IV. The diffraction of electric waves round a perfectly reflecting obstacle

In a previous communication it was verified that the effect at a point on a perfectly conducting sphere due to a Hertzian oscillator near to its surface was negligible in comparison with the effect that would have been produced at that point but for the presence of the sphere, when the point is at some distance from the oscillator and the radius of the sphere is large compared with the wave length of the oscillations. In what follows it is proposed to find the effect at all points produced by a Hertzian oscillator placed outside a conducting sphere whose radius is large compared with the wave length of the oscillations. For simplicity the axis of the oscillator will be assumed to pass through the centre of the sphere, but this assumption will not affect the generality of most of the results. An Appendix is added in which the more important mathematical relations required are established. 1. Let O be the centre of the conducting sphere of radius a, and let the oscillator be at a point O1 , the direction of the axis of the oscillator being OO1 , and the distance OO1 being r1 .In this case the lines of magnetic force are circles which have the line OO1 for common axis. If γ denotes the magnetic force at any point P, ρ the distance of P from OO1 , and z the distance of P from the plane through O perpendicular to OO1 , γρ satisfies the differential equation ∂2 /∂p 2 (γp )-1-p ∂/∂p (γp ) + ∂2 /∂z 2 (γp )+k 2 γp = 0, where 2 π/k is the wave length of the oscillations. Transforming to polar co-ordinates (r, θ ), where r is the distance OP andθ the angle POO1,z =r cosθ andp =r sinθ ; hence, writing cosθ = μ, the differential equation becomes ∂2 /∂r 2 (γp )+1-μ2 /r 2 (γp )+k 2 γp = 0 (1).