Ошибка invalid captcha

I have a PHP script that will process the Google Recaptcha, how can I send an error back to javascript to print an error like it does for the other fields when user does not interact with the Captcha?

PHP Script:

<?php
function validateRecaptcha($secret, $clientResponse, $clientIp)
{
    $data = http_build_query([
        "secret"   => $secret,
        "response" => $clientResponse,
        "remoteip" => $clientIp,
    ]);

    $options = [
        "http" => [
            "header" =>
                "Content-Type: application/x-www-form-urlencodedrn".
                "Content-Length: ".strlen($data)."rn",
            "method"  => "POST",
            "content" => $data,
        ],
    ];

    $response = file_get_contents(
        "https://www.google.com/recaptcha/api/siteverify",
        false,
        stream_context_create($options)
    );

    if($response === false)
    {
        return false;
    }
    else if(($arr = json_decode($response, true)) === null)
    {
        return false;
    }
    else
    {
        return $arr["success"];
    }
}

$errors         = array();      // array to hold validation errors
$data             = array();         // array to pass back data

// validate the variables ======================================================
    // if any of these variables don't exist, add an error to our $errors array

    if (empty($_POST['firstName']))
        $errors['firstName'] = 'First Name is required.';

    if (empty($_POST['lastName']))
        $errors['lastName'] = 'Last Name is required.';

    if (empty($_POST['companyName']))
        $errors['companyName'] = 'Company Name is required.';

    if (empty($_POST['companyAddress']))
        $errors['companyAddress'] = 'Company Address is required.';

    if (empty($_POST['city']))
        $errors['city'] = 'City is required.';

    if (empty($_POST['state']))
        $errors['state'] = 'State is required.';

    if (empty($_POST['emailAddress']))
        $errors['emailAddress'] = 'Email Address is required.';

    if (empty($_POST['comment']))
        $errors['comment'] = 'Comment is required.';

    if (empty($_POST['g-recaptcha-response']))
        $errors['captcha'] = 'Captcha is required.';


// return a response ===========================================================

    // if there are any errors in our errors array, return a success boolean of false

    if(empty($errors))
    {
        if(!validateRecaptcha($secret, $_POST['g-recaptcha-response'], $_SERVER["REMOTE_ADDR"]))
        {
            $errors['captcha'] = 'Captcha is required.';
        }
    }

    if ( ! empty($errors)) {

        // if there are items in our errors array, return those errors
        $data['success'] = false;
        $data['errors']  = $errors;
    } else {

        // if there are no errors process our form, then return a message

        // DO ALL YOUR FORM PROCESSING HERE
        // THIS CAN BE WHATEVER YOU WANT TO DO (LOGIN, SAVE, UPDATE, WHATEVER)

        // show a message of success and provide a true success variable
        $data['success'] = true;
        $data['message'] = 'Success!';
    }

    // return all our data to an AJAX call
    echo json_encode($data);

Javascript:

// Start
$(document).ready(function() {

    // process the form
    $('form').submit(function(event) {

        $('.form-group').removeClass('has-error'); // remove the error class
        $('.help-block').remove(); // remove the error text

        // get the form data
        // there are many ways to get this data using jQuery (you can use the class or id also)
        var formData = {
            'firstName'                 : $('input[name=firstName]').val(),
            'lastName'              : $('input[name=lastName]').val(),
            'companyName'               : $('input[name=companyName]').val(),
            'companyAddress'                : $('input[name=companyAddress]').val(),
            'city'              : $('input[name=city]').val(),
            'state'                 : $('input[name=state]').val(),
            'emailAddress'          : $('input[name=emailAddress]').val(),
            'comment'   : $('input[name=comment]').val(),
        };

        // process the form
        $.ajax({
            type        : 'POST', // define the type of HTTP verb we want to use (POST for our form)
            url         : 'formMaster.php', // the url where we want to POST
            data        : formData, // our data object
            dataType    : 'json', // what type of data do we expect back from the server
            encode      : true
        })
            // using the done promise callback
            .done(function(data) {

                // log data to the console so we can see
                console.log(data);

                // here we will handle errors and validation messages
                if ( ! data.success) {

                    // handle errors for name ---------------
                    if (data.errors.firstName) {
                        $('#firstName-group').addClass('has-error'); // add the error class to show red input
                        $('#firstName-group').append('<div class="help-block">' + data.errors.firstName + '</div>'); // add the actual error message under our input
                    }

                    // handle errors for name ---------------
                    if (data.errors.lastName) {
                        $('#lastName-group').addClass('has-error'); // add the error class to show red input
                        $('#lastName-group').append('<div class="help-block">' + data.errors.lastName + '</div>'); // add the actual error message under our input
                    }

                    // handle errors for name ---------------
                    if (data.errors.companyName) {
                        $('#companyName-group').addClass('has-error'); // add the error class to show red input
                        $('#companyName-group').append('<div class="help-block">' + data.errors.companyName + '</div>'); // add the actual error message under our input
                    }

                    // handle errors for Company Address ---------------
                    if (data.errors.companyAddress) {
                        $('#companyAddress-group').addClass('has-error'); // add the error class to show red input
                        $('#companyAddress-group').append('<div class="help-block">' + data.errors.companyAddress + '</div>'); // add the actual error message under our input
                    }

                    // handle errors for Company Address ---------------
                    if (data.errors.city) {
                        $('#city-group').addClass('has-error'); // add the error class to show red input
                        $('#city-group').append('<div class="help-block">' + data.errors.city + '</div>'); // add the actual error message under our input
                    }

                    // handle errors for Company Address ---------------
                    if (data.errors.state) {
                        $('#state-group').addClass('has-error'); // add the error class to show red input
                        $('#state-group').append('<div class="help-block">' + data.errors.state + '</div>'); // add the actual error message under our input
                    }

                    // handle errors for Email Address ---------------
                    if (data.errors.emailAddress) {
                        $('#emailAddress-group').addClass('has-error'); // add the error class to show red input
                        $('#emailAddress-group').append('<div class="help-block">' + data.errors.emailAddress + '</div>'); // add the actual error message under our input
                    }

                    // handle errors for superhero alias ---------------
                    if (data.errors.comment) {
                        $('#comment-group').addClass('has-error'); // add the error class to show red input
                        $('#comment-group').append('<div class="help-block">' + data.errors.comment + '</div>'); // add the actual error message under our input
                    }

                    // handle errors for recaptcha ---------------
                    if (data.errors.recaptcha) {
                        $('#recaptcha-group').addClass('has-error'); // add the error class to show red input
                        $('#recaptcha-group').append('<div class="help-block">' + data.errors.recaptcha + '</div>'); // add the actual error message under our input
                    }

                } else {

                    // ALL GOOD! just show the success message!
                    $('form').append('<div class="alert alert-success">' + data.message + '</div>');

                    // usually after form submission, you'll want to redirect
                    // window.location = '/thank-you'; // redirect a user to another page

                }
            })

            // using the fail promise callback
            .fail(function(data) {

                // show any errors
                // best to remove for production
                console.log(data);
            });

        // stop the form from submitting the normal way and refreshing the page
        event.preventDefault();
    });

});

HTML:

<!doctype html>
<html>
<head>
    <title>Form Master</title>
    <link rel="stylesheet" href="//netdna.bootstrapcdn.com/bootstrap/3.0.3/css/bootstrap.min.css"> <!-- load bootstrap via CDN -->

    <script src="//ajax.googleapis.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script> <!-- load jquery via CDN -->
    <script src='https://www.google.com/recaptcha/api.js'></script>
    <script src="formMaster.js"></script> <!-- load our javascript file -->
</head>
<body>
<div class="col-sm-6 col-sm-offset-3">

    <h1>Contact Form</h1>

    <!-- OUR FORM -->
    <form name="form" id="form" action="formMaster.php" method="POST">

        <!-- NAME -->
        <div id="firstName-group" class="form-group">
            <label for="firstName">First Name</label>
            <input type="text" class="form-control" name="firstName" placeholder="Henry Pym">
            <!-- errors will go here -->
        </div>

        <!-- NAME -->
        <div id="lastName-group" class="form-group">
            <label for="lastName">Last Name:</label>
            <input type="text" class="form-control" name="lastName" placeholder="Henry Pym">
            <!-- errors will go here -->
        </div>

        <!-- NAME -->
        <div id="companyName-group" class="form-group">
            <label for="companyName">Company Name:</label>
            <input type="text" class="form-control" name="companyName" placeholder="Henry Pym">
            <!-- errors will go here -->
        </div>

        <!-- NAME -->
        <div id="companyAddress-group" class="form-group">
            <label for="companyAddress">Company Address:</label>
            <input type="text" class="form-control" name="companyAddress" placeholder="Henry Pym">
            <!-- errors will go here -->
        </div>

        <!-- NAME -->
        <div id="city-group" class="form-group">
            <label for="city">City:</label>
            <input type="text" class="form-control" name="city" placeholder="Henry Pym">
            <!-- errors will go here -->
        </div>

        <div id="state-group" class="form-group">
      <label for="state">State</label>
      <select id="statea" name="state" class="form-control">
        <option selected>Choose...</option>
        <option>...</option>
      </select>
    </div>

        <!-- EMAIL ADDRESS -->
        <div id="emailAddress-group" class="form-group">
            <label for="emailAddress">Email Address:</label>
            <input type="text" class="form-control" name="emailAddress" placeholder="rudd@avengers.com">
            <!-- errors will go here -->
        </div>

        <!-- COMMENT -->
        <div id="comment-group" class="form-group">
            <label for="comment">Comment:</label>
            <input type="text" class="form-control" name="comment" placeholder="Ant Man">
            <!-- errors will go here -->
        </div>

        <div name="recaptcha" id="recaptcha-group" class="form-group">
            <div class="g-recaptcha" data-sitekey="6LePAV4UAAAAADzvGMCMHHJ47y9bUnmP4OhFJTax"></div>
        </div>

        <button type="submit" class="btn btn-success">Submit <span class="fa fa-arrow-right"></span></button>

    </form>

</div>
</body>
</html>

But when the form is submitted other fields are highlighted with their errors except for the recaptcha section. What am I doing wrong? What needs to be done to have an error print for the recaptcha as well?