When the potential difference applied to an X-ray tube is doubled, the cutoff wavelength will be doubled as well. This is because the cutoff wavelength is inversely proportional to the potential difference.

Option 1: This option suggests that the cutoff wavelength will remain the same. However, this is incorrect because the cutoff wavelength is directly influenced by the potential difference.

Option 2: This is the correct answer. Doubling the potential difference will lead to a doubling of the cutoff wavelength. This can be explained by the fact that increasing the potential difference increases the energy of the electrons, resulting in greater penetration of the X-rays and thus a longer wavelength.

Option 3: This option states that the cutoff wavelength will be halved. However, this is incorrect. Halving the potential difference would actually result in a shorter cutoff wavelength, as lower energy electrons would be produced.

Option 4: This option suggests that the cutoff wavelength will be four times the original wavelength. However, this is incorrect. Increasing the potential difference by a factor of 2 would only double the cutoff wavelength, not increase it fourfold.

In conclusion, when the potential difference applied to an X-ray tube is doubled, the cutoff wavelength will also be doubled.