I always give long answers, so get ready. First, before looking at coins, look at the evolution of money over time. First there was the barter system, in which we traded goods for other goods. After some time passed people realized that it was more efficient to trade with a common coin. By giving something a standard of value against which all good could be measured, we could have more efficient trade. No longer would a man trying to trade his goat have to find another man willing to buy the goat for the sugar the man with the goat wanted. each man could trade their goods for a coin and then give that coin to someone else in return for what they needed. but this coin system was not completely based on faith, that is people thought that if a standard were to be used, then the coin that was given as value had to actually be worth something in and of itself. hence coins were usually made of gold or silver and often continued to circulate only in the upper levels of society that could afford to trade in coin. however over time people and lenders figured out that the coins did not need to have inherent value, that as long as people generally agreed on a coins value as a currency it did not matter what the coin was made of. This breakthrough in monetary policy allowed governments to borrow, for lenders to give loans that were larger then its holdings, and a great manyo ther things. Today the switch to electronic money represents a further evolution of monetary policy. Just like all past evolutions, the acceptance of the system of trade will be based on acceptance (faith) in the mode of trade as well as in its effiency.
The faith in electronic systems is already well established. credi card companies have been providing electronic money to us as consumers for two decades now, and the system of credit-lending has only gotten more powerful. What a pure electronic system would do is further the efficiency of the system. think of it this way. if you hold a dollar in your wallet, it is not earning interest, it is not working for you. Nor is it safe in the way money in a bank is safe. A person may steal your wallet or you may lose it. electronic money makes these probems less likely. First of all, by having all currency stored as electronic date, every dollar in the account will be earning interest until the very moment that it is used to buy something. what this means is that people/businesses can maximize the value of their money. Secondly, by not having to carry the money the risk of losing it is eliminated. So electronic money both increases efficiency as well as eliminates the risks associated with any physical object of value (primarily the risk of its loss or destruction).
Electronic money however creates new risks as well. Computer errors may cause money to "disapear" since it has no physical existence. Also hackers may gain access to accounts which may lead to losses. However such problems are likely to decrease with time is biometric protections and other safeguards are integrated in trade. So for example in the near future a retna scan might be used to identify the holder of the account, and therefore the only one who would be able to gain access to the funds would be the original account holder.
The bottom line is that electronic systems increase efficiency and decrease risk in trade. They also create new issues of privacy and security. Whether such systems develop or not is not likely an issue of contention, they already exist, the only question is how long it will be before all paper currency is comepletely eliminated.