Introduction:
In this problem, we have a metallic rod of uniform diameter and length L. The rod connects two heat sources, each at a temperature of 500°C, and is exposed to an atmosphere temperature of 30°C. We need to determine the temperature gradient, dT/dL, at the center of the bar.

Understanding the problem:
To solve this problem, we need to consider the heat flow through the metallic rod. Heat flows from a higher temperature region to a lower temperature region. In this case, heat will flow from the heat sources at 500°C towards the atmosphere at 30°C.

Heat conduction equation:
The rate of heat conduction through a material is given by Fourier's law of heat conduction:

q = -kA(dT/dx)

Where:
q is the rate of heat conduction,
k is the thermal conductivity of the material,
A is the cross-sectional area of the rod,
(dT/dx) is the temperature gradient along the length of the rod.

Assumptions:
1. The rod is made of a homogeneous material with uniform thermal conductivity.
2. The rod is in steady-state conditions.

Solution:
1. Calculate the temperature difference between the heat sources and the atmosphere:
ΔT = 500°C - 30°C = 470°C

2. As the rod is of uniform diameter, the cross-sectional area is constant along its length.

3. Since the rod is in steady-state conditions, the rate of heat conduction through any cross-section of the rod is constant.

4. The rate of heat conduction through a cross-section of the rod can be calculated using Fourier's law:

q = -kA(dT/dx)

Since the rate of heat conduction is constant, we can write:

q = -kA(dT/dL)

Where (dT/dL) is the temperature gradient at the center of the rod.

5. Rearranging the equation, we get:

(dT/dL) = -q / (kA)

6. Substitute the values of q, k, and A:

(dT/dL) = -q / (kA)

7. Calculate the temperature gradient at the center of the rod using the given values.

Conclusion:
The temperature gradient, dT/dL, at the center of the bar can be calculated using the formula (dT/dL) = -q / (kA), where q is the rate of heat conduction, k is the thermal conductivity of the material, and A is the cross-sectional area of the rod. The temperature gradient indicates the rate of change of temperature with respect to the length of the rod.