Is it permissible for a woman to use the electricity of her married brother's house to charge her personal devices, given that she pays for the internet subscription, and would the money earned from her online work be unlawful or require permission, and do these usages affect the validity of her acts of worship, such as paying zakat and reciting the Quran, knowing that her brother and his wife share in the household expenses?

The customary use of electricity does not require verbal permission, as custom is authoritative, and customary permission is equivalent to verbal permission. The indications of customs and circumstantial evidence take the place of explicit statements. Therefore, customary permission for permissibility, ownership, or disposition through agency is like verbal permission. It is sufficient for the questioner to know, by customary indication, that her brother and his wife consent to her use of electricity.