Lab 210_ Magnetic Field of Helmholtz Coils --- Biot-Savart Law

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Lab 212: Measurement of e/m for an Electron Neel Gajera Group ID: Date of Experiment: 11/16/20 Date of Report Submission: 11/23/20 PHYS 121A-023 Professor Kurywczak Partners: Introduction The objective of this lab is to understand the equations of the Biot-Savart law about magnetic field created by a circular current loop or coil. It is also to measure the magnetic field strength of a single coil and a pair of coils as a function of the axial distance. Also to study the magnetic field on the axis of a coil and relate this to the geometry of a coaxial pair of coils applying the Biot-Savart Law. From the Biot-Savart Law, the magnitude of the magnetic field is created by a loop of the radius and carrying current, where the current is in the single turn coil. The magnitude of the magnetic field at the point created by a circular coil with n turns carrying current is given by the equation. Equations B = μ 0 IRsin(θ)/2r 2 B(x) = μ 0 NIR 2 /2(R 2 + x 2 ) 3/2 B net = B 1(D/2 + x) + B 2(D/2 - x) = μ 0 NIR 2 /2[(R 2 + (D/2 + x 2 )] 3/2 + μ 0 NIR 2 /2[(R 2 + (D/2 - x 2 )] 3/2 Experimental Procedure Part I. Magnetic Field of Single Coil A: Explore magnetic field created by a single coil ● Position the magnetic field sensor so that the sensor tip is located exactly at the center of the coil and press the “Tar button on the sensor for zeroing. ● Power on the DC power supply and adjust the output voltage to obtain a current of 0.6 A. ● Determine the current direction through the coil and direction of magnetic field using a compass. ● Use Capstone software to measure the magnetic field magnitude. ● Repeat previous step by changing the current direction through the coil. ● Measure the magnetic field strength at the center of the coil for electrical currents of 0.2A and 0.4A. B: Mapping Magnetic Field Magnitude in an Axial Direction of a Single Coil ● Make sure that the tip of the magnetic sensor is positioned radially and axially at the center of the single coil.

● Move the magnetic field sensor away from the coil by rotating the pulley on the rotary motion sensor until the magnetic sensor tip is about 10 cm from the coil. ● Turn on the DC power supply and adjust the output voltage to get 0.6A current through the coil. ● Begin recording on Capstone and slowly move the magnetic field sensor toward the coil until the magnetic field sensor tip is about 10 cm past the coil. Part II. Mapping Magnetic Field Magnitude in an Axial Direction of Double Coils ● With the DC power supply off, press the tare button on the magnetic field sensor. ● Move the magnetic field sensor by rotating the pulley on the rotary motion sensor until the magnetic field sensor tip in the half way between the two coils. ● Move the magnetic field sensor back away from the coils until it is about 10 cm from the first coil. ● Move the DC power supply on and set the current value of 0.6 A. ● Record on Capstone and slowly move the magnetic field sensor along the x-axis direction until the magnetic sensor is about 10 cm past the second coil. ● Change the separation between the two coils to 1.5 times the radius of the two coils. Results Part I. Magnetic Field of Single Coil A: Explore magnetic field created by a single coil B: Mapping Magnetic Field Magnitude in an Axial Direction of a Single Coil mu0 = 4*pi*10^-7; R = 6.8*10^-2; N = 500; I = 0.6; D = [8, 10.3, 12.6, 15]*10^-2; D(1) ans = 0.0800 x = -0.07:0.0005:0.07; B = (mu0*N*I*R^2)./(2*(R^2+x.^2).^1.5); Current (A) through Coil for Case I 0.2 A 0.4 A 0.6 A Magnetic Field Strength (T) 0.542 1.098 1.651

plot(x, B), xlabel 'x', ylabel 'B'

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Part II. Mapping Magnetic Field Magnitude in an Axial Direction of Double Coils mu0 = 4*pi*10^-7; R = 6.8*10^-2; N = 500; I = 0.6; D = [8, 10.3, 12.6, 15]*10^-2; D(1) ans = 0.0800 x = -0.07:0.0005:0.07; B = (mu0*N*I*R^2)./(2*(R^2+x.^2).^1.5); x1 = D(3)/2 - x; B1 = (mu0*N*I*R^2)./(2*(R^2+x1.^2).^1.5); x2 = D(3)/2 +x; B2 = (mu0*N*I*R^2)./(2*(R^2+x2.^2).^1.5); Btotal = B1 + B2; plot(x, Btotal), xlabel 'x', ylabel 'B total'

mu0 = 4*pi*10^-7; R = 6.8*10^-2; N = 500; I = 0.6; D = [8, 10.3, 12.6, 15]*10^-2; D(1) ans = 0.0800 x = -0.07:0.0005:0.07; B = (mu0*N*I*R^2)./(2*(R^2+x.^2).^1.5); x1 = D(1)/2 - x; B1 = (mu0*N*I*R^2)./(2*(R^2+x1.^2).^1.5); x2 = D(1)/2 +x; B2 = (mu0*N*I*R^2)./(2*(R^2+x2.^2).^1.5); Btotal = B1 + B2; plot(x, Btotal), xlabel 'x', ylabel 'B total

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Discussion & Analysis 1. Does the theoretical plot fit everywhere over the experimental plot on the graph? The theoretical plot does not fit everywhere over the experimental plot. This is due to the fact that the equations were graphed between certain distances which means the equations had to be rooted at the endpoints of the interval. 2. How does changing the coil separation affect the magnetic field? What coil separation creates the most uniform magnetic field between the two coils? Look at your plots and explain why the Helmholtz arrangement (D=R) is different from the other ones.

The change in separation of the coils, changes the magnetic fields. The coil separation that provided the most uniform, magnetic field between the two coils was R distance. The Helmholtz arrangement (D=R) is different from the other ones because there is more distribution of higher magnetic field strength towards the center of separation. Conclusion This experiment helps give a better understanding of the Biot-Savart regarding the magnetic field created by a circular loop (coil). It also helped find the magnetic field strength in a single coil and a pair of coils.

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