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I have an AT&T iPhone 4, which may have antenna problems when held in some positions (i.e. the "grip of death"). To help with this, Apple gave away bumper cases which worked around the problems when the phones were new. Unfortunately, my old iPhone case broke, so now I'm looking at new cases.

What properties must my new case have to be to avoid these problems with the iPhone 4's antenna?

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A piece of non-conductive scotch tape is all that is required to prevent your skin from bridging the gap in the metal band and changing the impedance of the antennas.

As long as your skin doesn't short the external metal band, you are good to go, so even extremely thin gloves that are dry would work without any case whatsoever.

For more reading, see:

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The only thing that matters is electrical isolation. A thin layer of non-conductive tape would probably be fine. It is not the distance from your skin but the possibility of your slightly-conductive fingers (depending on personal skin type, sweatiness, humidity etc.) bridging the electrically-insulated gap in the metal band antenna that causes a slight degradation in phone reception on the iPhone 4.

As long as the case stops your fingers from touching the metal I'd say any case is fine.

The AnandTech review of the iPhone 4 shows the actual metal pieces that function as the antenna. It is important for antennas to be the correct length to function correctly for the band they are designed to send or receive. If you bridge the gap then you change the effective length of the pieces of metal, and that's the first problem. The second problem is that you are electrically connecting a pair of circuits that are not designed to be connected. Your skin acts as a resistor of some finite value, electrically speaking. A gap, or insulation, is an infinite resistance.

The reason that the iPhone 4S does not have this issue is that it has two separate antennae for each band (a "diversity antenna" setup) and the phone will dynamically use whichever antenna has the best signal at any time. It is much harder for you to bridge all the gaps when you hold the phone in your hand so you can get by without a case with a 4S.

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