**Heat Transfer in Metallic Rod**

Heat transfer in a metallic rod can be analyzed using the principles of conductive heat transfer. The temperature gradient, defined as the change in temperature per unit length, is important in determining the rate of heat transfer along the rod. In this case, we have a metallic rod of uniform diameter and length L that connects two heat sources, each at 500°C, while the atmospheric temperature is 30°C. We need to find the temperature gradient at the center of the bar.

**1. Thermal Conductivity**

Thermal conductivity is a property of materials that determines their ability to conduct heat. It is denoted by the symbol k and is measured in units of watts per meter per degree Celsius (W/m·°C). The thermal conductivity of the metallic rod is required to determine the temperature gradient.

**2. Steady-State Heat Transfer**

In this scenario, we can assume that the heat transfer is in a steady state, meaning that the temperature distribution within the rod remains constant over time. This assumption is valid when the heat sources and the surrounding temperature do not change significantly.

**3. Heat Transfer Equation**

The rate of heat transfer through conduction along the rod can be determined using Fourier's law of heat conduction. The equation is given as:

Q = -kA(dT/dx)

Where Q is the rate of heat transfer, k is the thermal conductivity, A is the cross-sectional area of the rod, and (dT/dx) is the temperature gradient.

**4. Temperature Gradient Calculation**

To calculate the temperature gradient at the center of the rod, we need to determine the rate of heat transfer at that point. Since the rod is of uniform diameter, the cross-sectional area A remains constant. We can assume that heat transfer occurs only along the length of the rod and not across the cross-section.

At the center of the rod, the temperature difference (dT) is the difference between the heat source temperature (500°C) and the atmospheric temperature (30°C):

dT = 500°C - 30°C = 470°C

The length of the rod is given as L. Therefore, the temperature gradient can be calculated as:

dL/dT = L / dT

Substituting the values, we get:

dL/dT = L / 470°C

**5. Conclusion**

In conclusion, the temperature gradient at the center of the metallic rod connecting two heat sources at 500°C and an atmospheric temperature of 30°C can be determined by dividing the length of the rod, L, by the temperature difference between the heat source and atmospheric temperature, dT. The temperature gradient is given by the equation dL/dT = L / 470°C.