At temperature `0""^(@) C and 100 ""^(@) C,` currents passing through one conductor are resectively 1 A and 0.7 A . Find current through it when its temperature is `1200""^(@) C `. (Voltage source is same).

Explanation

We have,
`R_(t) = R_(0) { 1 + prop ( t - 0) } `
`therefore R_(t) = R_(0) + R_(0) prop t `
`therefore Rt - R_(0) prop ` t
`therefore prop = (R_(t) - R_(0))/(R_(0)t)`
`therefore prop = ((V)/(I_(t)) - (V)/(I_(0)) )/((V)/(I_(0)) xx t) `
` therefore prop = ((1)/(I_(t))-(1)/(I_(0)))/((t)/(I_(0))) = (I_(0))/(t) ((I_(0)-I_(t))/(I_(t) I_(0))) `
`therefore prop = (I_(0) - I_(t))/(I_(t) t) " " `... (1)
From equation (1),
`prop = (I_(0) - I_(t1))/(I_(t1) xx t_(1)) " " `.... (2)
Similarly, `prop = (I_(0) - I_(t2))/(I_(t2) xx t_(2)) " " ` ... (3)
From equations (2) and (3) ,
`(I_(0) - I_(t1))/(I_(t1) xx t_(1)) = (I_(0) - I_(t2))/( I_(t2) xx t_(2)) " " `... (4)
Substituting give values,
` (1- 0.7 )/(0.7 xx 100) = (1 - I_(t2))/( I_(t2) xx 1200)`
`therefore (0.3)/(0.7) - (I - I_(t2))/( I_(t2) xx 12)`
`therefore 3.6 I_(t2) = 0.7 - 0.7 I_(t2)`
4.3 `I_(t2) = 0.7`
`therefore I_(t2) = (0.7)/(4.3) = 0.1627 ` A
Note: Dear students, if current in a given conductor is `I_(0)` at reference temperature `t_(0)` then relation between currents temperature `t_(1) and t_(2)` is given by,
`I_(t2) = I_(t1) xx ( (I_(t1) - I_(0))/(I_(t2) - I_(0))) ( (t_(1) - t_(0))/(t_(2) - t_(0)) ) `
Above formula can be obtained by simplifying equation (4) and then by making `I_(t2)` as subject of equation.