Question

Two conductors having same resistance R first connected such a manner that their equivalent resistance becomes R / 2 , later on they were connected in such a way that their equivalent resistance becomes 2R . In what type of combination conductors were connected in first instant and in what type of combination in second instance ?
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Answer 1

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Solution :

⏭ Given:

✏ Two resistors are connected in series.

✏ Resistance of 1st resistor = R

✏ Resistance of 2nd resistor = 2R

✏ Temp. co-efficient of R1 = α

✏ Temp. co-efficient of R2 = 2α

⏭ To Find:

✏ Equivalent temp. co-efficient of the equivalent resistor for the series connection.

⏭ Formula:

✏ Change in resistance due to change in temp. is given by

ΔR = Ro(αΔT)

R'-Ro = Ro(αΔT)

R' = Ro(1 + αΔT)....(☆)

⏭ Terms indication:

R' denotes resistance at T°C

Ro denotes resistance at 0°C

α denotes temp. co-efficient

ΔT denoted change in temp.

⏭ Calculation:

✏ Equivalent resistance at 0°C

Ro = R1 + R2

Ro = R + 2R

Ro = 3R

✏ Equivalent resistance at T°C

R' = R'1 + R'2

R'1 = R(1 + αΔT)

R'2 = 2R(1 + 2αΔT)

R' = R + RαΔT + 2R + 4RαΔT

R' = 3R + 5RαΔT

✏ Putting all values in (☆) equation

3R + 5RαΔT = 3R[1 + (α)sΔT]

3R + 5RαΔT = 3R + 3R(α)sΔT

5RαΔT = 3R(α)sΔT

5α = 3(α)s

✏ (α)s = 5α/3