jquery, selector for class within id

Always use

//Super Fast
$('#my_id').find('.my_class'); 

instead of

// Fast:
$('#my_id .my_class');

Have look at JQuery Performance Rules.

Also at Jquery Doc

You can use the class selector along with descendant selector

$("#my_id .my_class")

Just use the plain ol' class selector.

$('#my_id .my_class')

It doesn't matter if the element also has other classes. It has the .my_class class, and it's somewhere inside #my_id, so it will match that selector.

Regarding performance

According to the jQuery selector performance documentation, it's faster to use the two selectors separately, like this:

$('#my_id').find('.my_class')

Here's the relevant part of the documentation:

ID-Based Selectors

// Fast:
$( "#container div.robotarm" );

// Super-fast:
$( "#container" ).find( "div.robotarm" );

The .find() approach is faster because the first selection is handled without going through the Sizzle selector engine – ID-only selections are handled using document.getElementById(), which is extremely fast because it is native to the browser.

Selecting by ID or by class alone (among other things) invokes browser-supplied functions like document.getElementById() which are quite rapid, whereas using a descendent selector invokes the Sizzle engine as mentioned which, although fast, is slower than the suggested alternative.