It is not permissible for a woman to be appointed as a judge

Shaykh Ibn Jibreen was asked whether it is permissible for a woman to be a qaadi. He replied: 

It is not permissible for a woman to be appointed to public positions which generally require speaking with men, mixing with them and going out frequently, asking questions of non-mahram men, and answering their questions. Such behaviour indicates that a woman is careless and audacious, which will lead to her giving up her modesty. This position also requires her  to raise her voice. That is contrary to her femininity and modesty. Similarly a woman cannot become an imam or khateeb or lawyer who has to frequent the courts and offices which are filled with men. 

This is also a kind of imitating men, and the Prophet (peace and blessings of Allaah be upon him) cursed women who imitate men. 

With regard to jobs which a woman may need, there is nothing wrong with a woman being appointed to them, such as teaching female students, being a doctor or nurse treating female patients. A woman may also work in offices that are visited only by women, so that women will not be forced to mix with men, which may be a cause of spreading wanton display (tabarruj) and unveiling, and other things that lead to evil and immorality. And Allaah knows best. 

Al-Lu’lu’ al-Makeen min Fataawa al-Shaykh Ibn Jibreen, p. 304. 

For more information on the evidence, please see question no. 20677.