b. Two metallic rods A and B with similar length and cross sectional area are joined together and all curved surfaces are insulated. At steady state, the temperatures at their ends are shown in the figure below. The thermal conductivity of A is twice that of B. Calculate the temperature at the junction of these rods.

thermal conductivity is equal on both A and B
area A and B EQUAL
TEMPERATURE A=90 degree celcius
temperature b = 30 degree celcius
temperature diff=60 degree celcius.

To calculate the temperature at the junction of the rods, we can use the concept of heat transfer by conduction.

First, let's define some variables:
- Tₐ: Temperature at end A (90°C)
- Tᵦ: Temperature at end B (30°C)
- ΔT: Temperature difference between Tₐ and Tᵦ (60°C)

Since the rods are in thermal equilibrium, the rate of heat transfer between them is equal. Now, the rate of heat transfer by conduction (Q) can be given by the equation:

Q = (k₁ * A₁ * (Tₐ - Tᵦ)) / L₁ = (k₂ * A₂ * (Tₐ - Tᵦ)) / L₂

Where:
- k₁, k₂: Thermal conductivity of rods A and B respectively.
- A₁, A₂: Cross-sectional areas of rods A and B respectively.
- L₁, L₂: Lengths of rods A and B respectively.

Given that the cross-sectional areas are equal (A₁ = A₂), the equation becomes:

Q = (k₁ * A * (Tₐ - Tᵦ)) / L₁ = (k₂ * A * (Tₐ - Tᵦ)) / L₂

Since k₁ = 2 * k₂ (the thermal conductivity of A is twice that of B), we can rewrite the equation as:

Q = (2 * k₂ * A * (Tₐ - Tᵦ)) / L₁ = (k₂ * A * (Tₐ - Tᵦ)) / L₂

Now, we can cancel out the common terms:

2 * (Tₐ - Tᵦ) / L₁ = (Tₐ - Tᵦ) / L₂

Let's solve this equation for the temperature at the junction (Tʲ):

2 * (Tₐ - Tᵦ) / L₁ = (Tₐ + Tʲ) / L₂

(Tₐ - Tᵦ) / L₁ = Tʲ / L₂

Tʲ = [(Tₐ - Tᵦ) / L₁] * L₂

Plugging in the given values:
- Tₐ = 90°C
- Tᵦ = 30°C
- ΔT = 60°C
- k₁ = 2k₂ (thermal conductivity of A is twice that of B)
- A₁ = A₂ (cross-sectional areas are equal)

Let's assume the length of both rods is the same (L₁ = L₂):

Tʲ = [(90 - 30) / L₁] * L₁

Tʲ = 60°C

Therefore, the temperature at the junction of these rods is 60°C.