The thermal powered density u is generated uniformly inside a uniform sphere of radius R and thermal conductivity Find the temperature distribution in the sphere when the steady state temperature at the surface is `T_(0)`.

Answer

To find the temperature distribution in a uniform sphere with a uniform thermal power density \( U \) generated inside it, we can follow these steps:

### Step 1: Understand the Problem
We have a uniform sphere of radius \( R \) and thermal conductivity \( k \). Inside this sphere, there is a uniform thermal power density \( U \). The steady-state temperature at the surface of the sphere is given as \( T_0 \). We need to find the temperature distribution \( T(r) \) within the sphere.

### Step 2: Set Up the Heat Equation
In a steady state, the heat generated inside the sphere must equal the heat conducted to the surface. The heat conduction can be described by Fourier's law:

\[ \frac{dq}{dt} = -kA \frac{dT}{dr} \]

where \( A \) is the surface area of a sphere at radius \( r \), which is \( 4\pi r^2 \).

### Step 3: Heat Generation Inside the Sphere
The heat generated inside the sphere can be expressed as:

\[ \frac{dq}{dt} = U \cdot V \]

where \( V \) is the volume of the sphere of radius \( r \), given by \( \frac{4}{3}\pi r^3 \).

### Step 4: Equate Heat Generation and Heat Conduction
Setting the two expressions for heat equal gives:

\[ -k \cdot 4\pi r^2 \frac{dT}{dr} = U \cdot \frac{4}{3}\pi r^3 \]

### Step 5: Simplify the Equation
We can simplify this equation by canceling \( 4\pi \) from both sides:

\[ -k r^2 \frac{dT}{dr} = \frac{U}{3} r^3 \]

Rearranging gives:

\[ \frac{dT}{dr} = -\frac{U}{3k} r \]

### Step 6: Integrate the Equation
Integrating both sides with respect to \( r \):

\[ \int dT = -\frac{U}{3k} \int r \, dr \]

This results in:

\[ T = -\frac{U}{6k} r^2 + C \]

where \( C \) is the constant of integration.

### Step 7: Apply Boundary Conditions
We know that at the surface of the sphere (where \( r = R \)), the temperature \( T(R) = T_0 \):

\[ T_0 = -\frac{U}{6k} R^2 + C \]

Solving for \( C \):

\[ C = T_0 + \frac{U}{6k} R^2 \]

### Step 8: Write the Final Temperature Distribution
Substituting \( C \) back into the equation for \( T \):

\[ T(r) = -\frac{U}{6k} r^2 + T_0 + \frac{U}{6k} R^2 \]

This simplifies to:

\[ T(r) = T_0 + \frac{U}{6k} (R^2 - r^2) \]

### Final Answer
The temperature distribution in the sphere is given by:

\[ T(r) = T_0 + \frac{U}{6k} (R^2 - r^2) \]