Magnetic Potential due to a Solenoidal Magnet

We have seen that if P be a point at distances r₁, r₂ from the north and south poles, N, S, respectively, of a solenoidal magnet NOS (fig. 44) of strength m, the magnetic potential at P is

We will now put this expression into another and more useful form, to which it is for our purposes practically equivalent. Let O, the middle point of the line NS, be the centre of the magnet; let OP=r, ON = OS = l, so that 2l is the length of the magnet, and let the angle between the magnetic axis and the radius vector OP be θ, this angle being measured from the north pole to the south, so that in the figure NOP = θ.

Draw NR, ST perpendicular to PO or PO produced, and suppose that OP is so great compared with ON that we may neglect the square and higher powers of the ratio of ON/OP. Then PRN is a right angle, and PNR differs very little from a right angle, for ON is small compared with OP, so that PxN = PR very approximately, and similarly PS = PT.

Also OR = OT = ONcos(PON) = lcosθ.

Thus

and

and, if v denote the magnetic potential at P, we have

But we are to neglect terms involving l²/r², etc.; thus we may put

if M be the moment of the magnet.

We shall see next how to obtain from this expression the magnetic force at P.