C

Step 1: Use the formula for Young's modulus, which is defined as:
Y = \frac{F/A}{\Delta L/L},
where Y is Young's modulus, F is the force applied, A is the cross-sectional area, \Delta L is the change in length, and L is the original length.

Step 2: Rearranging gives:
\Delta L = \frac{F \cdot L}{A \cdot Y}.

Step 3: Calculate the area in square meters:
A = 2 \text{ cm}^2 = 2 \times 10^{-4} \text{ m}^2.

Step 4: Substitute values:
Y = 10^4 \text{ N/m}^2,
F = 2 \times 10^{5} \text{ dynes} = 2 \times 10^{5} \times 10^{-5} \text{ N} = 2 \text{ N}.

Step 5: Now, substituting in the equation:
\Delta L = \frac{2 \cdot L}{2 \times 10^{-4} \cdot 10^4}.

This simplifies to:
\Delta L = \frac{2L}{2} = L.

Hence, the new length \(L + \Delta L = L + L = 2L\).
Therefore, the correct option is C: 2L.