  The 30 Dor C superbubble is unique for its synchrotron X-ray shell, as well
as being the first superbubble to be detected in TeV $\gamma$-rays, though the
dominant TeV emission mechanism, i.e., leptonic or hadronic, is still unclear.
We aim to use new Chandra observations of 30 Dor C to resolve the synchrotron
shell in unprecedented detail and to estimate the magnetic ($B$) field in the
postshock region, a key discriminator between TeV $\gamma$-ray emission
mechanisms. We extracted radial profiles in the 1.5-8 keV range from various
sectors around the synchrotron shell and fitted these with a projected and
point spread function convolved postshock volumetric emissivity model to
determine the filament widths. We then calculated the postshock magnetic field
strength from these widths. We found that most of the sectors were well fitted
with our postshock model and the determined $B$-field values were low, all with
best fits $\lesssim 20~\mu$G. Upper limits on the confidence intervals of three
sectors reached $\gtrsim 30~\mu$G though these were poorly constrained. The
generally low $B$-field values suggests a leptonic-dominated origin for the TeV
$\gamma$-rays. Our postshock model did not provide adequate fits to two
sectors. We found that one sector simply did not provide a clean enough radial
profile, while the other could be fitted with a modified model where the
projected profile falls off abruptly below $\sim0.8$ times the shell radius,
yielding a postshock $B$-field of 4.8 (3.7-11.8) $\mu$G which is again
consistent with the leptonic TeV $\gamma$-ray mechanism. Alternatively, the
observed profiles in these sectors could result from enhancements around a
shock-cloud interaction as suggested in previous works. The average $B$-field
determined around the X-ray synchrotron shell of 30 Dor C suggests the leptonic
scenario as the dominant emission mechanism for the TeV $\gamma$-rays.
