WordPress path url in js script file

According to the Wordpress documentation, you should use wp_localize_script() in your functions.php file. This will create a Javascript Object in the header, which will be available to your scripts at runtime.

See Codex

Example:

<?php wp_localize_script('mylib', 'WPURLS', array( 'siteurl' => get_option('siteurl') )); ?>

To access this variable within in Javascript, you would simply do:

<script type="text/javascript">
    var url = WPURLS.siteurl;
</script>

You could avoid hardcoding the full path by setting a JS variable in the header of your template, before wp_head() is called, holding the template URL. Like:

<script type="text/javascript">
var templateUrl = '<?= get_bloginfo("template_url"); ?>';
</script>

And use that variable to set the background (I realize you know how to do this, I only include these details in case they helps others):

Reset.style.background = " url('"+templateUrl+"/images/searchfield_clear.png') ";

    wp_register_script('custom-js',WP_PLUGIN_URL.'/PLUGIN_NAME/js/custom.js',array(),NULL,true);
    wp_enqueue_script('custom-js');

    $wnm_custom = array( 'template_url' => get_bloginfo('template_url') );
    wp_localize_script( 'custom-js', 'wnm_custom', $wnm_custom );

and in custom.js

alert(wnm_custom.template_url);

If the javascript file is loaded from the admin dashboard, this javascript function will give you the root of your WordPress installation. I use this a lot when I'm building plugins that need to make ajax requests from the admin dashboard.

function getHomeUrl() {
  var href = window.location.href;
  var index = href.indexOf('/wp-admin');
  var homeUrl = href.substring(0, index);
  return homeUrl;
}