If the cold junction is at 0^oC and the hot junction is at any temperature say T^oC, then equation of thermo-e.m.f. can be written as

E = p_AB T + 1 q_AB T²
                 2

where p_AB and q_AB are characteristic contents of combination of metals A and under conderation and are known as thermo-electric coefficient and E is e.m.f. in micro-volts.

However if cold junction instead of being at 0^oC  is at T₁^oC and the hot junction is at T₂^oC then the above equation is written as

E = p_AB ( T₂-T₁) + 1 q_AB ( T₂² -T₁² )

As discussed above the e.m.f. is at neutral temperature and e.m.f.

will be maximum when dE is zero.
                                 dT

Differentiating expression T we get

dE   = p_AB + q_AB T
dT

The temperature T, at which dE will be the neutral temperature.
                                         dT

Hence                0 = p_AB + q_AB T_n

or                       T_{n =  -}p_AB
                                 q_AB

The temperature of inversion T , is the temperature at which the e.m.f. is zero and starts reversing. Hence substituting E= 0 in equation we get

0 =  p_AB T₁ + 1q_AB T₁²

or         T₁ =  -2p_AB
                     p_AB

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