Introduction:
When two uniform brass rods, A and B, are heated to the same temperature, they will undergo thermal expansion. The change in length for each rod will depend on its original length and radius. In this scenario, rod A has a length of l and a radius of 2r, while rod B has a length of 2l and a radius of r.

Thermal Expansion:
When a material is heated, its atoms and molecules gain energy and start to vibrate more rapidly. This increased molecular motion causes the material to expand. The amount of expansion depends on the material's coefficient of linear expansion, which is denoted by α.

Linear Expansion:
Linear expansion refers to the change in length of a material when its temperature changes. It can be calculated using the formula:

ΔL = α * L * ΔT

Where:
ΔL = Change in length
α = Coefficient of linear expansion
L = Original length
ΔT = Change in temperature

Ratio of Increase in Length:
To find the ratio of the increase in length of rods A and B, we need to compare their respective change in lengths.

For rod A:
ΔLA = α * l * ΔT

For rod B:
ΔLB = α * 2l * ΔT

Simplifying the Ratio:
To simplify the ratio, we can divide the change in length of rod A by the change in length of rod B:

Ratio = ΔLA / ΔLB
Ratio = (α * l * ΔT) / (α * 2l * ΔT)
Ratio = l / (2l)
Ratio = 1/2

Conclusion:
The ratio of the increase in length of rod A to rod B is 1:2. This means that rod B will experience twice the increase in length compared to rod A when both are heated to the same temperature.