I have simple HTML form with submit button. After hitting this button I would like to the see div#my_id
which is not visible before.
Your submit button will submit the form it is placed in using the defined action and method. Any arguments / fileds in the form will be included as query parameters.
The reason you are not seeing your div appear, is because the click results in the page being reloaded. After the reload the div will be hidden again.