Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/221

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where F is a function of the direction of r, and is a numerical quantity the square of which may be neglected.

Let the potential due to the external electrified system be expressed, as before, in a series of solid harmonics of positive degree, and let the potential U be a series of solid harmonics of negative degree. Then the potential at the surface of the conductor is obtained by substituting the value of r from equation (74) in these series.

Hence, if C is the value of the potential of the conductor and B_0 the charge upon it,

\begin{array}{ll}
C= & A_{0}+A_{1}aY_{1}+\dots+A_{i}a^{i}Y_{i},\\
\\ & \qquad+A_{1}aFY_{1}+\dots+iA_{i}a^{i}FY_{i},\\
\\ & +B_{0}\frac{1}{a}+B_{1}\frac{1}{a^{2}}Y_{1}+\dots+B_{i}a^{-(i+1)}Y_{i}+\dots+B_{j}a^{-j+1}Y_{j},\\
\\ & -B_{0}\frac{1}{a}-2B_{1}\frac{1}{a^{2}}FY_{1}+\dots-(i+1)B_{i}a^{-(i+1)}FY_{i}+\dots-(j+1)B_{j}a^{-(j+1)}FY_{j}.\end{array} (75)

Since F is very small compared with unity, we have first a set of equations of the form (72), with the additional equation

\begin{array}{ll}
0= & -B_{0}\frac{1}{a}F+3A_{1}aFY_{1}+\mathrm{etc}.+(i+1)A_{i}a^{i}FY_{i}\\
\\ & +\sum\left(B_{j}a^{-(j+1)}Y_{j}\right)-\sum\left((j+1)B_{j}a^{-(j+1)}FY_{j}\right).\end{array} (76)

To solve this equation we must expand F, FY_{1}\dots FY_{i} in terms of spherical harmonics. If F can be expanded in terms of spherical harmonics of degrees lower than k, then FY_i can be expanded in spherical harmonics of degrees lower than i+k.

Let therefore

B_{0}\frac{1}{a}F-3A_{1}aFY_{1}-\dots-(2i+1)A_{i}a^{i}FY_{i}=\sum\left(B_{j}a^{-(j+1)}Y_{j}\right), (77)

then the coefficients B_j will each of them be small compared with the coefficients B_{0}\dots B_{i} on account of the smallness of F, and therefore the last term of equation (76), consisting of terms in B_{j}F, may be neglected.

Hence the coefficients of the form B_j may be found by expanding equation (76) in spherical harmonics.

For example, let the body have a charge B_0, and be acted on by no external force.

Let F be expanded in a series of the form

F=S_{1}Y_{1}+\mathrm{etc}.+S_{k}Y_{k}. (78)

Then

B_{0}\frac{1}{a}S_{1}Y_{1}+\mathrm{etc}.+B_{0}\frac{1}{a}S_{k}Y_{k}=\sum\left(B_{j}a^{-(j+1)}Y_{j}\right), (79)