238 MAGNETOCHEMICAL INVESTIGATIONS

MAGNETOCHEMICAL INVESTIGATIONS. P A R T VII. CALCULATIONS O N T H E MAGNETIC FIELD.

BY (THE LATE) DONALD V. TILSTON.

Received 10th April , 1946.

In earlier parts the experimental calibration 1, of the magnetic field and its comparison with the calculated curves relating field strength to the distance from the centre of the pole-gap have been discussed. This paper will describe the method of calculation. An expression for the field strength at a point on the vertical line through the centre of the pole-gap is derived and integrated between finite limits related to the extent and shape of the pole-piece. The number of cases capable of solution becomes limited by mathematical difficulties to the following four : (a) magnet with point poles ; (b) magnet with line poles ; (c) magnet with square poles (vertical line parallel to the side of the square) ; and (d) magnet with square poles (vertical line parallel to diagonal of the

FIG. I.

squari) . The generaltechnique of integration is similar for all except (a) which requires no integration. Consequently, it is proposed to discuss here only case ( d ) , which, whilst the most complicated of the four cases, does contain all the details of the other integrations.

Calculation of the Fieldldis- tance Curve along the Diagonal of a Square Pole-Piece.-Consider a unit North pole a t the point P (Fig. I ) , which is situated on the diagonal y = z. EFGH is a square pole-piece (North) of a magnet, side zb, parallel to, and distant -a from, the zOy plane; and there is a similar (but South) pole-piece lying parallel a t x = a. Let the magnetisation per unit area of EFGH be I . The force, z d F , in

the direction parallel to Ox which is exerted on the unit pole at P bythe element of area dydz at A on the strip MN, width dz, distance z from yOx,

Angus and Hill, Trans. Faraday Soc., 1943, 39, 185 (Part I). Angus and Tilston, ibid., preceding paper.

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D. V. TILSTON 239 and by A' (its image with respect to the zOy plane) will also be the field strength d H 2 due to A and A' and can be expressed as

The total field strength at P, H,, will be the double definite integral of this function, taken between the limits y = 6 b and z = &- b.

i b i b

( 2 ) H , = ! I d H z = ZSJdF, = - 2aI dz dY 1 [ { (h - y)2 + a2 + ( k - ~ ) ~ } 3 / 2 D D - b - b

It is more convenient to consider the two integrals separately : + b

dY [ { (h - Y ) ~ + a2 + ( k - ~ ) ~ } 3 / 2 ;

- b

Let

Then

B u t it can b

equation (3), r

a2 + ( k - = A2

(y - h) = X tan 8.

dy = A sec2 8 . d8 and

X sec5 8 . d8 y = + b y = + b

1 = I ( X 2 tan20 + X2)3/2 = - $ rsin '1 '

y= - b y = - b

shown that sin 0 = - and hen {(y - hI2 + X 2 P

1 J = -A[ (b - 4 (b + 4 X2 ( (b - h ) 2 + h2}* + { (b + h)2 + h2}* .

e, from

Substituting for A, and returning to equation ( z ) , we may reconstruct the integral with respect to z, splitting it into two portions. We then obtain

+b dz

H , = z a I ( b - h ) I (a2 + ( k - 2 ) 2 } {(b - h)2 + a2 + ( k - 2 ) 2 } ) + (44 -6

+ b

Consider the first of these integrals, (4a) ; let k - z = w, then - dz = dw : when z = b, w = k - b and when z =- b, w = k + b. On substitution the integral now becomes

k f b

Now, let w = a tan 4, dw = a sec2 4 . d4, and a2 + w2 = a2 sec2 4. integral becomes

The

w=k+b a sec2 4 . d+ s= aaI (b -

h)Ia2 sect 4{(b - h)2 + a2 sect +I* W = k - b

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240 MAGNETOCHEMICAL INVESTIGATIONS

Similarly for (4b)

Now

and as w = a tan 4, sin 4 = sin tan-1 w/a . (8)

Hence from (4), (5) , (6) , (7) and (8) and substituting the limits of ( 5 ) and (6)

k + b H , = 21 sin-1 sin tan-' (b+) . sin tan-1 ( 7 ) } c i b - h k - b )

- sin-'{ sin tan-' (,-> . sin tan-' ( ,-> } ) . sin tan-'( k + b y), 1

- sin-l{sin tan-'( b + h y) . sin tan-1 (y)}] k - b . . (9)

Reverting to the original argument it will be seen that, for all such points as P, y = 2, and, hence, h = K = 8, where 5 = 2/0P and, since

sin tanL1(- x) = sin (- tan-1 x ) = - sin tan-lx

equation (9) becomes

Equation (10) relates H,, the resultant field strength at P, to 6, a linear proportional function of OP, the distance from the centre of the pole-gap. Table I shows the corresponding €unctions for the other types of pole- pieces studied.

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24 1 D. V. TILSTON Comparison of these curves with the experimental curve shows good

agreement, except at large distances, where the theoretical values ap- proach the distance axis much more rapidly than do the experimental.

TABLE I.-FIELD-DISTANCE CURVES, CASES (a) TO (c).

Type of Pole-Piece.

(a) 2u apart, pole strength = I

(b) zu apart, length 2b, pole strength per unit length = I

Distance Variable.

y

y

I- (c) zu apart, side 2b, pole

strength per unit area = I

z (c) zu apart, side 2b, pole

strength per unit area = I

2aI (a2 + y2)3/2

z 41[ sin-1 {sin 4 sin tan-1 -

- sin-l(sin 4 sin tan-1 - (" ; ")> (" : .")}I*

Type of Magne tis ation.

This may be pieces is not the values of the variation

H,.

explained by the fact that magnetisation with real pole- confined to the area within the pole-gap. Table I1 gives

H , calculated for case (b) with various assumptions regarding of the magnetisation with distance along the pole-piece,

and assuming that magnetisation on the top and bottom of the pole-piece can produce a field within the pole-gap. I(=, = pole strength per unit length at points (x , f b) such that

x < - a, a < x .

1, = magnetisation at the four points ( f a, f b) .

* sin + = b / ( b 2 + a2)+.

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242 MAGNETOCHEMICAL INVESTIGATIONS

the experimental when this last effect is included. The graphs (Fig. 2-4) show how closely the theoretical values support

FIG. 2.-Field-distance curves.

(i) Exptl. curve (ref. I , Table

(ii) Calc. curve from Table I, case (b ) , with a = 0.05 cm., b = 2-54 cm.

(iii) Difference of curves (i) and (ii).

11).

An extension of this work to disc-shaped pole-pieces was attempted, using the methods of both Cartesian and radial analysis, but the integrals

FIG. 3 .-Field-distance curves.

(i) As in Fig. 2 (i). (ii) As in Fig. 2 (i).

(iii) Calculated curve, Table I, case (c), with a = 0.5 cm., b = 2-54 cm.

proved too complex for solution. An idea of the curve which would be produced by a disc is obtained by comparing the curves for squares

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D. V. TILSTON 243 inscribed within (field strength along a line parallel to a diagonal) and outside the circle (field strength along a line parallel to a side). No very great difference exists between these two curves.

Recent papers by Bates and co-workers give an account of a very elegant method of calculating the shape of pole-pieces required to produce

FIG. 4 .-Field-distance curves.

(i) As for (i) in Figs. 2 and 3. (ii) Calc. curve, Table I, case (b ) . (iii) Calc. curve Table 11, I,,, b ) =

(iv) Calc. curve, being sum of curves

(Curve (iii) was calculated using values of 1, derived from the y = 3.0 t o y = 6.0 region of curve (iii), Fig. 2. The curve (ii) was derived using a value of I calculated from the field a t y = o after subtracting that produced by curve (iii) above.

I,a/x.

(ii) and (iii).

a field with given characteristics but the method has little in common with the present work.

This work was carried out during the tenure of a Department of Scientific and Industrial Research maintenance grant.

Summary. The calculation of curves relating field strength to the distance from

the centre of the pole-gap is described for four types of pole-piece, viz. point poles, line poles, square poles with the vertical line (a) parallel to the side of the square and (b) parallel to the diagonal of the square.

R6sum6. On calcule les courbes de l’intensitk du champ, en fonction de la dis-

tance 8. partir du centre de l’entrefer, pour quatre types de pihces polaires : pales points, p6les lignes, pales carrks avec la verticale parallhle : ( I ) au c6t6 du card et (2) 8. sa diagonale.

Zusammenfassung. Die Berechnung der Kurven fur die Beziehung zwischen Feldstarke

und Entfernung von der Mitte der Liicke zwischen den Polen wird fur vier verschiedene Formen von Magnetpolen beschrieben, u.zw. punkt- formige, linienformige, qusdratische mit der senkrechten Linie parallel zu einer Seite des Quadrats oder parallel zur Diagonale des Quadrats. Department of Chemistry,

Bangor. University College of North Wales,

Bates, Baker and Meakin, Proc. Physic. Soc., 1940, 52, 425 ; Bates and Somekh, ibid., 1944, 56, 182 ; Davy, Phil. Mag., 1942, 33, 575.

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