30-pin cable compatibility issue for iPod Nano

I have an iPod Nano 6th generation which I purchased new a few years ago. The cable has gotten lost, so I have tried using the cable that came with my iPod Classic. This Classic was not purchased new, I bought it off eBay. The cable that came with it does not seem to be compatible with my Nano - it looks the same as the cable that was lost, but when I use it to attach the Nano to my Macbook, nothing happens.

Can I just assume this is due to a defect in the iPod Classic which I got on eBay? Or does a cable have to be engineered to be compatible with a specific model of iPod? Looking on the Apple web site, I only see one type of 30-pin to USB cable listed there. So I assume I should be able to buy one of these and it will work with my iPod Nano?

Grateful for any information that anyone can share.

Enjoy your day!