Is there any way I can connect a USB external hard drive to your computer via an eSATA port instead of USB?

There is a difference in speed. With a USB connection its 480 Mb / s, eSATA although its 1000Mb / s. Is there some kind of adapter that can make my USB external hard drive connected to eSATA port in my pc?

No, but even if it were possible, would remain limited to the speed of USB. Any operation of a compuer is not limited to the component Slower operation

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