Sort posts by category (specific problem)

Hello everybody,

I’ve been trying for a while now, to sort all my posts by category. More precisely, to have a list of all the categories, with all the relatives posts’ titles sorted by name, under the relative category.
For that purpose I wrote this:

<?php wp_reset_query();
		$categories = get_categories( 'hide_empty=1' ); 

		foreach($categories as $category){
			$currentCat = get_cat_id($category);
			echo '<div class="category"><ol>'.$category->category_nicename;
			$myquery = new WP_Query( 'cat=$currentCat', array ( 'orderby' => 'rand', 'posts_per_page' => '1' ) );
			while ( $myquery->have_posts() ) : $myquery->the_post(); ?>
					<li><a href="<?php the_title_attribute(); ?>"><?php the_title(); ?></a></li>
			<?php endwhile;
			echo '</ol></div>';
			wp_reset_postdata();
		} ?>

It works only partially: i get a correct alphabetical list of all the categories, but the sub-lists of posts show always the same 10 posts, the first 10 posts sorted by date.
(LIKE THIS)
I’ve already checked other people’s solution to similar problems, but I can’t figure out what I’m doing wrong.
Thanks in advance…