The smallest travel chargers / adapters available these days do not contain a magnetic transformer.
Instead, isolation and AC coupling is achieved by using piezoelectric transformers: These are really tiny, and flat, in comparison to the conventional magnetic coil based transformer, and work at greater efficiency as well. The frequencies involved are not AC power line frequencies, they are much higher.
See the image below:
The design is the same as described in the answer by Nick Alexeev (switched mode), with power line AC being rectified to DC, then high frequency AC generated, isolated via the Piezoelectric transformer, and rectified again for DC.
The use of high frequencies also reduces the required size of capacitors for smoothing the DC voltage, another key factor contributing to the bulk of older chargers / wall warts.
For additional information about such tiny travel chargers, see this answer.
This is some sort of RF test connector with a built-in switch. Possibly an MS-156 or MS-147. There is a trace going under the connector from the bottom, and another one from the top. One side goes to the radio in the phone, the other goes to the antenna. Generally the test connector is installed so that when something is plugged in, the antenna gets disconnected and the external connector is connected to the radio in its place. This allows easy testing of the radio during assembly to ensure that everything works correctly. This is necessary because if the antenna is not bypassed, the whole phone would need to be placed in a faraday cage to isolate it. My phone (a galaxy S3) has three of them under small round stickers.
They're not really designed for use with external antennas, but that doesn't mean it's not possible, just not convenient. Generally they're designed to work with pogo pins in an automated tester, it may be very difficult to find the correct adapter. And then you would need to know what frequencies the radio needs to use, and find an antenna that wil work across those frequencies. Modern smart phones have radios for GPS, Bluetooth, and Wifi in addition to the cell network radios LTE, GSM, Edge, 3G, etc. These radios operate in different bands and in some cases share the same antenna. There may also be complex RF switching to select the correct antenna and/or switch antennas on the fly, complicating the process of properly adding an external antenna. You may need more than one antenna to get everything to work correctly.
From looking at the images of the S4 in ifixit, there is definitely more than one antenna. Samsung does an excellent job integrating everything, though, so they are not very obvious. Looks like there are at least 4 antennas in the back cover - two on the top right, one on the top left, and one at the bottom. You can see two of them after taking the battery cover off; they are under the two small plastic covers on the sides at the top. Then there is another one that curves around the inside of the top right corner. Then there is at least one more down at the bottom with the speaker.
Best Answer
That "little white thing" is a thermal cutoff. It is a sort of fuse that reacts to external heat.
They are used to prevent things from burning up. The one you have is rated for 130 degrees celsius.
Measure it with an ohmmeter. It should read as a complete short circuit (very nearly zero ohms.)
If it reads as an open circuit (high resistance) then it was exposed to too much heat and opened the circuit. This probably saved you from having a fire.
Replace it with a part rated for the same temperature.
Under no circumstances should you leave it out or replace it with a wire. If it has opened up, then it is needed.
Try not to place as heavy a load on the transformer, or maybe not run it as long. Also make sure not to cover it up - it needs air circulation to keep cool.