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Physics Laboratory 2
last update: 2009. 8. 31
Exp. 5. 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.
Outline of Experiment
A rectangular coil with flowing currents makes magnetic field around. In this
experiment, the objective is to confirm formation of magnetic field. We use the
equipment called Hall Sensor from many magnetic-measuring equipment to measure
and indicate the magnetic field.
To use hall sensor, we should do calibration whenever program is initiated. We contact
calibrated hall sensor around the radial plane and measure. Objective of experiment is
measuring the magnetic field of each point varying angle and distance.
Experimental Method
These equipments are prepared in the laboratory. (Parentheses mean the number of them.)
Rectangular coil(1)
Power supply(1)
Solenoid(500 winded, 1)
Magnet(1)
Compass(1)
Hall Sensor(1)
Hall Sensor source(1)
Computer(1)
Radial plane(1)
30cm ruler
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.
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
Experimental Setup (video link)
(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).
(1) Get a winding number density from length and winding number of solid, and
calculate the current making magnetic field inside of the solenoid to be 2mT.
(2) Inside of the solenoid, flow the current to make a 20 G magnetic field with outwardto-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.
 Using the compass, confirm the direction of magnetic field made by solenoid.
(3) 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.
 Use DC mode in the power source, don’t exceeds 1.5A currents flowing solenoid and
don’t keep a long time with exceeded 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.
 You could know the direction of currents using compass.
(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.
To write the experiment note, that method is encouraged :
1.
Calibration of hall detector or 3CH Hall Sensor AMP, and measuring of magnetic field on the
center of solenoid.
 Current of solenoid
 Winding number of solenoid
 Length of solenoid
2. Measuring of magnetic field varying the place on the axis of solenoid.
3. Measuring of magnetic field varying the place on the axis of rectangular coil.
Background 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.
This is Biot-Savart law and expressed by :
[cf : Considerate this law is inverse square of distance r]. (
) we call it
magnetic permeability.
We can get magnetic field from whole portion of wire from sum magnetic field of the each
portion vector :
Specially, a magnitude of magnetic field from infinitely long wire is :
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.
We can confirm constant by thinking long infinite currents I and closed curve horizontal with
wire.
The magnetic field from each point of curve is given (3) and direction is tangential, so you
can confirm from :
Although Ampere’ s law tells same thing as Biot-Savart law, it can be used
conveniently when distribution of magnetic field has symmetry because de
We can expect that the magnetic field at point P on the axis of center of the
rectangular solenoid almost same as a case of circular solenoid if point P is
sufficiently longer than size of the solenoid. It is calculated – in Physics book –
that the magnetic field at perpendicular distance z from circular solenoid with
radius R, winding number N and currents I is parallel to axis and magnitude is
given by – as fig 6(b) ;
Let’ s calculate the magnetic field at point on the axis of center at perpendicular
distance z from rectangular solenoid with length L, winding number N.
When currents I is applied in wire, the magnetic field at P in y-axis is given by:
Including vertical magnetic field at other side, the magnetic field at the point P
outside a loop is given by;
Inside a loop is given by;
Also, the magnetic field at the point P outside a loop contributed by two horizontal
part is given by;
Inside a lop is given by;
Can you induce these formula using Biot-Savart law? How about other arbitrary
points?
-3-dim Hall sensor
To use 3-dim Hall sensor used this experiment, attach three same small hallsensors to the plane of a cube perpendicular, sense the each element of magnetic
field. I.e, each element of x,y,z axis sets Bx,By,Bz then vector B is given by;
And expressing that as spherical coordinate like fig (b) it is given by;
So if you know x,y,z direction of Hall sensor, you can know magnitude and
direction of magnetic field without rotate the hall sensor. You should use a same
size – when same currents is applied, make same voltage – hall sensors, know
the direction of axis when measure.
References

Constant current(voltage) power supply

Measurement of specific heat of an object - calorimeter

Measurement of temperature by the thermocouple and the digital thermometer

Treatment of measurement data

Analysis method based on the graph

James Joule - The importance of precise measurement
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