Helmholtz Tangent Galvanometer

It consist of two parallel co-axial coils C₁ and C₂ having the same radius (a) and the same number of turns (N). The distance between the two coils is equal to the radius (a) of each coil. The coils are connected in series so that the current through them is in the same direction. At the midpoint of this pair, we have a situation where the field B₁ due to one coil is decreasing linearly with x, while B₂ due to the other coil is increasing linearly with x at the same rate. B₁ and B₂ add up to produce a filed which is almost uniform in this region needle pivoted in a compass box is placed at a point P midway between the coils. Hence the magnetic field, in which the deflection magnet is situated, is uniform.

Theory. If the rate of variation of the field due to a coil at any point should be constant then

dB / dx = constant or, d² B / dx² = 0.

B = μ₀ Nia² / 2 (a² + x²)^3/2
dB / dx =  - μ0 Nia² / ² . 3x (a² + x²) ^{– 5/2}
d²B / dx² = - 3 μ₀ Nia² / [(a² + x²) ^{– 5/2} + x (- 5/2) (a² + x²)- ^7/2 . 2x]
    = - 3 μ₀ Nia² / 2(a² + x²)^7/2. (a² – 4x²)
d²B / dx² = 0 when a² – 4x² = 0 or x = ± a/²

The resultant field at the midpoint O due to the two coils is

B = 2 X μ₀ Nia² / 2 . 1 / {a² + (a/2)²}^3/2        (. : x =  a/²)
   = 8 μ₀ Ni / 5 √5 a

In order to determine an unknown current, the planes of the coils are made to lie in the magnetic meridian. Let θ be the deflection of the magnetic needle. By tangent law,

        8 μ₀ Ni / 5 √5 a = B_H tan θ

where B_H = horizontal component of the induction due to the earth’s magnetic field.
    . : i = (5 √5 a B_H / 8 μ₀ N) tan θ = K tan θ

where K is the reduction factor of the galvanometer.

Advantage. The main defect in the tangent galvanometer is that the magnetic field due to the current in the coil is not uniform in the space occupied by the magnetic needle. But in a Helmholtz galvanometer, the field is uniform over the region occupied by the magnetic needle.