THEORY OF ELECTRICAL IMAGES

Excerpts

• THEORY OF ELECTRICAL IMAGES

  The point B with its imaginary charge is called the *electric image* of A.

• THEORY OF ELECTRICAL IMAGES

  It can only exist when A is insulated, and it is everywhere of the same sign as P_a.

• THEORY OF ELECTRICAL IMAGES

  the negative electrification is more concentrated than the positive, so that the neutral line which separates the positive from the negative electrification is not the equator of the sphere, but lies nearer to B.

• THEORY OF ELECTRICAL IMAGES

  This method has the great advantage of being intelligible by the aid of the most elementary mathematical reasoning, especially when it is considered in connection with the diagrams of equipotential surfaces described in Arts. 93-96.

• THEORY OF ELECTRICAL IMAGES

  But if we stand in front of a plane mirror and make observations on the apparent direction of the objects reflected therein, we find that these observations are consistent with the hypothesis that there is no mirror, but that certain objects exist in the region beyond the plane of the mirror. These hypothetical objects are geometrically related to certain real objects in front of the plane of the mirror, and they are called the *images* of these objects.

• THEORY OF ELECTRICAL IMAGES

  When Adams and Leverrier discovered the hitherto unknown planet Neptune, they did so by ascertaining the direction and magnitude of the gravitating force due to the unseen planet at certain points of space.

• THEORY OF ELECTRICAL IMAGES

  It appears, therefore, that when a spherical surface is uniformly electrified, the electric phenomena in the region outside the sphere are exactly the same as if the spherical surface had been removed, and a very small body placed at the centre of the sphere, having the same electric charge as the sphere.

• THEORY OF ELECTRICAL IMAGES

  This is a simple instance in which the phenomena in a certain region are consistent with a false hypothesis as to what exists beyond that region.

• THEORY OF ELECTRICAL IMAGES

  Hence, if an electrified point A be placed outside a spherical conductor which is at potential zero, the electrical action at all points outside the sphere will be equivalent to that due to the point A together with another point, B, within the sphere, which is the inverse point to A, and whose charge is to that of A as -1 is to m. The point B with its imaginary charge is called the *electric image* of A.

• THEORY OF ELECTRICAL IMAGES

  By this method he has solved problems in electricity which have never been attempted by any other method, and which, even after the solution has been pointed out, no other method seems capable of attacking.

• THEORY OF ELECTRICAL IMAGES

  The surface-density is negative on the side next to B and positive on the side furthest from B, but though the total quantities of positive and negative electrification are equal, the negative electrification is more concentrated than the positive, so that the neutral line which separates the positive from the negative electrification is not the equator of the sphere, but lies nearer to B.

• THEORY OF ELECTRICAL IMAGES

  When the potentials of the two spheres are equal the force is always repulsive.

Equations

• THEORY OF ELECTRICAL IMAGES

  \overline{CA} = ma

  The point A is placed at distance ma from the centre C, where m is the ratio of CA to the sphere's radius a.

• THEORY OF ELECTRICAL IMAGES

  \overline{AP} : \overline{PB} : : \overline{AC} : \overline{PC}

  Triangles APC and PCB are similar, so AP is to PB as AC is to PC.

• THEORY OF ELECTRICAL IMAGES

  \overline{AP} = m \overline{BP}

  The distance from A to any point P of the sphere is m times the distance from B to P.

• THEORY OF ELECTRICAL IMAGES

  V &= \frac{e}{\overline{AP}} + \frac{e'}{\overline{BP}}

  The potential at P due to the charge e at A and the image charge e' at B is the sum of the two charges divided by their distances from P.

• THEORY OF ELECTRICAL IMAGES

  AD = \frac{a^2}{c}

  The image of B in the sphere a lies at D, at distance a^2/c from A.

• THEORY OF ELECTRICAL IMAGES

  BE = \frac{b^2}{c}

  The image of A in the sphere b lies at E, at distance b^2/c from B.

• THEORY OF ELECTRICAL IMAGES

  AF = \frac{a^2}{AE} = \frac{a^2 c}{c^2 - b^2}

  The image of E in the sphere a, a third-order image, lies at F at the stated distance from A.

• THEORY OF ELECTRICAL IMAGES

  BG = \frac{b^2}{DB} = \frac{b^2 c}{c^2 - a^2}

  The image of D in the sphere b, a third-order image, lies at G at the stated distance from B.

• THEORY OF ELECTRICAL IMAGES

  q_{aa} &= a + \frac{a^2 b}{c^2 - b^2} + \text{\&c.,}

  The coefficient q_aa, summing the images that contribute to the charge of sphere a, is a series whose first terms are given.

• THEORY OF ELECTRICAL IMAGES

  q_{ab} &= - \frac{ab}{c} - \frac{a^2 b^2}{c (c^2 - a^2 - b^2)} - \text{\&c.,}

  The coefficient q_ab, the cross-coefficient linking the two spheres' charges and potentials, is a series whose first terms are given.

• THEORY OF ELECTRICAL IMAGES

  q_{bb} &= b + \frac{ab^2}{c^2 -a^2} + \text{\&c.,}

  The coefficient q_bb, summing the images that contribute to the charge of sphere b, is a series whose first terms are given.

• THEORY OF ELECTRICAL IMAGES

  E_a = q_{aa} P_a + q_{ab} P_b

  The total charge of sphere a is the sum of its potential terms weighted by the coefficients q_aa and q_ab.

• THEORY OF ELECTRICAL IMAGES

  E_b = q_{ab} P_a + q_{bb} P_b

  The total charge of sphere b is the sum of its potential terms weighted by the coefficients q_ab and q_bb.

• THEORY OF ELECTRICAL IMAGES

  &P_a = \frac{1}{a} E_a + \frac{1}{c} E_b

  Neglecting terms in b^3, the potential of sphere a is expressed in terms of the two charges.

• THEORY OF ELECTRICAL IMAGES

  &P_b = \frac{1}{c} E_a + \left\{\frac{1}{b} - \frac{a^3}{c^2 (c^2 - a^2)}\right\} E_b

  Neglecting terms in b^3, the potential of sphere b is expressed in terms of the two charges.

• THEORY OF ELECTRICAL IMAGES

  \frac{1}{2} (E_a P_a + E_b P_b) = \frac{1}{2} \frac{1}{a}\, E_a^2 + \frac{1}{c}\, E_a E_b + \frac{1}{2} \left\{\frac{1}{b} - \frac{a^3}{c^2 (c^2 - a^2 )}\right\} E_b^2

  The electric energy of the two-sphere system is half the sum of charge times potential for each sphere, written as a quadratic form in the charges.

• THEORY OF ELECTRICAL IMAGES

  R =\frac{ E_b}{c^2}\left\{E_a - E_b\frac{a^3 (2c^2 - a^2)}{c(c^2 - a^2)^2}\right\}

  The repulsion between the two spheres is the rate at which the electric energy diminishes as the distance c increases.

• THEORY OF ELECTRICAL IMAGES

  E_a \text{ must be greater than } E_b\frac{ a^3(2c^2 - a^2)}{c(c^2 - a^2)^2}

  For the force to be repulsive, the charge of sphere a must exceed the stated multiple of the charge of sphere b.

• THEORY OF ELECTRICAL IMAGES

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• THEORY OF ELECTRICAL IMAGES

  \sigma = - \frac{1}{4\pi} \frac{ea}{r^3} (m^2 - 1)

  The surface density of the induced charge on the sphere due to a point A outside it is negative everywhere and falls off as the inverse cube of the distance from A.

• THEORY OF ELECTRICAL IMAGES

  4 \pi \sigma = R

  By Coulomb's law, 4π times the surface density equals the resultant force acting outwards at the surface.

Problems

No exercises in this chapter.