you forget to open the Connection while executing the first Query
DataTable dt = new DataTable();
string sql = "select Field_ID,Field_name_english FROM Field_TB";
if (con.State == ConnectionState.Closed)
{
con.Open();
}
SqlDataAdapter da = new SqlDataAdapter(sql, con);
da.Fill(dt);
DropDegree.DataSource = dt;
DropDegree.DataTextField = "Field_name_english";
DropDegree.DataValueField = "Field_ID";
DropDegree.DataBind();
DropDegree.SelectedValue = "your degree";
if (con.State == ConnectionState.Open)
{
con.Close();
}