How do I store the user uid to firebase database after I register an account?

Here's my code in .js

$("#loginBtn").click(
function(){


var email = $("#loginEmail").val();
var password = $("#loginPassword").val();

if(email != "" && password != ""){
  $("#loginProgress").show();
  $("#loginBtn").hide();
  $("#registerBtn").hide();

  firebase.auth().signInWithEmailAndPassword(email, 
password).catch(function(error) {
    // Handle Errors here.
    var errorCode = error.code;
    var errorMessage = error.message;

    $("#loginError").show().text(errorMessage);
    $("#loginProgress").hide();
    $("#loginBtn").show();
    $("#register_account_Btn").hide();
    $("#back_btn").hide();

  });
 }
}
);

/* REGISTER PROCESS */

$("#register_account_Btn").click(
 function () {
var email = $("#regEmail").val();
var password = $("#regPassword").val();

if(email != "" && password != ""){
  $("#loginProgress").show();
  $("#loginBtn").hide();

  firebase.auth().createUserWithEmailAndPassword(email,
   password).catch(function(error) {
    // Handle Errors here.
    var errorCode = error.code;
    var errorMessage = error.message;

    $("#loginError").show().text(errorMessage);
    $("#loginProgress").hide();
    $("#loginBtn").show();
    $("#register_account_Btn").hide();
    $("#back_btn").hide();

  });
}

I found this on the internet but I don't know where to insert it. Is this correct?

var uid = firebase.auth().currentUser.uid;
  firebase.database().ref().child('accounts').child(uid).set({
    email: user.email,
    userId: uid
  })

Should I create another function?

and these are the rules in my firebase database. Is it advisable to use this? Additional Question: I want to create a user roles: teacher and student. Teacher can create classroom, add students, view the list of students. While the student can view his/her classmates and teacher. How should I do this?

    {
  "rules": {
    "users": {
      "$uid": {
        ".read": false,
        ".write": "$uid === auth.uid"
       }
      }
    }
   }

Sincerest apology but I'm new to this ;(