However, if you are not in a hurry, I highly recommend that you take the smaller roads across the Negev, going towards Mizpe Ramon (it is a small town up on the mountain ridge and a major intersection of all the roads in this region, and is therefore clearly marked on all the road signs). The roads wind through the desert, and the scenery you see would most likely exist on Mars. No camera could do it justice, though I tried...
On this particular trip, we had some extra entertainment. In Israel, of course, you are inseparable from your cell phone (and generally there is service, except in the Golans). We rented one, and it helped us stay connected with friends and find places. So at one point, driving through the desert, we got a call from a concerned friend asking us whether we've been flooded yet. I thought it very funny. But he wasn't joking. When it rains in the mountatins (and it can be quite far away, we haven't seen any rain around) the water finds its way down, carrying stones and sand, and flooding roadways as it flows through the gullies. Sure enough, soon we came to one such flooded crossing.
As you follow the road, at some point you see a huge mountain ridge ahead, stretching all across. The first time we were driving this way, in 1992, we started wondering: where would the road go? Is there a tunnel? Now we know that there isn't. The road starts climbing up the ridge, and up and up, and then there is Mizpe Ramon. If you start from Eilat around 4 or 5 in the afternoon, you'll reach Mizpe Ramon between 7 and 8, just as the sun is setting.