The temperature coefficient of resistance of tungsten is 4.5 × 10⁻³ °C⁻¹ and that of germanium is -5 × 10⁻² °C⁻¹. A tungsten wire of resistance 100 Ω is connected in series with a germanium wire of resistance R. The value of R for which the resistance of the combination does not change with temperature is:
(A) 9 Ω
(B) 1111 Ω
(C) 0.9 Ω
(D) 111.1 Ω

To find the value of R for which the resistance of the combination does not change with temperature, we need to make sure that the change in resistance of the tungsten wire cancels out the change in resistance of the germanium wire as the temperature changes.

The resistance of a material changes with temperature according to the formula:

ΔR/R₀ = αΔT

where ΔR is the change in resistance, R₀ is the initial resistance, α is the temperature coefficient of resistance, and ΔT is the change in temperature.

For the tungsten wire:

ΔR_tungsten/R_tungsten = 4.5 x 10^-3 ΔT

For the germanium wire:

ΔR_germanium/R_germanium = -5 x 10^-2 ΔT

Now, since the two wires are in series, the total resistance of the combination is:

R_total = R_tungsten + R_germanium

To ensure that the total resistance does not change with temperature, we must have:

ΔR_total/R_total = 0

So, let's substitute the expressions for ΔR_tungsten and ΔR_germanium into the equation for ΔR_total:

ΔR_total/R_total = (ΔR_tungsten/R_tungsten) + (ΔR_germanium/R_germanium) = (4.5 x 10^-3 ΔT)/R_tungsten - (5 x 10^-2 ΔT)/R_germanium

Now, plug in the values:

R_tungsten = 100Ω (given)

We want to find R_germanium. Let's solve for ΔT = 1°C (a common choice for temperature coefficients):

0 = (4.5 x 10^-3 x 1)/100 - (5 x 10^-2 x 1)/R_germanium

Simplify:

0 = 0.045 - 0.05/R_germanium

Now, solve for R_germanium:

0.05/R_germanium = 0.045

R_germanium = 0.05/0.045

R_germanium = 1.111Ω

So, the value of R for which the resistance of the combination does not change with temperature is approximately 1.111Ω, which is closest to option (D) 111.1Ω.