In order to change the default search engine or the order of search engines in the top right search box in Firefox, please perform the following steps:

1. Move the mouse cursor inside the search box at the top right of the Firefox window and click the down arrow next to the provider's logo.
2. If you would like to select another search engine, simply pick the engine from the dropdown list.

In order to change the order of the search engines, do this:

1. From the dropdown list, select Manage Search Engines....
2. In the upcoming window, you have all your search engines listed. To change the order, simply click one search engine and then select Move up or Move down from the buttons on the right side in order to change the order of the search engines.

Thanks, but I but my question pertains to the URL bar, as I stated, which is different from the top right search bar.

The Right search bar and URL Search are integrated.