A thin copper rod of uniform cross section A square metres and of length L metres has a spherical metal sphere of radius r metre at tis one end symmetrically attached to the copper rod. The thermal conductivity of copper is K and the emissivity of the spherical surface of the sphere is `epsi`.The free end of the copper rod is maintained at the temperature T kelving by supplying thermal energy from a P watt source. Steady state conditions are allowed ot be established while the rod is properly insulated aginst heat loss from its lateral surface. Surroundings are at `0^@C` Stefan's constant`=sigma W//m^(2)K^(4)`.
If the metal sphere attached at the end of the copper rod is made of brass, whose thermal conductivity is `K_b lt K`, then which of the following statements is true?

Options

A. The temperature of the sphere will, under steady state conditions, continue to be `T_B`
B. The power that will be radiated out from the sphere will still be `P_S`.
C. It will take smaller time for steady state conditions to be reached
D. The rate of thermal energy transmitted across the copper rod, under steady state, will reduced.

Explanation

To solve the problem, we need to analyze the heat transfer in the system consisting of a copper rod and a brass sphere. Let's break it down step by step.

### Step 1: Understand the System
We have a copper rod of length \( L \) and cross-sectional area \( A \) connected to a brass sphere of radius \( r \). The copper rod is maintained at a temperature \( T \) by a power source supplying \( P \) watts. The surroundings are at \( 0^\circ C \) (or \( 273 \, K \)).

**Hint**: Identify the materials involved and their properties (thermal conductivity, emissivity).

### Step 2: Steady State Conditions
In steady state, the heat flow into the system equals the heat flow out of the system. This means that the power supplied \( P \) must equal the power radiated by the brass sphere.

**Hint**: Recall that in steady state, the temperature at every point remains constant, and the heat flow is steady.

### Step 3: Heat Transfer Mechanisms
1. **Conduction in the Copper Rod**: The heat from the power source flows through the copper rod by conduction. The rate of heat transfer \( Q \) through the rod can be expressed using Fourier's law:
\[ Q = \frac{K \cdot A \cdot (T - T_b)}{L} \]
where \( T_b \) is the temperature at the interface between the copper rod and the brass sphere.

2. **Radiation from the Brass Sphere**: The brass sphere radiates heat to the surroundings. The power radiated can be expressed using the Stefan-Boltzmann law:
\[ Q_{rad} = \epsilon \cdot \sigma \cdot A_{sphere} \cdot T_b^4 \]
where \( A_{sphere} = 4\pi r^2 \) is the surface area of the sphere, \( \epsilon \) is the emissivity, and \( \sigma \) is the Stefan-Boltzmann constant.

**Hint**: Understand the difference between conduction and radiation and how they apply to the materials involved.

### Step 4: Equating Heat Transfer
At steady state, the heat conducted through the copper rod equals the heat radiated from the brass sphere:
\[ P = \epsilon \cdot \sigma \cdot (4\pi r^2) \cdot T_b^4 \]
This equation indicates that the power radiated by the brass sphere is equal to the power supplied to the system.

**Hint**: Set up the equations based on the principles of heat transfer and solve for the unknowns.

### Step 5: Analyze the Statements
Now, we can evaluate the statements provided in the question:
- **Statement A**: The temperature \( T_b \) of the sphere under steady state condition will be constant. (True)
- **Statement B**: The power that will be radiated from the sphere will be \( P \). (True)
- **Statement C**: It will take smaller time for steady state conditions to be reached. (True, as both materials are good conductors)
- **Statement D**: The rate of thermal energy transmitted across the copper rod under steady state will reduce. (False, it remains constant)

### Conclusion
The correct statements are A, B, and C. The incorrect statement is D.

**Final Answer**: The true statements are A, B, and C.