To solve this problem, we need to understand the relationship between energy, momentum, and radiation.$\\$When radiation of energy $E$ falls on a perfectly reflecting surface, it transfers momentum to the surface.$\\$The momentum $p$ of a photon is given by:$\\ p = \frac{E}{c} \\$Since the surface is perfectly reflecting, the photon will be reflected back,$\\$which means the change in momentum is $2p. \\$Thus, the momentum transferred to the surface is:$\\ \Delta p = 2 \times \frac{E}{c} = \frac{2E}{c} \\$Therefore, the correct option is $\frac{2E}{c}. \\$This corresponds to Option 1.