I recently stated several relations in lecture without providing any proof. You will work out the derivations in this homework. We will use only Ohm’s law, the junction rule, and the ideas from previous chapters such as electric potential, work, energy, etc.

Problem 1: The first task is to prove the relation I gave in lecture for the power dissipated by a resistor. Consider a single resistor with a known value of R connected to a battery of voltage V , as in Fig. 1

i ) Given that we know the resistance and the voltage, we can find the current using Ohm’s law. How much charge flows through the resistor is a short amount of time dt? You may write this either in terms of R and V or by using the current I.

ii ) Every charge that goes through the resistor must go through a potential difference of magnitude V . Given this idea, how much total energy is lost by the charge carriers in a time dt?

iii ) The total energy lost, divided by the time it took gives the power. So for this last step, simply divide your result from ii by dt to get the power dissipated by the resistor. For completeness, use Ohm’s law to write the power in three ways, one without R, one without V and finally one without I.