Experiment 2-6. Magnetic Field Induced by Electric Field - Biot-Savart law and Ampere s Law - Purpose of Experiment We introduce concept called charge to describe electrical phenomenon. The simplest electrical phenomenon is electrostatic(deal with only static charge) problem. Before 19C, we recognized that electrical and magnetic phenomenon don t have any relation each other because what we could do is only dealing with electrostatic problem. But, in 1820, Orsted observed that a magnetic needle around electric wire which is flowing current is affected. Thus the currents were sources of magnetic field. We can get direction and magnitude of magnetic field induced around wire which is flowing currents by Biot-Savart law. But if wire s shape is slightly complicated, it is almost impossible to calculate magnetic field. In this experiment, we examine how the magnetic field is formed around the most simple shaped conductor when there is current flowing through it. You do not need to check it quantitatively, but you can check to see if the magnetic field follows the Biot-Savart law while changing the distance from the wire or changing the current flowing. Outline of Experiment
When a current flow through a square coil, a magnetic field is formed around the square coil. In this experiment, it is aimed to confirm what kind of magnetic field forms on the radial plane. There are various ways to measure the magnetic field. In this experiment, the Hall sensor is used to measure the magnetic field and display it on the screen. In order to use the Hall sensor, it is necessary to go through a process called calibration every time the program is initialized. It is the goal of the experiment to measure the magnetic field at each point while varying the angle and distance. Experimental Method These equipments are prepared in the laboratory. (Parentheses mean the number of them.) - Square coil (1) - Power supply (1) - Solenoid (500 turns, 1) - Magnet (1) - Compass (1) - Hall sensor (1)
- Hall sensor source (1) - Computer (1) - Radial plane (1) - 30cm ruler (1) If you need more stuff, inquire to your teaching assistant or experiment preparation room (19-114), or prepare yourself. The following is a recommended experiment method. The recommended standard test method is as follows. Ⅰ. Calibration Unlike with Gauss meter, Hall sensor doesn t measure precise magnitude of magnetic field as an absolute value in initiated phase. You should specify the initial value and this course is called the calibration a) Offset calibration : : this is exist to make sensor output 0 in the non-magnetic field region. (1). Switch the hall sensor source and computer, and drive a measuring program. Fix the position of hall sensor far from electric equipment(computer, monitor, power source, etc.). (2). Confirm the x, y, z number appearing monitor are below 0.1V and press Zero in the Calibration menu, then computer remember this value and will subtract from measured value. (3). If number exceeds 0.1V, rotate the corresponding adjustment grid to make number below 0.1V and calibrate.
CH1, CH2, CH3 correspond x, y, z component respectively. Caution : If you use a sensor which numbered differently with amp you may couldn t blow 0.1V, so be careful. If you can t make below 0.1V, inquire the experiment preparation room and take complementary measure. Already offset calibration may did. But if someone handled the power source, you should this course again. b) Hall sensor calibration : Although hall sensor generates the hall voltage in proportion to magnetic field, proportion constant can be differ by kind of sensor, current flowing in the sensor, and temperature of sensor. So calibration is needed to convert voltage to magnetic field. Objective of this course is to output 20G magnitude when 20G magnetic field is applied. Hall sensor calibration should be done after doing (a). ⅰ. Get a winding number density from length and winding number of solienoid(500), and calculate the current making magnetic field inside of the solenoid to be 2mT((=2 10-3 T = 20 G). ⅱ. Inside of the solenoid, flow the current to make a 20 G magnetic field with outward-to-hall sensor direction, put a hall sensor inside solenoid parallel with axis of solenoid and press Z-axis in the Calibration menu in the monitor. [Video : Magnetic Field] Using the compass, confirm the direction of magnetic field made by solenoid. ⅲ. Wait 2~3 seconds until z-direction magnetic field indicates 20G. Reading that fields value, computer remember the constant which transfer z- direction magnetic field measuring hall-sensor voltage to 20, and it will apply this constant measuring of x, y, z hall-sensor. These measured values will have G(gauss) unit. If you calibrate other magnetic field value is not 20G in any reason, computer also read that wrong magnetic field as 20G. So to get right magnetic field value you should multiply the ratio of initial magnetic value(computer remember as 20G) and
measuring value. Ⅱ. Measuring the magnetic field in the solenoid a) Measure the magnetic field at center of the solenoid with varying currents and confirm dependent of currents. Measure with varying direction of currents Measure the magnetic field at center of the solenoid with varying currents and confirm dependent of currents. Measure with varying direction of currents. b) Fix the current and measure the magnetic field on the solenoid axis. Is it applied well with theoretical result? Ⅲ. Measuring the magnetic field on the radial plane. a) Connect the wire to the rectangular coil equipment and make the currents flowing. Connect the wire to the rectangular coil equipment and make the currents flowing. b) Take the hall sensor which calibrated any desired point on the radial plane and click the corresponded point on the monitor using mouse. If you take the sensor on the radial plane without consideration, you couldn t get a good result. You can see the three projections if you see the tip of sensor, then you can think that direction of projection indicates x, y, z direction, respectively See the below, left picture) Like the below, right picture, when you measure, at any point make the y point indicates an experimenter and you can get good result on the monitor.
c) Do a course of (b) varying angle and distance. Measure as many as data points. [Axis setting] -> Axis configuration menu can vary the interval of length and angle. If vector sign is too big, adjustment in the [Axis setting] -> Axis configuration menu. Ⅳ. Measuring the magnetic field on the center of rectangular coil. a) Lay down hall sensor horizontally and place at one point of axis of rectangular coil. Read the z-direction magnetic field value. b) Measure with varying point and confirm the change of magnetic field from the distance of plane of the coil. Compare with magnetic field of infinitely long wire. Is there any difference? If there exists difference, can you explain it?? Compare with magnetic field of finitely long wire. Is there any difference? Does difference get smaller compared with case of infinitely long wire? Where comes from additional difference? Compare with magnetic field of two serial long wire in perpendicular direction. If you can, compare with magnetic field of outside of solenoid, magnetic field of at any point on the rectangular coil, and magnetic field of magnet..
It is recommended to write experiment notes in the following way. 1. Calibration of hall detector or 3CH Hall Sensor AMP, and measuring of magnetic field on the center of solenoid Current of solenoid I = A (@Magnetic field B = 20 G) Solenoid turns N = Length of solenoid L = cm Current i(a) Axis-component magnetic field B z(g) 2. Measurement of magnetic field according to the position on the solenoid axis Current of solenoid I = A (@Magnetic field B = 20 G) Solenoid turns N = Length of solenoid L = Distance from center d(cm) cm angle θ l( o ) angle θ r( o ) Axis-component magnetic field B z(g)
3. Measurement of magnetic field according to position on the axis of a square wire loop Current of leading wire I = 1 A Wound number of leading wire N = Distance from ring face d(cm) Axis-component magnetic field B z(g) Magnetic field according to theoretical formula B(G) [cf] [cf : Use the equation for the circle (9) or for the square ring.] Backgrounds theory Investigating the character of magnetic field made around the currents flowing wire, you can know that like pic 1., magnitude of the magnetic field db in the point P from portion of wire ds is proportionate to currents i flowing in the wire, inverse proportionate to square of distance from wire(ds) r, and proportionate to sine value of angle theta between currents and displacement vector. And, direction of magnetic field is the propagation direction of righthanded screw when turn a screw in the direction of displacement vector.
Fig.1 This is Biot-Savart law and expressed by : 0 ids r db 3 4 r [cf : Considerate this law is inverse square of distance r]. magnetic permeability. (1) 0 410 7 T.m/A we call it We can get magnetic field from whole portion of wire from sum magnetic field of the each portion vector : B db 0 ids r 3 4 r (2) Specially, a magnitude of magnetic field from infinitely long wire is : 0i B 2 d (3) And it depends only displacement from wire d and independent of other. And, direction of magnetic field is direction of tangent line of circle with d radius. It is right-hand screw propagation direction. Character of this magnetic field can be understand easily from geometrical symmetry had infinitely long wire currents.
Otherwise, characteristic of inverse-square-law magnetic field has convenient nature called Ampere s law. Ampere s law means that if you think arbitrary closed curve, whole currents pass inside of curve is proportionate to integrate the magnetic field vector tracing that curve, as shown in Fig 2 Bds i i i 0 0 1 2 (4) Fig.2 We can confirm constant by thinking long infinite currents I and closed curve horizontal with wire. Fig.3 The magnetic field from each point of curve is given (3) and direction is tangential, so you can confirm from : i 2 r 0 Bds ds 0i (5)
Although Ampere s law tells same thing as Biot-Savart law, it can be used conveniently when distribution of magnetic field has symmetry. Using Ampere's law, you can easily find the magnetic field inside an infinitely long (ideal) solenoid. If a current i flows through an infinitely long and infinitely tight solenoid as shown in Fig. 4, the magnetic field inside the solenoid will be uniform regardless of its position, and the direction will be the axial direction to which the right-hand rule applies to the current. In addition, since the magnetic field outside the solenoid is 0, if we choose the rectangular ring with the length h as closed circuit, as shown by the dotted line in Fig. 4, we can derive Fig.4 b B ds Bds Bh nhi a 0 (6) So, we can know that the size of the magnetic field inside the solenoid is B ni 0 (7) Where n is the winding number per unit length, that is, the winding density of the solenoid. Since the actual solenoid is not infinitely long, the magnetic field outside the solenoid is not zero and is not uniform inside as shown in Fig. 5 (a).
Fig.5 At this time, the size of the magnetic field at the point P on the axis can be obtained easily by considering the solenoid as a group of circular rings. As shown in Fig. 5 (b) we can derive 1 B 0ni cosr cosl 2 (8) Here, the angles θr and θl are angles formed by the line segment connecting the right end and the left end of the solenoid to the center axis at point P, respectively. It can now be expected that the magnetic field at the point P on the vertical axis passing through the center of the bundle of square conductors will not be significantly different from that of the bundle of circular conductors if the position of the point P is very far from the ring size. In a general physics textbook, as shown in Fig. 6 (a), when a current I flows through a
bundle of circular conductors having a radius R and a number N of turns, the magnetic field at a distance of z from the conductor is parallel to the axis, As shown in Fig. 6 (b) B z 0INR 2 2 2 R z 2 3/2 (9) is derived. Fig.6 For a square wire bundle of square shape with length L of each side and number N of turns, calculate the magnetic field at a distance z from the center axis as shown in the above equation (9). When the current I flows through this square wire bundle, the value of y-axis magnetic field at the point P spaced by a distance d on the bisector of one vertical straight line portion on the ring surface is 0INL B 4 d L / 2 d 2 2 1/2 (10) When we consider the magnetic field by the opposite direction current on other vertical straight portion part, for a point P outside the ring plane, the magnetic field is
0INL 1 1 B 4 d L /2 d L d L /2 L d 2 2 2 2 1/2 (11) And for the point P in the ring plane, the magnetic field is 0INL 1 1 B 4 d L /2 d L d L /2 L d 2 2 2 2 1/2 (12) And the magnetic field contributed by two straight lines in the horizontal direction is 0IN L d d B L L d L d /2 /2 2 2 2 2 1/2 (13) on the outside the ring plane. On inside the ring plane the magnetic field is 0IN L d d B L L L d L d /2 /2 2 2 2 2 1/2 (14) Is it possible to derive these equations by Biot-Sabert's law? What about other arbitrary points? Things to think about 3D Hall-sensor
The three-dimensional Hall-effect magnetic field detector used in this experiment is a method in which three Hall-sensors of the same size are attached to the surface of a small rectangular parallelepiped perpendicularly to each other to detect each magnetic field component. That is, if the components in the x, y, and z directions in the figure (a) are Bx, By, and Bz, respectively, the magnetic field vector B is B Bxi By j Bzk (15) If we denote the spherical coordinate system (B, θ, φ) for the magnetic field as shown in Figure (b), B is 2 2 2 1/2 x y z B B B B (16) B 1 z cos 2 2 2 Bx By Bz 1/2 (17) 1 tan By / B x (18) Therefore, if you know only the x, y, and z directions of the hall sensor, you can find out the size and direction of the magnetic field without turning the Hall sensor. At this time, the same standard Hall sensors should be used as possible (i.e. When the same current flows, the same
Hall voltage is applied to the same magnetic field), and the direction of the three axes of the sensor should be known when measuring. References Hall magnetic field censor와 Hall effect Measured data processing method Analysis method by graph Andre-Marie Ampere Role model of theoretical physicist Edwin Herbert Hall - Unfortunate Modern Heroes of American Physics Jean-Baptiste Biot - An outstanding student who was interested in all aspects of physics Felix Savart Supporting role of Biot-Savart s law(?) History of compass The Magnetic Field Magnetic Field Measurements