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Derivation of γ
This small element of the wire subtends an angle dβ at the centre of the coil.
The maximum local strain in the material (at the surface of the wire) is:
where θL is the angular twist per unit length induced by the torque.
The torque acting on this element causes it to twist through an angle dθ.
Therefore, the twist per unit length is:
The vertical deflection of the coil due to the twisting of this element is:
The vertical deflection associated with the twist in one turn of the coil is:
The spacings between adjacent turns in the coil is thus related to the peak local shear strain in the wire by:
It may also be noted that the average local strain, γ, is proportional to the peak strain, so that:
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