Question

Two rods of same area of cross-section when connected in series
have a combined thermal resistance of 9SI units. When connected in
parallel, they have a combined thermal resistance of 20 Sl units. If
their lengths are in the ratio 7 : 10, then their thermal conductivities
may be in the ratio​

Answer 1

Given:

Thermal resistance when connected in series = 9 SI units

Thermal resistance when connected in parallel = 20 SI units

Ratio of length = 7:10

To find:

Ratio of thermal conductivities.

Solution:

In series thermal resistance:

R = R1 + R2

9 = R1 + R2

In parallel, thermal resistance:

1/R = 1/R1 + 1/R2

20 = (R1 + R2)/ R1 R2

Since R1 + R2 = 9

1/20 = 9 / R1 R2

R1 R2 = 180

R1 = 180/R2

R = L/Ap

where p = thermal conductivities

L = length

A = area

Area is same in both the rods:

p = L/R

p1 =  7L/ R1

p2 = 10L/ R2

p1 / p2= 7/10(R2/R1)

Therefore, the ratio of thermal conductivities is 7/10(R2/R1).