servlet清空cookie时遇到的很奇怪的问题，求教

我自己写的登出代码：

 public void logout(HttpServletRequest request, HttpServletResponse response) {
        HttpSession session = request.getSession();
        // 从session中将user删除
        session.removeAttribute("user");

        // 将存储在客户端的cookie删除掉
        Cookie cookie_username = new Cookie("cookie_username", null);
        cookie_username.setMaxAge(0);
        cookie_username.setPath(request.getContextPath());
        // 创建存储密码的cookie
        Cookie cookie_password = new Cookie("cookie_password", null);
        cookie_password.setMaxAge(0);
        cookie_password.setPath(request.getContextPath());
        response.addCookie(cookie_username);
        response.addCookie(cookie_password);

        try {
            response.sendRedirect(request.getContextPath() + "/index.jsp");
        } catch (IOException e) {
            e.printStackTrace();
        }
    }

当我把 cookie_username.setPath设置为"/"的时候，登出就没有效果了，有没有大神能告诉我是为什么吗

https://blog.csdn.net/abc86319253/article/details/9212565