How to Measure the Horizontal Component of Earth's Magnetic Field

The aim of this experiment is to find the magnitude of the magnetic field produced by a current carrying circular coil. Using the experimental values obtained from the following experiments, it is possible to estimate the horizontal component of the earth’s magnetic field.

In Part A of the experiment, a function is derived that represents the tangent of the change in the angle shown on a compass positioned in the center of a circular coil in terms of the number of turns in the coil.

Part B of the experiment is intended to derive a function that represents the tangent of the change in angle of a compass needle positioned in the center of a circular coil in terms of the input current.

The variables being used in each experiment are as follows:

Θ = The number of degrees the compass needle is displaced when the coil is charged.
i = current in Amperes.
N = the number of turns of the coil.
Bh = estimation of the horizontal component of the earth’s magnetic field
Be1 = An estimation of the horizontal component of the earth’s magnetic field based on part A
Be2 = An estimation of the horizontal component of the earth’s magnetic field based on part B.
D = Diameter of the coil

M = 1.257*10^-6 (The permeability of free space)
m1 = Slope of tan(Θ) vs N
m2 = Slope of tan(Θ) vs i

 
Experimental Method

A tightly wound circular coil, with a diameter of 0.39m was attached to a power source capable of producing a current ranging from 0 to 1 Ampere. The circuit was designed in such a way that the number of active turns in the coil can have a value of 5, 10, 15, 20, or 25.

A compass was placed on the North-South plane, positioned as close to the center of the coil as possible so that the compass needle pointed north at 0 degrees.

Part A:

1. The coil was charged with a fixed current of 0.092 A.
2. For values of N ranging from 5 to 25, increasing by multiples of 5 each time, values of Θ were recorded.
3. Values of tan(Θ) were recorded to retrieve a linear set of data.
4. Linear regression, using the ANOVA method, was performed on tan(Θ) vs N to get a function that returns tan(Θ) given N.
5. Calculate the horizontal component of the earth’s magnetic field from the equation:
   Be1 = Mi/m1D

Part B:

1. Set N with a fixed value of 25 turns.
2. Record values of Θ while the coil is charged at various currents ranging from 0.2 A to 1.0 A, increasing the current by 0.2A each time.
3. Values of tan(Θ) were recorded to retrieve a linear set of data.
4. Linear regression, using the ANOVA method, was performed on tan(Θ) vs i to get a function that returns tan(Θ) given i.
5. Calculate the horizontal component of the earth’s magnetic field from the equation:
   Be1 = MN/m2D

Results:

Data Analysis

The data for Part A was stored in a data array consisting of two columns, x1 and y1. X1 is a list of the values of N used in Part A of the experiment. Y1 is a list of the tan(Θ) results from part A.

The data for Part B was stored in a data array consisting of two columns, x2 and y2. X2 is a list of the values of i used in Part B of the experiment. Y2 is a list of the tan(Θ) results from part B.

Both sets of data were linear and could be represented by the standard linear equation y = mx+c. The ANOVA (analysis of variance) method was used to determine several useful pieces of information in terms of the each regression model. The following set of equations were used to obtain the slope for each part of the experiment.

meanx = sum(x)/len(x)                               

meany = sum(y)/len(y)                              

meanxx = sum(x*x)/len(x)                            

meanx = sum(x*y)/len(y)                          

m = ((meanx*meany)-meanxy)/((meanx**2)-meanxx)    

c = meany-m*meanx

Determining the Horizontal component of the earth’s magnetic field:

Beh = cos(dip)*(permeability of free space)

        = cos(56)* 5.0*10^-5

        = 4.26610053861292e-05

be1 = (M*i)/(D*m1)

        = (1.27*10^-6 * 0.092)/(0.39*0.00826)

        = 3.5898677593592835e-05

be2 = (M*N)/(D*m2)

        = (1.27*10^-6 * 25)/(0.39*2.0715)

        = 3.8897862938413255e-05

 

Experimental Discussion

Comments concerning consistency of the data with theory: Most of the variance in the data was accounted for in both parts of the experiment, but the model resulting from part A proved to be a tighter fit than that of part B, while Part B had a more realistic representation of the y intercept. Part A’s y intercept was above zero even though theoretically it should have been on zero because a coil with zero turns isn’t a coil anymore. This is probably due to a combination of errors, for example, by  relying on humans to place the compass in the center of the coil without any obvious guidelines it is possible that the compass be offset from the center of the plane in the x or y direction by some amount. Also, while the current is not charged, the compass is still receiving input from the earth’s magnetic north, so it is necessary that the compass be placed so that it reads zero, or at least the experiment is run to take into account the inherent possibility of error in compass placement.

The horizontal component of the earths magnetic field was estimated to be 4.266*10^-05 T based on a permeability of free space of 1.257.0* 10^-6  and a dip of 65 degrees.

The horizontal component of the earths magnetic field was estimated to be 3.740*10^-05 T based on the average of both Part A and Part B’s results. 

Part A and Part B were both different estimates of the Beh , and it is assumed that if the data was taken again on a larger coil and with a compass capable of more accurate degree readings, the given value of beh might be closer to the experimental values, yet would still have a cartain low level of variance. This is because the equipment that was used to record the data was all hand positioned and hand designed. The magnetic field lines produced by a coil can differ drastically depending on the way it was built. A tighter coil with more turns and a larger diamerer would reduce the possibility of human error regarding placement of equipment. 

Conclusion

The coefficient of determination of the Part A of the experiment was 0.995 showing that the data matched the linear model with a high level of accuracy. 

The coefficient of determination of the Part B of the experiment was 0.983 showing that the data matched the linear model with a high level of accuracy, yet an r squared value of 0.99 or greater would be preferred.

Both models fit the data well and were adequate in producing an amateur estimation of the earth’s magnetic field based on current input and number of coil turns.