Magnetic Field of a Coil

Your text probably derives the equation for the field in a very long coil:
B = m_oni, where n is number of turns per length,
or B = m_oNi/L, where N is the total number of turns, and L is length. I will use the latter expression. For this to be correct, the length must be very long compared to the radius, and then the formula is Ok for the center, but it drops off to half of this at the end.

I always figured that since a more general equation is not found in the typical text, it must be too complex. Not so! You can write the exact equation for the field on the axis of a circular loop, treat it as a continuous series of loops (the number of loops in distance dx is [N/L]dx), integrate, and find that

B = (m _o Ni/2L)(cosa + cosb ),

where the angles are shown below. This is valid anywhere on the axis, inside or outside a coil with circular windings. If you are outside the coil, note that one of the angles is greater that 90^o so the cosine is negative. The big rectangle below represents the coil.

It can be of any length. If it is very long, the cosines are each approximately 1 if you are near the center, and it reduces to the familiar long coil formula. At the center of a very short coil, the sum of the cosines is approximately L/R, and the formula reduces
to B = m_oNi/2R, which is the familiar expression for such a coil.

On the axis outside the coil at a large distance x from the nearer end of the coil, the sum of the cosines is the difference between two numbers that are almost the same. With a Taylor's series expansion, it can be shown that
B = m_oNiR²/[2x(x+L)²] for this case. Notice the inverse cube behavior for large x. Another approach to this is to let x = distance to the center of the coil, and the small difference is cosines = -sinqdq. But q in radians is R/x, so dq = - Rdx/x². With x large, replace dx with coil length L. Combine these and get B = m_oNiR²/2x³. (For large x, essentially the same as the previous result.)

Note also that NipR² is the magnetic dipole moment, analogous to the electric dipole moment, which is charge times distance apart. At off-axis locations, one can find the electric field for an electric dipole, then to determine the magnetic field for a coil, simply replace the electric dipole moment with the magnetic one, and replace the Coulomb’s law constant 1/(4pe_o) with the corresponding magnetic constant, m_o/4p.

It can be shown that for large r from an electric dipole, the potential (voltage) is approximately
V = kpcosq/r², where k is Coulomb’s law constant, p=dipole moment, r is the distance from the center of the dipole to any far away point, and q is the angle between p and r. The electric field component in any direction is minus the change of V per distance in that direction. Thus we find
E_r = 2kpcosq/r³ and E_q = kpsinq/r³. (q is constant in the r direction, and r is constant in the ^ direction in which the change of cosq per distance is
d(cosq)/rdq = -sinq/r.
The magnitude of E is the square root of E_r²+E_q², or E = (kp/r³)(4cos²q+sin²q) = (kp/r³)(3cos²q+1)

As noted earlier, we replace the electric dipole moment with the magnetic dipole moment (m = NiA) and Coulombs law constant with m_o/4p:
B_r = (m_o/2p)NiAcosq/r³ and B_q = (m_o/4p)NiAsinq/r³

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