# How to Rigorously Derive P(t) = V(t)I(t) in an Electric Circuit?

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In summary: But in a single point in space, the power is the magnitude of the Poynting vector, which has units of power per area, and is also the magnitude of the cross product of the electric and magnetic field vectors.In summary, the conversation is about deriving the equation for instantaneous power in an electric circuit, P(t) = V(t) I(t). The process involves starting with the equation P = {\bf F} \cdot {\bf v} and using the concepts of force, voltage, and electric field to arrive at the desired result. However, there is a distinction between power in a circuit and power at a single point in space, and it is important to clarify which is being calculated in order to derive the equation accurately
a4b3c2d1e0f
I kindly ask for assistance in derivation of the equation for instantaneous power in an electric circuit, P(t) = V(t) I(t). I want to derive it as rigorously as possible. Here's what I got:
We start with $P = {\bf F} \cdot {\bf v}$, where ${\bf v} = \frac{d\bf r}{dt}$
We know that the force exerted on a test charge q is given by ${\bf F} = {\bf E} q$, and for voltage we know $dV = - {\bf E} \cdot dx$.
Inserting F in equation for power, we get $P = {\bf E}q \cdot \frac{d\bf x}{dt} = {\bf E}\cdot dx \frac{q}{dt} = - dV \frac{q}{dt} .$
How would I go from this, to the desired result, without taking a "quantum leap"?
Is there a better way to actually derive this mathematically impeccably?

You need first to decide whether you want the power in a circuit, as you described,

or the flow of power through a single point in space, which you appear to be trying to calculate.

The old fashioned definition of EMF made this quite clear for a circuit.

The EMF of a circuit develops the power in that circuit, equal to the EMF times the current flowing in that circuit.

## 1. What is the formula for P(t) = V(t)I(t)?

The formula for P(t) = V(t)I(t) is a mathematical representation of power, which is equal to the voltage (V) multiplied by the current (I) at a specific time (t). This formula is commonly used in electrical and electronic engineering to calculate the amount of power consumed by a circuit or device.

## 2. How is P(t) = V(t)I(t) derived?

P(t) = V(t)I(t) is derived from Ohm's law, which states that the current through a conductor is directly proportional to the voltage and inversely proportional to the resistance. By rearranging the formula V=IR (voltage = current x resistance), we can substitute I for V/R and get P = VI = (V/R) x V = V^2/R. This leads to the final formula P(t) = V(t)I(t).

## 3. What are the units of measurement for P(t) = V(t)I(t)?

The units of measurement for P(t) = V(t)I(t) are watts (W), which is the standard unit for power. This formula can also be expressed in different units depending on the values of voltage and current being used, such as kilowatts (kW) or megawatts (MW).

## 4. Can P(t) = V(t)I(t) be used for both AC and DC circuits?

Yes, P(t) = V(t)I(t) can be used for both AC (alternating current) and DC (direct current) circuits. However, for AC circuits, the values of voltage and current are constantly changing, so the formula is usually written as P(t) = V(t)I(t)cosφ, where φ is the phase angle between voltage and current.

## 5. How is P(t) = V(t)I(t) used in practical applications?

P(t) = V(t)I(t) is used in various practical applications, including calculating the power consumption of electrical devices, determining the size and capacity of power sources such as batteries or generators, and analyzing the efficiency of electrical systems. It is also an important concept in renewable energy systems, as it helps to understand and optimize the power output of solar panels, wind turbines, and other renewable sources.

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