Is it permissible for a woman traveling with her husband to uncover her ear for a photograph if it is an essential condition of the country she will be traveling to for study and work, and it is a non-Islamic country?

For a woman to expose her ear in front of a foreign (non-mahram) photographer is forbidden, because the ear is awrah (private parts) and must be covered from all men except her husband and mahrams. If it is possible for her husband or a mahram to photograph her, then that is obligatory. Many contemporary scholars have forbidden a non-mahram man from looking at awrah even if it is in a photograph. It is not permissible to commit a forbidden act in pursuit of financial gain, and there is no obedience to a creature in disobedience to the Creator. Allah has opened many doors of lawful provision for the Muslim. A Muslim must seek Allah's help in all their affairs.