Experiment 2-6. Magnetic Field Induced by

Electric Field

- Biot-Savart law and Ampere’s Law -

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.

Purpose of Experiment

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.

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)

Experimental Method

- 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 Bz(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 = cm

Distance from center

d(cm)

angle θl(o) angle θr(o) Axis-component

magnetic field Bz(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 Bz(G)

Magnetic field according to

theoretical formula B(G)[cf]

[cf : Use the equation for the circle (9) or for the square ring.]

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 right-

handed screw when turn a screw in the direction of displacement vector.

Backgrounds theory

Fig.1

This is Biot-Savart law and expressed by :

(1)

[cf : Considerate this law is inverse square of distance r]. T.m/A we call it

magnetic permeability.

We can get magnetic field from whole portion of wire from sum magnetic field of the each

portion vector :

(2)

Specially, a magnitude of magnetic field from infinitely long wire is :

(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.

3

0

4

ids rdB

r

7

0 4 10

0

34

ids rB dB

r

0

2

iB

d

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

(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 :

(5)

0 0 1 2B ds i i i

00

2

iB ds ds i

r

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

(6)

So, we can know that the size of the magnetic field inside the solenoid is

(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).

0

b

aB ds Bds Bh nhi

0B ni

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

(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

0

1cos cos

2r lB ni

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)

(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

(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

2

0

3/22 22

INRB z

R z

0

1/22 24 / 2

INLB

d L d

(11)

And for the point P in the ring plane, the magnetic field is

(12)

And the magnetic field contributed by two straight lines in the horizontal direction is

(13)

on the outside the ring plane. On inside the ring plane the magnetic field is

(14)

Is it possible to derive these equations by Biot-Sabert's law? What about other arbitrary

points?

3D Hall-sensor

0

1/22 2 2 2

1 1

4 / 2 / 2

INLB

d L d L d L L d

0

1/22 2 2 2

1 1

4 / 2 / 2

INLB

d L d L d L L d

0

1/22 2 2 2/ 2 / 2

IN L d dB

L L d L d

0

1/22 2 2 2/ 2 / 2

IN L d dB

L L L d L d

Things to think about

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

(15)

If we denote the spherical coordinate system (B, θ, φ) for the magnetic field as shown in

Figure (b), B is

(16)

(17)

(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

x y zB B i B j B k

1/2

2 2 2

x y zB B B B

1/2

1

2 2 2cos z

x y z

B

B B B

1tan /y xB B

Hall voltage is applied to the same magnetic field), and the direction of the three axes of the

sensor should be known when measuring.

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

References