Labotorio experimentacion

Experiment II: Magnetic Fields due to Currents
Yue-Kin Tsang
(Dated: March 15, 2004)

1

I.

INTRODUCTION

Electric currents generate magnetic fields. The magnitude and direction of the magnetic field generated depend on the specific geometry of the wire in which the current is flowing. In this experiment, we study three different types of geometry: (1) a circular coil, (2) two circularcoils with current flowing in either the same or opposite directions and (3) a torus. Using the Biot-Savart Law and Amp`re’s Law, the magnetic field B of these configurations e can be calculated: 1. a circular coil of radius a with current I, B= µ0 NI a2 2 (a2 + x2 )3/2 (1)

where µ0 is the magnetic permeability of vacuum, N is the number of loops in the coil and x is the distance from the center ofthe coil along the axis of the coil. 2. two circular coils separated by a distance of 2b, B= µ0 NI 2 a2 [a2 + (x − b)2 ]3/2 ± a2 [a2 + (x + b)2 ]3/2 , (2)

the origin is taken to be at the midpoint between the two coils. 3. a torus, µ0 NI 1 2π R where R is the distance from the center of the torus. B=

(3)

We experimentally verify Eq. (1) – Eq. (3) and also use them to determine some systemparameters.
II. APPARATUS AND PROCEDURE

Apparatus 1. Battery eliminator power supply 2. Circular coils 3. Toroidal coil 4. Gaussmeter 5. Digital multimeters Procedure The circuit shown in FIG. 1 was setup.

2

+ Power Supply _

A

Switch

Coil

FIG. 1: The circuit used to generate magnetic field in the experiment A. A single circular coil

The Gaussmeter was first calibrated forthe longitudinal Hall probe. With the Hall probe placed at the center of the coil, the strength of the magnetic field was measured as a function of current. The direction of the current was reversed and the direction of the magnetic field was noted. The current was then fixed at 7.5 A. The center of the coil was located as the position where the magnetic field strength was maximum. Then, the magneticfield strength was measured as a function of position along the axis of the coil. The number of turns in the coil was counted.
B. Two circular coils

The circuit was setup so that a current of 5.05 A was flowing through the two coils in the same sense. The magnetic field strength as a function of position along the axis of the coils was measured. The circuit was then modified so that a current of 5.05A was flowing through the two coils in the opposite sense. The magnetic field strength as a function of position along the axis of the coils was again measured.
C. A toroidal coil

A current of fixed magnitude 10.15 A was setup to flow through the torus. The Gaussmeter was then calibrated for the transverse Hall probe. Using the transverse Hall probe, the magnetic field strength was measured as afunction of R. The number of turns in the torus was also counted.
III. A. DATA AND NUMERICAL ANALYSIS A single circular coil

The measured values of the magnetic field strength B and current I, along with their uncertainties σB and σI are shown in TABLE I. FIG. 2 shows the plot of B versus I. We fit the data with a straight line. The y-intercept is forced to be zero in the fitting process.

30.020 data linear fit 0.015

B (T)

0.010

0.005

0.000

0

2

4

6 I (A)

8

10

FIG. 2: Plot of B versus I. The red line is a linear fit to the data

From Eq. (1), the number of turns N is related to the slope m of the linear fit as follow, N= 2am . µ0 σm = 0.00004 T/A σa = 0.0005 m . (4)

The measured values of m and a and the corresponding uncertainties σm and σa are,m = 0.0019 T/A , a = 0.034 m , By Eq. (4), we obtain N = 101 (to the nearest integer) 2 2 2 σN = a2 σm + m2 σa = 3 . µ0
TABLE I: magnetic field strength B at different current I for a single circular coil I (A) 1.15 1.98 3.17 3.75 4.95 6.03 6.96 8.30 8.90 10.07 σI (A) 0.04 0.05 0.07 0.08 0.09 0.11 0.12 0.14 0.15 0.17 B (×10−4 T) 21 35 58 71 92 114 130 155 166 190 σB (×10−4 T) 4 4 4 4 4 12 12 12...