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 (definition)) 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.