Introduction:
In this problem, we are given two wires made of the same material. The length of the wires is in the ratio 2:3, and the radii of the wires are in the ratio 8:9. We need to determine if the two wires will have the same potential difference when connected to a voltage source.

Explanation:
To understand whether the two wires will have the same potential difference, we need to consider the factors that affect the potential difference across a wire.

Potential difference:
The potential difference across a wire is given by Ohm's law, which states that V = IR, where V is the potential difference, I is the current flowing through the wire, and R is the resistance of the wire.

Resistance:
The resistance of a wire is given by the formula R = ρ(L/A), where ρ is the resistivity of the material, L is the length of the wire, and A is the cross-sectional area of the wire.

Length:
Given that the lengths of the two wires are in the ratio 2:3, let's assume the lengths of the wires are 2x and 3x.

Radius:
Similarly, the radii of the wires are in the ratio 8:9. Let's assume the radii of the wires are 8y and 9y.

Resistance ratio:
The resistance of a wire is inversely proportional to its cross-sectional area. Since the wires are made of the same material, the resistivity ρ is constant.

Using the formula for resistance, we can write the ratio of the resistance of the two wires as:

R1/R2 = (ρ(2x)/(8y)^2) / (ρ(3x)/(9y)^2)
R1/R2 = (2x/64y^2) / (3x/81y^2)
R1/R2 = (2x/64y^2) * (81y^2/3x)
R1/R2 = 54/64
R1/R2 = 27/32

Conclusion:
From the above calculation, we can see that the ratio of the resistance of the two wires is 27:32. Since resistance is directly proportional to the potential difference, we can conclude that the potential difference across the two wires will not be the same when connected to the same voltage source. The wire with the larger resistance (R2) will have a lower potential difference compared to the wire with the smaller resistance (R1).