A rotten egg smell that appears only on the hot tap is one of the most diagnosable problems in domestic plumbing, because the location of the symptom identifies the cause. The supply reaching the house is fine; something in the water heater is producing hydrogen sulfide. There are three things to check, in order, and two of them cost nothing.
Confirm it is really only the hot side
Run a cold tap and smell it, then run the hot and smell that, at more than one fixture. Smell on both sides means the source is the supply, which on a well points at sulfur bacteria or dissolved hydrogen sulfide in the water itself and needs treatment ahead of the house. Smell on the hot side alone puts the cause inside the water heater. Do the check at a tap far from the heater as well, since a single fixture that smells may simply have a problem in its own tailpiece or trap.
The anode rod and the bacteria
A tank heater contains a sacrificial anode rod, usually magnesium, which corrodes in preference to the tank and is what stops the tank rusting. In the warm, oxygen poor environment of a heater, sulfate reducing bacteria can use the hydrogen released by that corrosion to produce hydrogen sulfide, which is the smell. The two consequences follow directly: the bacteria need dealing with, and the anode material may need changing to an aluminium zinc alloy that supports the reaction less readily. Removing the anode entirely is not the answer, since the tank then corrodes instead.
The fix, in order
First, raise the heater temperature to the level the manufacturer permits for a disinfection period, following their instructions, which kills the bacteria. Be careful about scald risk afterwards and return the setting. Second, flush the tank thoroughly to carry out the sediment the bacteria live in. Third, if the smell returns, replace the magnesium anode with an aluminium zinc alloy rod, which very often ends it for good. Fourth, if it still returns, chlorinate the tank per the manufacturer's method. Work in that order rather than jumping to the anode.
When the problem is the water, not the heater
If the cold side smells too, the heater is innocent and the well is the issue. Hydrogen sulfide dissolved in the water needs oxidation and filtration, or aeration, or continuous chlorination with a contact tank, depending on the load. Sulfur bacteria in the well need disinfection as well as treatment, because they regrow. In either case the treatment sits ahead of the house rather than at the heater. Our softener installation guide covers where the stages sit in a well water train.
Questions people ask about hot water smells like rotten eggs how to fix
Why does only the hot water smell?
Because the hydrogen sulfide is being produced inside the water heater, where sulfate reducing bacteria interact with the sacrificial anode rod.
Should I just remove the anode rod?
No. It is what protects the tank from corroding. Replace magnesium with an aluminium zinc alloy rod instead of removing it.
Will raising the temperature fix it permanently?
It kills the bacteria present, which often clears the smell, but it can return. Flushing the tank and changing the anode material is the durable fix.
What if the cold water smells as well?
Then the source is the supply rather than the heater, and the well needs oxidation, aeration or chlorination ahead of the house.