We investigate the possiblity to detect Earth-like planets, in the visible and the near infrared domains, with ground based Extremely Large Telescopes equipped with adaptive systems capable of providing high Strehl ratios. From a detailed analysis of the speckle noise, we derive analytical expressions of the signal to noise ratio on the planet flux, for direct and differential imaging, in the presence of the speckle noise and the photon noise of the residual stellar halo. We find that a 100 m telescope would detect an Earth at a distance of 10 pc, with a signal to noise ratio of 5, in an integration time of 12 h. This requires to control the instrumental aberrations with a precision better than 1 nanometer rms, and to reach an image dynamics of 1.2 × 106 at 0.1 arcsec radius. Under the same conditions, a telescope of 30 m would require a dynamics of 1.3 × 107 for a positive detection.