A wire of $$cross-sectional$$ are d A, length $$L1$$, resistivity ρ$$1$$ and temperature coefficient of resistivity α$$1$$ is connected in series to a second wire of length $$L2$$, resistivity ρ$$2$$, temperature coefficient of resistivity α$$2$$ and the same area A, so that wires carry same current. Total resistance R isindependent of temperature for small temperature change if (Thermal$$ expansion effect is $$negligible)

Question

A wire of $$cross-sectional$$ are d A, length $$L1$$, resistivity ρ$$1$$ and temperature coefficient of resistivity α$$1$$ is connected in series to a second wire of length $$L2$$, resistivity ρ$$2$$, temperature coefficient of resistivity α$$2$$ and the same area A, so that wires carry same current. Total resistance R isindependent of temperature for small temperature change if (Thermal$$ expansion effect is $$negligible)

Infinity Learn · Accepted Answer

Let initial resistances of the wires are $$R1$$ and $$R2$$ respectively. Then R'$$1$$ $$+R$$'$$2=R1+R2$$⇒$$R11+$$α$$1$$ΔT $$+R21+$$α$$2$$Δ$$T=R1+R2$$⇒$$R1$$ α$$1+R2$$α$$2=0$$⇒ ρ$$1L1A$$ α$$1+$$ρ$$2L2A$$α$$2=0$$ρ$$1L1$$α$$1+$$ρ$$2L2$$α$$2=0$$

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