We report X-ray observations of the powerful radio galaxy Cygnus A with the Chandra X-ray Observatory .
This letter focuses on the radio hot spots , all four of which are detected in X-rays with a very similar morphology to their radio structure .
X-ray spectra have been obtained for the two brighter hot spots ( A and D ) .
Both are well described by a power law with photon index \Gamma = 1.8 \pm 0.2 absorbed by the Galactic column in the direction of Cygnus A .
Thermal X-ray models require too high gas densities and may be ruled out .
The images and spectra strongly support synchrotron self-Compton models of the X-ray emission , as proposed by Harris , Carilli & Perley on the basis of ROSAT imaging observations .
Such models indicate that the magnetic field in each of the brighter hot spots is 1.5 \times 10 ^ { -4 } gauss , with an uncertainty of a few tens of percent .
This value is close to the equipartition field strengths assuming no protons are present .
The possibility that the X-rays are synchrotron radiation is briefly discussed , but not favored .
We speculate that production of the \gamma \sim 10 ^ { 7 } electrons necessary for X-ray synchrotron radiation from hot spots is inhibited when the external gas density is high , as is the case when the radio galaxy is within a cooling flow .