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# HELMHOLTZ COILS

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```									FYSP105 / K2                   HELMHOLTZ COILS

Introduction

In the exercise you’ll measure the spatial distribution of magnetic field of Helmholtz
coils at various currents applied to them. Electromagnetics in general as well as
Helmholtz’s coils can be found in:

•   Grant, I. S. & Phillips, W. R. Electromagnetism. 2nd edition, John Wiley &
Sons, West Sussex, UK, 1990. Ch. 4 (especially pages 139-144)

•   Young, H. & Freedman, R,. University Physics with Modern Physics, 11th
ed., Addison Wesley Longman, 2000. Ch. 28.5 and example 29.10.

1. Theory

1.1. Magnetic field of a wire

Ampère’s law defines the relation between the integral of a magnetic field B around a
closed path C and the current intercepted by the area. A spanning the path C:

r       r             r   r
∫ B ⋅ ds = µ ∫ j ⋅ da
C
0
A
(1)

r
Here j is the density of the electric current and µ 0 = 1,26 *10 −6 Vs Am is the

permeability constant. Thus there is a relation between the total current I

r r
I = ∫ j ⋅ da                                                   (2)
A

r
and the magnetic field B caused by it. The relation
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→    →

∫ B⋅ da = 0
A'
(3)

→
(Gauss’ Law) on the other hand indicates that the flux lines of field B are continuous
when A’ is a closed surface. Near a wire, the flux lines act as in Fig. 1, they have no
beginning or end.

→
Figure 1: Magnetic field B , close to a wire with current I.

Biot-Savart’s law applied to a finitely long piece of an electric wire gives the formula (4)
→
for the magnetic field at a distance r (compare with Fig. 2)..

r r r
r µ 0 Idl × (r − r ')
dB =      r r3                                                              (4)
4π r − r '

r                               r
Figure 2: Magnetic field      dB at point P caused by a piece dl at point Q of an electric wire.
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1.2. Magnetic field of an infinitely long electric wire

The magnitude of the magnetic field B at a distance r from an infinitely long electric
wire can be derived from eq. (4). The result is

µ0 I
B=         .                                                              (5)
2π r

1.3. Magnetic field of electric wire loop

Biot-Savart’s law applied to an electric wire loop with radius R and current I, gives the
magnetic field at distance x at point Q (Fig. 3)

µ 0 IR 2
B=                          .                                            (6)
2( R 2 + x 2 ) 3 / 2

If the loop (a coil or a solenoid) has N turns, the magnetic field is the sum of fields of
individual loops i.e. the total field is N times eq. (6).

Figure 3: Electric wire loop with radius R

1.4. Magnetic field of axial coils

The total magnetic field of two identical axial electric coils (separation L, fig. (4), notice
the selection of positive x-direction) at a distance x can be derived from eq (6):

µ 0 NIR 2         1                         1                 
B=                ( R + x 2 ) 3 / 2 + ( R 2 + ( L − x) 2 ) 3 / 2
 2                                                .
    (7)
2                                                        
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Eq. (7) is valid between the coils as well. Point Q must lie on the axis.

Figure 4. Two identical, axial coils with separation L

Problem: Two axial, identical coils with radius R at a distance L=R form a s.c.
Helmholtz coils. In the experiment you'll use such a pair. Work out the problem a)
before coming for measurements. To solve problem b) is not mandatory but to ponder
about it is very useful. Can you observe any analogia?

a) Sketch the magnitude of the magnetic field B(x), on the axis of Helmholtz coils
between x = [–2L, 3L]. The origin is selected as shown in fig. 4. Use units
µ 0 NIR 2
[B] =               .
2

b) How does B behave at a non-axial point Q?

2. Equipment

The experiment will be done with coils at distance L = 150 mm having diameter 155
mm, turns N =125 each and with equal currents I. The direct current can be obtained e.g.
from Metric LPS 303. Measure B with e.g. Phywe Teslameter. Notice that the active
part of the probe is its tip, functioning properly only in the direction of its axis. In other
directions with respect to direction of B you'll observe the component of B in the
corresponding angle.
FYSP105 / K2                                - 5 -

3. Measurements

First set the reading of the magnetic field probe to zero using the fine tuning button on
the right hand side of Phywe Teslameter. If this is not enough, use the small button on
lower left corner. Use settings for uniform field and the most sensitive scale (20 mT).
Then switch on the current for coils. Notice that the coils may heat up and thus the
current may not stay constant. Keep eye on the current to avoid systematic errors.

Perform the following measurements:

•   Magnetic field B on the axis, in the middle with several currents I ≤ 3 A.

•   Magnetic field B in several positions Q on the axis, between the coils with a
constant current e.g. I = 3 A. Outside the coils you need to measure B only in one
axial side (why?).

•   Magnetic field B in several non-axial points Q around the middle of the coils and in
1 - 2 interesting non-axial points Q. You may choose points 1 - 2 outside the coils as
well.

4. Analysis of results

Verify how well eq. (7) is valid for different values of x and I keeping one of them
constant and varying the other. Compare measured values to theoretical ones. Draw the
theoretical flux densities B(I) (x = constant) and B(x) (I = constant). To the same
co-ordinates plot also the measured values. Do not forget the estimates for calculated
and experimental errors.

How does the non-axial field behave? Try to explain the phenomenon theoretically. Can
you figure out any applications for this?

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