To determine the percentage error in Young's modulus, we need to first understand the propagation of errors in the given formula.

Given the equation:

$$ \mathrm{Y}=49000 \frac{\mathrm{M}}{\mathrm{l}} \frac{\mathrm{dyne}}{\mathrm{cm}^2} $$

where:

• $$M$$ is the mass (with its value given as $$500 \mathrm{~g}$$)
• $$l$$ is the extension (with its value given as $$2 \mathrm{~cm}$$)

The errors in the measurements are determined by the smallest scale divisions on the graph paper, which are:

• $$5 \mathrm{~g}$$ for the load axis
• $$0.02 \mathrm{~cm}$$ for the extension axis

To find the percentage error in Young's modulus ($$Y$$), we need to compute the relative errors in the measurements $$M$$ and $$l$$, and then propagate these errors through the given formula.

The relative error in $$M$$ is:

$$ \frac{\Delta M}{M} = \frac{5 \mathrm{~g}}{500 \mathrm{~g}} = 0.01 $$

The relative error in $$l$$ is:

$$ \frac{\Delta l}{l} = \frac{0.02 \mathrm{~cm}}{2 \mathrm{~cm}} = 0.01 $$

Since $$Y$$ is proportional to $$M$$ and inversely proportional to $$l$$, the overall percentage error in $$Y$$ is the sum of the percentage errors in $$M$$ and $$l$$:

$$ \frac{\Delta Y}{Y} = \frac{\Delta M}{M} + \frac{\Delta l}{l} = 0.01 + 0.01 = 0.02 $$

To express this as a percentage, we multiply by 100:

$$ \text{Percentage error in } Y = 0.02 \times 100 = 2\% $$

Thus, the percentage error in Young's modulus $$Y$$ is:

Option A: 2%