Single Loop Circuit Energy Conservation

[breez]

My physics textbook states the following about a single loop circuit with a resistor and ideal battery.

"From the principle of conservation of energy, the work done by the (ideal) battery must equal the thermal energy that appears in the resistor"

Can anyone explain why energy must be conserved? Isn't the battery producing an applied force on the charges? How can we be sure this applied force is a conservative force? Also even if energy is conserved, why must the thermal dissipation equal the work done by the battery?

 

[dst]

Energy is always conserved by definition. If energy is not conserved then we create a new type of energy so it is.

In this case, current is never "lost", but voltage (electromotive force) is. That "voltage" is "converted" into heat energy and so it remains conserved. It's a simple application of ohm's law.

 

[breez]

But couldn't some of the voltage have gone into the kinetic energy of the charges? How would you write out mathematically the conservation of energy in this case?