. The expected range of error the magnetic field in both experiments was

calculated by combining the uncertainty parameters for the theoretical and experimental values to calculate a “worst case scenario” or maximum possible expected error.

The first set of data points is for an experiment wherein the the point along the x axis where the magnetic field was measured was a constant (x=0, or the midpoint between the two coils), and the applied current was varied. The equation used to predict the magnetic field was equation 17 from the predictions section. Graph 1 plots the experimental magnetic field (measured by a Hall probe) in blue against the theoretical magnetic field predicted by equation 17 in red. While he standard deviation of the magnetic field with varying applied currents at ±3.4x10-5T (with an average percent error of ±63%), was found to fall outside the parameters of ±1.01x10-4T, graph 1 shows that the predicted and measured results followed a similar linear trend.

The data from the second table was collected using in experiment that kept the applied magnetic field constant at 0.05 amps, while changing the location along the x axis where the magnetic field was measured. Equations 16 and 18 were used to predict the magnetic field. The prediction suggests that the magnetic field between two Helmholtz coils is relatively constant. This is depicted in the line showing the theoretical results in Graph 3, which remains mostly level between 0 and 10 cm (the area between the coils), and then falls from 10 to 15cm (outside the area between the coils) . The results were consistent with the predictions, both quantitatively and qualitatively- the magnetic field should increase with proximity to the current producing it, which explains why it falls with distance outside the two coils. The standard deviation of magnetic field at different points along the x axis, at ±3.5x10-

6T (with an average percent error of ±4.53%), was found to within outside the uncertainty parameters of ±1.01x10-4T.

In this experiment, the Helmholtz coils were oriented so that current flowed counterclockwise in both coils, giving a magnetic field pointing out of the page (positive z direction) by the right hand rule. Several sources of consistent human error existed in this experiment, which may explain the the high percentage of error in measurement of magnetic field at different applied currents. The main source of error was in determining by eye the placement of the magnetic field sensor (Hall Probe) to determine the magnetic field. Any deviation from the center (which was determined by eye) would result in a change in a different value. It is also notable that the the magnetic field values for the space between the Helmholtz coils (0~10cm) matched more closely with the theoretical values, while most of the error came from the measurements outside of the coils. A more thorough and accurate experiment would include a mechanism to ensure that the Hall probe was accurately placed for each measurement.

Conclusion

To develop a suitable configuration for the research of magnetotactic bacteria behavior in uniform magnetic fields, the magnetic field between two identical coils of wire separated by a distance equal to

their radii was predicted and experimentally confirmed. A system of Helmholtz coils was used to produce a magnetic field pointing out of the page. The magnetic field prediction at varying points along the x axis was confirmed to within experimental error, with an average percent error of ±4.53%. The percentage error in magnetic fields for varying applied currents, however, fell well outside the expected parameters given by the experimental and theoretical uncertainty values. This can be explained by possible error in the placement of the Hall Probe when recording the magnetic field. In general, however, the magnetic field between the coils confirmed the trends as predicted.