What is the ruling on displaying the ears in a passport photograph, if that is in accordance with the prevailing law and it cannot be obtained by any other means, and the passport will be used to travel to an Arab country?

The ear is part of the 'awrah (parts of the body that must be covered) that a woman is not permitted to expose to non-mahram men. However, necessity or pressing need – such as for a passport photo or ID card – makes it permissible, provided that one limits it to the extent of the need, for necessities are measured by their بقدرها.