There is no electromagnetic field (and certainly also no "electric field") if there is no current flowing. A battery per se is a chemical device, not an electrical device.
And there’s no general current flow direction with AC, it alternates, as the name implies.
And furthermore, an electromagnetic field that has an effect on an audio signal is known as an interference. It is important to keep an audio signal free from any electromagnetic interference, since this is always a distortion. Why should a signal that has been distorted by electromagnetic interference ever sound better than the undistorted original signal?