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 Its 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 kelvin by supplying thermal energy from a P watt source. Steady state conditions are allowed to be established while the rod is properly insulated against heat loss from its lateral surface. Surroundings are at `0^@C` Stefan's constant`=sigma W//m^(2) K^(4)`.
After the steady state conditions are reached,the temperature of the spherical end of the rod, `T_S` is

Options

A. A.`T_S=T-(PL)/(KA)`
B. B.`T_S=0^@C`
C. C.`T_S=(PL)/(KA)`
D. D.`T_S=T-(P(L+r))/(KA)`

Explanation

To find the temperature \( T_S \) at the spherical end of the copper rod, we can follow these steps:

### Step 1: Understand the Setup
We have a copper rod of length \( L \) and cross-sectional area \( A \) with a spherical metal sphere of radius \( r \) attached at one end. The free end of the rod is maintained at temperature \( T \) by a power source supplying \( P \) watts. The rod is insulated, meaning no heat is lost from its lateral surface.

### Step 2: Apply the Heat Conduction Equation
In steady-state conditions, the heat supplied to the rod must equal the heat conducted through it. The heat conduction through the rod can be described by Fourier's law:

\[ P = \frac{K \cdot A \cdot (T - T_S)}{L + r} \]

Where:
- \( P \) is the power supplied (in watts),
- \( K \) is the thermal conductivity of copper,
- \( A \) is the cross-sectional area of the rod,
- \( T \) is the temperature at the free end of the rod,
- \( T_S \) is the temperature at the spherical end,
- \( L \) is the length of the rod,
- \( r \) is the radius of the sphere.

### Step 3: Rearrange the Equation to Solve for \( T_S \)
Rearranging the equation gives:

\[ P \cdot (L + r) = K \cdot A \cdot (T - T_S) \]

Now, we can isolate \( T_S \):

\[ T - T_S = \frac{P \cdot (L + r)}{K \cdot A} \]

Thus,

\[ T_S = T - \frac{P \cdot (L + r)}{K \cdot A} \]

### Step 4: Final Expression for \( T_S \)
The final expression for the temperature at the spherical end of the rod is:

\[ T_S = T - \frac{P \cdot (L + r)}{K \cdot A} \]

### Step 5: Conclusion
This equation gives us the temperature \( T_S \) at the spherical end of the copper rod in terms of the known quantities \( T \), \( P \), \( L \), \( r \), \( K \), and \( A \).

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