In my textbook there are 2 formulas for electric power:
$$\begin{array}{cccr} P &=& E/t &\hspace{10pt} (1) \\ P &=& VI. &\hspace{10pt} (2) \end{array}$$
What is the difference between the 2 formulas? Do they both calculate electrical power? If so, how do you know which formula to use in a given situation?
Power is defined as:
$$P(t)=\frac{dE(t)}{dt}$$
This is valid for any system. If energy is constant, then:
$$P(t)=\frac{E(t)}{t}$$
If you're dealing with a resistance in a circuit, the dissipated power is given by Joule's law:
$$P=VI$$
So the last one is a particular case of the first one.
Picking up on jinawee's answer, current is charge per unit time:
$$ I = \frac{Q}{t} $$
So substituting for $I$ in your second equation gives:
$$ P = V \frac{Q}{t} $$
But $VQ$ is just the work done, i.e. the energy, in moving a charge $Q$ through a voltage difference of $V$. So substituting $E$ for $VQ$ gives us:
$$ P = \frac{E}{t} $$
which shows that the two equations are equivalent.
The first equation is the definition of power which is, in words, the rate of energy conversion:
$$P \equiv \frac{dE}{dt}$$
Where it is understood that $E$ is the amount of energy converted. For example, in a mechanical system where gravitational potential energy is converted to mechanical kinetic energy and vica versa.
In electrical circuits, the power associated with any circuit element is simply the product of the voltage across and the current through:
$$P = VI$$
Check the units: the volt is joules per coulomb and the amp is coulombs per second so their product is joules per second or watts, the unit of power.
For a resistor, electrical energy is converted to thermal energy and via Ohm's Law
$$V_R = RI_R $$
we can write these formulas for the power associated with a resistor
$$P_R = \frac{V^2_R}{R} = RI^2_R $$
For a capacitor, the energy is converted to or from electric potential energy and, via the defining capacitor equation
$$I_C = C \frac{dV_C}{dt} $$
we can write these formulas for the power associated with a capacitor
$$P_C = CV_C\frac{dV_C}{dt} = \frac{I_C}{C}\int_{-\infty}^tI_C(\tau)\ d\tau$$
For an inductor, the energy is converted to or from magnetic energy and, via the defining inductor equation
$$V_L = L \frac{dI_L}{dt} $$
we can write these formulas for the power associated with an inductor
$$P_L = LI_L \frac{dI_L}{dt} = \frac{V_L}{L}\int_{-\infty}^tV_L(\tau)\ d\tau$$
