<script type="text/javascript">
$(function () ) {
$('#category_id').change(function(){
var id = $(this).find(':selected')[0].id;
$('#something').val(id);
alert(id); // this is just for testing remove this :|
});
</script>
<input type="hidden" name="category_id_s" value="cars" id="something" />
etc..
http://www.wjunction.com/92-development-area/159368-print-selected-value-list-menu-php.html
http://pastebin.com/5U5Xedp8
Not sure why you would want to get the selected value to another html element.
You can get the same thing by adding name="category_id" attribute to the select tag. Value of category_id when submitted would be equal to the selected item's value