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Q units will at unit distance be measured by Q dynes; and, at

r

any other distance from the centre O, by dynes.

Let V. and V, represent the numerical value of the potentials at A and B respectively; these being measured by the number of ergs work done in bringing our unit from infinity up to the points A and B respectively. Then the formulæ given in § 7 tell us that

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An example or two will make the meaning clearer.

(i.) Let there be + 24 units of electricity at the point O; let rв be 4 cms., and let be 6 cms..

Then at unit distance (or 1 cm.) from O our unit will

be repelled with a force of 24 dynes; at B with a force of

=

3 dynes; and at A with a force of 24 = 2 dynes.

2

62 3

24

42

The potential of B will be greater than that of A by an amount measured by

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Or it will take 2 ergs work to move our + unit from A up to B.
The absolute potential of B will be measured by = 6; and

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24

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16. take 6 ergs to move our + unit

from infinite distance up to B, and 4 ergs to move it from infinity up to A.

(ii.) Let there be several quantities of electricity, as + 20 at O, 16 at P, + 10 at Q. These will give rise to a somewhat complex field of force. But still the results arrived at in § 7 and in the present section enable

us to find easily the potentials of any points as B and A in the field, and the difference of potentials.

For the total work due to this system of electric charges will be the algebraic sum of the works due to each charge separately.

+

20

Thus the absolute potential of B will be measured by BO

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the difference of potential, VB - VA, is easily found by subtraction. (iii.) Where there are a whole series of quantities q!, 9 29 43 ・ ・ ・ situated at distances r1, 2, 3 . . . respectively from a point B, then the potential of B as due to these quantities will be 91 92 93 + + + ... ; which is conveniently represented by Γι r2 r3

Σ 9

B

Here, of course, attention must be paid to the signs of the

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(iv.) Where the quantities are distributed continuously over a

conductor, or over any surface, this adding up 91
of +

&c. Τι becomes in general a matter for the application of the integral calculus.

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Note. It may occur to the reader that if r be zero, or if we touch the charge Q, then the potential becomes infinite, since = ∞. In answer to this we may remark that it is physically impossible to be at zero distance from any finite mass, or electrical or magnetic quantity. We can be at zero distance only from a geometric point. Hence, we cannot have Q finite and zero simultaneously.

§ 10. Equipotential Surfaces.

(i.) If we so move our + unit of electricity as not to move up or down the lines of force, then we do no work against the electrical forces. Hence, by definition, all the positions thus arrived at must have the same potential.

Let us consider the case of a simple field of force due to a single charge of Q units situated at the point O (see § 7, figure). If we move our unit over any sphere that has O as centre, we

do no work; and hence every point on such a sphere has the same potential.

Thus if we consider the sphere that has O as centre and that passes through the point A, it is clear that all points on this sphere have the same potential as A; a potential that is measured by

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It is necessary to do work to the amount of Q (-;-)

ergs to move our unit of electricity from any point on the sphere passing through A to any point on that passing through B. (ii.) Equipotential surfaces in general.—Where there are many centres of force, there both the lines of force and the equipotential surfaces will be complex, and will not have the simple forms that they had in the last case.

But in all cases certain general results hold good. The equipotential surfaces are always perpendicular to the lines of force (see 11); they may be mapped out into a series of marked' surfaces such that we do one erg work in moving our + unit between two consecutive surfaces; these marked surfaces will be closer together where the field is stronger, further apart where the field is weaker, and at a constant distance where the field is uniform.

§ 11. Lines of Force are Perpendicular to Equipotential Surfaces. This has been already proved implicitly; or rather it is a consequence of the very definition of an 'equipotential surface.' But it may be well to give a formal demonstration.

Let P F be a line of force; and let A P B be a section of the equipotential surface that passes through P ; and let PT be a section of the tangent-plane to this surface, at the point P.

A

H

B

T

Then shall PF be perpendicular to the tangent P T. For if possible let it lie as P F'; making with PT an angle less than 90°. We can now resolve the force PF into two components, along the tangent P T and perpendicular to it respectively. For the former component we have PF cos F'PT; and this is not zero, since F'P T is not 90°. Since then there is a tangential component, we should do work in moving our + unit along

the surface near P. But this is contrary to hypothesis, since AP B is an equipotential surface. Therefore, F P must be perpendicular to AP B.

§ 12. Field-strength; and Rate of Change of Potential. The diagram represents a portion of a field of force; this portion being so small that we may consider it to be uniform, or the lines of force to be parallel (see § 13, coroll.). Let A and B be any two points in this field, and let them be separated by a distance of A B cms.

If A B be not perpendicular to the lines of force, i.e. if A and B be not on the same equipotential surface, there will be a force on a unit of electricity acting along the line AB. We may represent the force on a + unit that acts from B to A by the symbol F; this will be measured in dynes, and may be + or as it acts from B to A or from A to B. [See § 1 (i.); and compare

Chapter II. § 8.]

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By the definition of 'field-strength,' &c., it is clear that FR measures that component of the field-strength that has the direction B A.

Then, by the definition of work, we have

The work done in moving

our unit from A to B.

==

FB x AB ergs.

But, by the definition of potential, this work measures the difference of potential between A and B; or we have (V - V1) = FR × A B.

B

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Now since A B is measured in centimètres, this last expression represents the change of potential per unit length, or the rate of change of potential, along A B.

So we may translate this formula (FR =

somewhat as follows.

VB - V

into words

AB

In a field of force, the component in any direction of the force acting upon a + unit, i.e. the component in any direction of the total field-strength, is measured by the space-rate of change of potential in that direction.

When the potential is constant there is zero field-strength, and conversely.

It is very important to remember that a field of force implies varying potential, and conversely.

Examples.-(i.) In the inside of a charged vessel we may havea potential of very great magnitude; but it is constant throughout, and we have zero field of force.

(ii.) Half-way between two equal charges of the same sign we have a certain potential; but there is a point of zero force.

(iii) Half-way between two equal charges of opposite signs we have zero potential, but not zero field of force.

§ 13. The Mapping Out of Lines of Force; Simple Case.Since we can draw a line of force through every point in the field --such a line being defined as the direction in which a + unit of electricity is urged-it follows that the lines of force are infinite in number.

But a little further consideration will show us that here, as in the case of equipotential surfaces, we can mark a certain number out of this infinite crowd of lines, in such a way as to represent the various strengths of different parts of the field.

Let us as before consider the simple case of a single charge of Q units collected at the point O (see fig. § 7); and let us consider first a spherical surface, described about O as centre, with radius 1 centimètre. At any point on this sphere the force exerted on a + unit will be Q dynes; or the strength of the field will be measured by Q.

The reader must remember that in our present case the lines of force, which are infinite in number, radiate from O; and that they cut at right angles the surfaces of all spheres which have O

as centre.

Now let us mark out (we may suppose that we paint them red) such a number of lines of force that exactly Q of them pierce each I square centimètre of this spherical surface. Then the number of lines piercing unit area placed perpendicularly to the lines of

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