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is a magnet run through a cork which is
inserted in the lower end of a short and
broad glass tube. The annular space around
the projecting pole N is filled with mer-
cury. A current is led down by the wire A,
through the flexible joint and B into the
mercury cup and out by C. While the
current flows B is surrounded by lines of
force which viewed from A are clockwise.
If B were stationary and N were free to
move, N would travel around B in a clock-
wise direction, that is, N would move out
towards the observer. However, N being
fixed, B moves back from the observer and
travels around AT in a clockwise direc-
Fig. 151. tion.

352. Left Hand Rule for Direction of Motion. The con-
ductor described in the preceding paragraph is in the field of
the magnet and owes its motion to the interaction of this field
with its own. Any conductor carrying a current and placed
in a magnetic field will move if it be ' free to do so. It is
useful to have a rule by which the direction of this motion
can be foretold. The following is such a rule. Place the palm



of the left hand upon the wire, the extended fingers pointing in
the direction of the flow of the current (Fig. 152) and the palm
turned to receive the lines of force of
the field; the extended thumb will point
in the direction of the motion of the con-

353. Intensity of Field About a
Straight Conductor. A magnetic field
is known when we have determined its
direction and intensity. We have
shown above (Par. 347) how to deter-
mine the direction of the field about
a conductor carrying a current. The
intensity may be measured as ex-
plained in Pars. 148-150. In two sim-
ple cases (which fortunately are the
ones most frequently encountered), it
may be calculated. These are, first,
when the conductor is straight, and
second, when it is bent into the arc
of a circle.

In Fig. 153 let A B represent a por-
tion of a straight wire of indefinite length carrying a current of
strength /. (The unit in which / is measured is for the moment
R held in abeyance; see Par. 355.) Let m

represent a unit pole at a distance r from
the wire. The force exerted upon m will
measure the intensity of the field at that
point (Par. 136). Let A represent an
infinitely small section of the wire, its
length being dy. It has been shown by
Laplace that the force exerted upon a
magnet pole by an infinitely short element
of a conductor carrying a current is
directly proportional to the strength of
the pole, to the strength of the current,
to the length of the element, and to the
sine of the angle which this element makes
with the line joining its center and the pole; it is also inversely
proportional to the square of the length of this line. In the case

Fig. 152.



Fig. 153.



represented, the force exerted by A upon the unit pole at m is

* >y2* ^ *

This expression integrated between proper limits will give the
intensity of the field produced at m by the corresponding lengths
of AB.


From the figure x =
also y


tan a

, hence dy = r ( cosec 2 a da)

Substituting these values in (I) and remembering that
cosec a = -r , we obtain



df = - . sin a da
f = - . cos a + a constant.

Taking this between the limits a = and a = 180, we have


or the field at any point

about an indefinitely long straight wire is directly proportional to
the current and inversely proportional to the simple distance from
the wire.

354. Field on the Axis of a Circular Coil. The field produced
at a point on the axis of a circular coil may be determined as


Fig. 154.

follows: With a current of strength 7 flowing as indicated by the
arrow (Fig. 154), the infinitely small portion of the coil at A exerts


upon a unit north pole at P a force in the direction PF which is

^-, dl being the length of A. This may be divided into two

components, one PD = -^- . sin 0, and the other PE. The

diametrically opposite element of the coil at B likewise exerts a
force upon P which may be divided into two components, one in
the direction PD, the other opposite and equal to PE and hence
counterbalancing it. Every element of the coil therefore exerts
in the direction PD a force upon P equal to



The sum of these elementary forces is

1.2 TTT

f = 2" . sm 6

Substituting for sin 6 its value r/x



or, the field at any point

on the axis of a circular coil varies directly with the current and
inversely as the cube of the slant distance.

If the point P be moved to the center of the coil, x becomes
equal to r and the above expression becomes

f = 1.2*

Should the coil consist of n turns, the field produced is n times
as strong as that produced by. one turn, therefore, the above
expressions for the field must be multiplied by n.

An important consequence follows from the foregoing. Since
the field at the center of a circular coil varies directly with the
current, the measure of the field may be used as a measure of the
current. This will be shown in the following paragraph.

355. Absolute Unit of Current. Since, as has just been seen,
we obtain the intensity of the field at the center of the coil by
adding up the effects produced by each infinitesimal section of
the coil, the field produced by a portion of the coil must be


directly proportional to the length of this portion, or, if this
length be I


If in this expression we make r and I each one centimeter, we


and if / be one dyne, / is

unity, whence we derive at once the definition of the absolute unit
of current as that current, which flowing through one centimeter of
a conductor bent into the arc of a circle whose radius is one centi-
meter, exerts a force of one dyne upon a unit pole placed at the
center of the circle.

If in Fig. 155 the length of the
conductor from a to 6 be one
centimeter and if it be bent into
the arc of a circle of one centi-
meter radius, the current which
flowing through this conductor
exerts a force of one dyne upon
the unit pole at m, has a strength
of one absolute unit.

The absolute unit of current, as will be explained later (Chap.
39), is ten times as great as the practical unit, the ampere, or,
one absolute unit equals ten amperes. Therefore, in applying the
expressions in Pars. 353 and 354, if / be given in amperes, it must
be reduced to absolute units or divided by ten in order that /
should be in dynes.

356. Force Exerted by a Magnetic Field upon a Conductor
Carrying a Current. The force exerted upon a unit pole at m by
the field of ab (Fig. 155) is shown to be


If the strength of the pole be m instead of unity, the force is

._ m.I.l

J ~ ~2



If the current flows as shown in the figure, and if m be a north
pole, this force acts upward. An equal downward force acts


upon ab. In the above expression -y is the field along ab due to
the pole m (Par. 136) and is uniform. Calling this H, we have


or, the force exerted by a

magnetic field upon a 'con due tor carrying a current and at right
angles to the field is proportional to the current, to the intensity
of the field and to the length of the conductor. This force is at
right angles to the field and to the conductor and, as explained
in Par. 355, is expressed in dynes when I is in absolute units.

Fig. 156 represents a cross-section of such a conductor lying
in a field NS. If the current is flowing away from the observer,


Fig. 156.

the lines of force about the wire are clockwise, therefore, on the
upper side of the wire they coincide in direction with those of the
field but on the lower side they are opposite in direction. The
field is therefore distorted as shown, the lines thickening up above
the wire and thinning out below. Since lines of force have a ten-
sion in the direction of their length, or a tendency to shorten, the
result is that the wire is urged downward. Application of the left
hand rule (Par. 352) indicates this downward motion.

357. Work Done in Moving Across a Magnetic Field a Con-
ductor Carrying a Current. From the preceding paragraph, the
force exerted upon a conductor carrying a current and lying at
right angles to the field is I. H.I dynes.


If the conductor be moved at right angles to the field and to its
own length, it will move either against this force or with it. In
the first case, work must be done upon the conductor; in the second
case, work is done by the conductor. In either case, if the dis-
tance moved be x centimeters the work, being force X path, is
W = I.H.l.xergs

But l.x is the area in square centimeters swept over by the con-
ductor in its movement, H is the number of lines of force per square
centimeter (Par. 145), therefore H.l.x is the total number of lines
of force cut by the moving conductor. Placing this equal to N we


W = LNergs

or the work done in mov-
ing across a magnetic field a conductor carrying a current of I
absolute units is equal to the product of the current into the num-
ber of lines of force cut.

358. Work Done in Moving Across a Magnetic Field a Coil
Carrying a Current. This is merely a particular case of the fore-
going but furnishes conceptions which facilitate the application
of the principle in certain deductions which we shall make later on.
Suppose the moving conductor to be in the form of a closed coil
and, for the sake of simplicity, suppose this to be rectangular and
to be moved so that while two sides cross the field at right angles
to the lines of force the other two sides move lengthwise through
the field. Since these latter cut no lines of force they perform no
work. If the field be uniform each of the other two sides performs
an equal amount of work, but the current in them flows in opposite
directions so that in one IN is positive while in the other it is
negative. The net result is zero, or, no matter how it may be
moved, if in its successive positions in a uniform field a coil
remains parallel to its original position, no work is done.

If, however, the field be not uniform, the work done by one of
these sides will be IN ergs, that by the other IN' ergs, and the
total work is 7 ( N N') ergs, or the work done in moving a coil
in a magnetic field is equal to the product of the current in the coil
into the change in the number of lines of force embraced by the
coil. This is general, that is, it is true whatever the shape of the
coil and whether its motion be one of translation or of rotation.
It also follows that the same amount of energy is expended if


the coil be kept ^motionless and the field embraced be moved or

If two separate and similar coils be moved in succession across
the field, the work done by each is, from the foregoing, IN ergs,
in which N is the change in the number of lines of force embraced
by the coil, the total work being 21 N ergs. If they be moved
simultaneously the work will be the same. Finally, they need
not be separate coils but may be two turns of the same coil
and .still the work will be 21 N ergs. In general, therefore, if the
field within a coil of n turns carrying a current I be increased
or decreased by N lines of force, the work done will be nIN

359. Energy Expended upon an Electro- Magnetic Field. The

conclusions in the preceding paragraph are irrespective of the
origin of the field. It may therefore be produced in any way, even
by the current itself. When a current is sent around a coil, N
lines of force are produced in the coil. By a similar method to
that followed in Par. 96, or by an application of the integral cal-
culus, it may be shown that if the current starts at zero and in-
creases to a value /, the energy expended in establishing the field
is \ IN ergs over and above that spent in the mere heating of the
conductor. This energy is absorbed in the field and restored when
the circuit is broken. This fact explains why the current never
rises instantly to its full strength when the circuit is closed and
also why the current always lingers after the circuit is broken,
revealing itself as a spark. This subject will be referred to again
when the discussion of induction is reached.

360. Electro-Dynamics. In Par. 356 it was shown that a con-
ductor carrying a current and placed in a magnetic field is acted
upon by a force at right angles to the field and to the conductor.
Since conductors carrying currents are surrounded by magnetic
fields (Par. 346), it follows that if two such conductors be placed
near together, each will lie in the magnetic field of the other and
each will be subjected to a force. Ampere, who made this dis-
covery in 1820, applied the term electro-dynamics to that branch
of electricity which treats of the forces exerted between currents,
and formulated the laws given in the following paragraphs.

361. Force Exerted Between Conductors Carrying Currents.

Two parallel conductors attract one another if the currents in them



flow in the same direction but repel each other if the currents flow
in opposite directions.

A and B, Fig. 157, are two such conductors. Considering B as
lying in the field about A, application of the left hand rule (Par.
352) will show that B is urged at right angles to its length and

Fig. 157.

towards A. Similarly, A is urged towards B. Had the currents
flowed in opposite directions the wires would have repelled each

It may be shown by Laplace's law (Par. 353) that if the two
wires be not parallel, the electro-magnetic effect of either current
can be resolved into two components, one parallel to the remain-
ing current, the other perpendicular to it and contributing noth-
ing to the forces between the two wires. In the most general
case, therefore, if two .conductors cross, those portions in which
both currents flow either towards or from the point of crossing
attract each other while those portions in which one current flows
towards and the other from the point of crossing repel each

It is not necessary that the two conductors be parts of different
circuits. The same law applies to portions of a single circuit. If,
for example, a current be passed through a helical coil, the adjacent
turns attract each other and the coil tends to shorten.


362. Intensity of Force Between Parallel Conductors Carrying
Currents. If the wire A, Fig. 157, be of indefinite length and if
there be flowing in it a current of strength /', the intensity of the
field produced by it at any point along B is (Par. 353)

r being the distance between

the two wires. But we have seen (Par. 356) that the force exerted
by a field H upon a wire of length I carrying a current of strength
/ is / = /. H .1. Substituting in this the value of H from above, we

f _ 2in

J= r

or the force exerted upon

the second wire B is directly proportional to the product of the
currents in the two wires and to the length of B and inversely
proportional to the simple distance between the wires.



363. Galvanoscopes. Oerstedt's discovery affords us a means
of determining whether or not a current is flowing in a conductor,
and if flowing, in what direction. For example, in the case of an
electric wire crossing the ceiling of a room, it is only necessary to
hold a magnetic needle an inch or so below the wire when, if a cur-
rent is flowing, the needle will be deflected and the direction of
this deflection, in conjunction with the right hand rule, will reveal
the direction of flow of the current. Instruments designed to give
information of this character are called galvanoscopes.

364. Increase of Sensitiveness. We frequently have to deal
with currents so small that the deflection they produce in an
ordinary needle is imperceptible. In such cases the only remedy
is to increase the sensitiveness of the instrument. A needle when
in use is acted upon by two forces; first, the deflecting force which
causes it to move and, second, the controlling force which resists
deflection. We therefore have two expedients; we may multiply
the effect of the deflecting force or we may weaken the controlling
force. The highest degree of sensitiveness is attained by com-
bining these two. We shall now examine these in detail.

365. Schweigger's Multiplier. Suppose a needle to be placed at
the center and to lie in the plane of a vertical coil. Application of
the right hand rule will reveal the fact (shown already in Par.
354), that when the circuit is closed, the top, the sides, the bottom
of the coil, all contribute to produce a deflection of the needle in
one and the same direction. As the palm of the right hand is slid
along the coil, the thumb points constantly in the same direc-
tion which is that of every line of force enclosed by the coil. If,
therefore, instead of simply passing the wire by the needle, as in
Oerstedt's experiment, we take a turn entirely around it, the de-
flecting force is very much increased. Finally, we need not stop at
one loop. Every succeeding turn adds its lines of force to those
already in the field and we may, therefore, use a coil of a great



many turns and multiply by just so much the effects of the cur-
rent. Such is the principle of Schweigger's multiplier (Fig. 158)
which consists of a suitable frame which may be rectangular, oval
or circular and around which are wrapped many turns of insulated

Fig. 158.

wire. The frame must be of some non-magnetic material such as
wood, ebonite, brass, etc., otherwise it would acquire magnetism
from the current. In the center of the coil is pivoted the needle
whose deflection is to be observed.

A multiplier should be used with feeble currents only. With a
strong current it is not necessary; moreover, the resistance of the
many turns of wire would cut down a large current. It is true
that it also reduces a small current but not so much proportionally.
The general rule is that a multiplier is used when the circuit
already contains great resistance but should not be used if the
resistance be small.

366. Methods of Weakening Controlling Force. The second
method of rendering a needle more sensitive, the weakening of the
controlling force, may be applied in two ways:

(a) Haiiy's method. The earth's controlling force may be
very nearly neutralized, there being left a small excess just suffi-
cient to control the needle.

(b) Astatic combinations. The earth's controlling force may
be entirely neutralized, and some very feeble force, such as the
torsion of a silk fibre, substituted therefor.

367. Haiiy's Method. Hauy's method of weakening the earth's
control is shown diagrammatically in Fig. 159 as being applied to
the needle of Schweigger's multiplier. The needle is suspended in



the center of the multiplier, the plane of the coil being in the
magnetic meridian. A brass rod AB projects upward from the
top of the coil and upon this rod there slides a bar magnet NS,
usually curved as shown. Now, as this bar magnet ir slid down
towards the coil, its north pole repels with an increasing force the

Fig. 159.

north pole of the needle. A point is finally reached where NS
exactly counterbalances the earth's controlling force upon the
needle and if this point be passed the needle is reversed. By stop-
ping the bar magnet just above this critical point, the effect of the
earth's control may be reduced to a minimum. This method is
employed in Thompson's mirror galvanometer (Par. 377).

368. Astatic Combinations. Two needles of equal size and
strength fastened rigidly together in reversed positions and with
their axes parallel constitute an astatic pair (Fig. 160). This com-
bination is independent of the earth's control and the controlling
force is generally the torsion of a fine silk fibre by which the needles
are suspended. They are usually mounted so that the lower needle
swings in the center of a multiplier, the upper needle travelling
over a graduated scale on the top of the coil and thus serving as a
pointer. Application of the right hand rule will show that the



current in the portion of the coil between the two needles will
cause them both to rotate in the same direction. By using a coil
wrapped like a figure eight, both needles may be surrounded by


Fig. 160.

There are a number of other astatic combinations. A suspended
horseshoe magnet is astatic and may be used as an astatic pair.

369. Magnetic Shells. Should we be able to cut from the end
of a bar magnet a thin slice, and should this slice preserve its
original polarity, we would have a magnetic shell, a thin piece of
metal, one face of which would be of north polarity, the other

Another conception of a magnetic shell is to suppose that we
had a great number of very small magnets, like minute type, and
that we should arrange them over the area of a circle side by side

Fig. 161.

like the cells of a honeycomb (Fig. 161), the north poles all point-
ing up, the result would be a magnetic shell. If a coil of wire be
bent into a circle of the same size as the shell, a current could be
sent through the wire which would produce inside the coil as many
lines of force as emerged from the shell. Since we have shown (Par.
365) that these lines all emerge from one face of the coil and in the
same direction, the coil and the shell are magnetically equivalent
to each other. Coils carrying a current behave in many ways as



if they were magnets. They have polarity and will attract or
repel a magnet, depending upon the pole of the magnet and the
face of the coil to which it is presented. They also attract or repel
each other.

This conception of a magnetic shell is used in mathematical dis-
cussions of electricity. We will not have occasion to use it further
but there follows from it a very important principle which we shall
now develop.

370. De La Rive's Floating Battery. One form of De La Rive's
floating battery is represented in Fig. 162. It consists of a turnip-
shaped glass cell with a constricted lower part containing dilute
acid. Upon a cork in the mouth of the cell is mounted a vertical

Fig. 162.

coil of wire of a number of turns, the ends of this coil extending
below the cork and terminating, one in a copper, the other in a
zinc plate which dip into the acid. The arrangement is therefore
seen to be only a simple cell, the coil constituting the external cir-
cuit. The cell is placed in a basin of water so that it floats freely.
If the current flows around the coil in the direction indicated by
the arrow, the lines of force of the coil pass through from right to
left, or, from what we have just seen, the coil is equivalent to a
magnetic shell whose north face is to the left.

Suppose that, as represented in the figure, the north pole of a
bar magnet be presented to the north face of the coil. The float-



ing cell, as was to be expected, will back away or recede, but, more
than this, it will turn around until its south face is presented and
will then approach the magnet and, instead of stopping when it
has reached the pole, will continue to advance and will thread
itself upon the magnet until it has reached the middle point. Its
lines of force and those of the magnet now coincide in direction.
It is now in a position of stable equilibrium for if it be pushed
towards either end of the magnet and released it will immediately
return to its median position. On the other hand, suppose the
coil to be held and the magnet thrust into it in reversed direction,
that is, with its lines of force opposite in direction to those of the
coil. If the coil be released when exactly at the center of the
magnet it will remain, but it is in unstable equilibrium, for if dis-
placed ever so slightly in either direction from this central position
it will slip off the magnet, turn around and return.

One of these cells floating freely in a vessel of water will finally
come to rest with the axis of the coil in a north and south position,
that is, with its field coinciding in direction with that of the earth.
Two such cells will move about until the fields of their coils coin-
cide in direction.

371. Maxwell's Law. The principle in accordance with which

Online LibraryWirt RobinsonThe elements of electricity → online text (page 22 of 46)