PHP reCaptcha未在网站上显示

我正在尝试让reCaptcha为我的表单工作。没有reCaptcha，表单本身可以完美地工作。下面是我的.html和.php文件中的相关代码。

问题是reCAPTCHA小部件根本没有显示在页面上。我可以让提交按钮工作，但当然它总是失败，因为缺少小部件。

sign_up.html

<div id="form">
    <script type="text/javascript">
        var RecaptchaOptions = {
        theme : 'white'
    };
    </script>
    <form class="pure-form" action="myprocessingscript.php" method="post">        
        <fieldset class="pure-group">
            <input name="field1" type="text" class="pure-input-1-2" placeholder="Name">
            <input name="field2" type="text" class="pure-input-1-2" placeholder="Username">
            <input name="field3" type="email" class="pure-input-1-2" placeholder="Email">
        </fieldset>
        <?php
            require_once('recaptchalib.php');
            $publickey = "xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx(censored)"; 
            echo recaptcha_get_html($publickey);
        ?>
        <button name="submit" type="submit" class="pure-button pure-input-1-2 pure-button-primary">Skicka</button>      
    </form>
</div>

myprocessingscript.php

<?php
  require_once('recaptchalib.php');
  $privatekey = "xxxxxxxxxxxxxxxxxxxxxxxxxxxx(censored)";
  $resp = recaptcha_check_answer ($privatekey,
                                $_SERVER["REMOTE_ADDR"],
                                $_POST["recaptcha_challenge_field"],
                                $_POST["recaptcha_response_field"]);
  if (!$resp->is_valid) {
    die ("The reCAPTCHA wasn't entered correctly. Go back and try it again." .
         "(reCAPTCHA said: " . $resp->error . ")");
  } else {
    if(isset($_POST['field1']) && isset($_POST['field2']) && isset($_POST['field3'])) {
    $data = $_POST['field1'] . '-' . $_POST['field2'] . '-' . $_POST['field3'] . "'n";
    $ret = file_put_contents('submissions.txt', $data, FILE_APPEND | LOCK_EX);
    if($ret === false) {
        die('There was an error writing this file');
    }
    else {
        echo "$ret bytes written to file";
    }
}
else {
   die('no post data to process');
}
  }
?>

谢谢你的帮助！