## Voltage and Current

As described in the Introduction to this book, familiarity with the physics of electricy & magnetism is a prerequisite. See that section for further references.

Here is a brief conceptual review:

### Electric Potential

As discussed in the Electrons at Rest chapter, there are three ways to describe a system of electric charges in space.

To be more precise, there are three ways to calculate the force on a test charge introduced into an existing system of charges:

1. Distribution of Charges: a local density or point concentration of charge, with Coulomb's Law attraction/repulsion to a test charge.
2. Electric Field: a vector field of the total electric force a test charge would experience at each point in space, $\vec{F} = Q \vec{E}$.
3. Electric Potential: a scalar field whose (negative) gradient is the electric field; the (positive/negative) test charge would feel a force pushing toward (lower/higher) potential.

These three are effectively equivalent ways of describing the same situation.

Electric potential is also called voltage.

The three descriptions are strictly mathematically related:

• To go from a distribution of charges to an electric field, just sum up the vector forces on a unit test charge ($q_t=1 \ \text{C}$) positioned at $(x,y,z)$.
• To go from a electric field to an electric potential, just add up the (negative) dot product of electric field and length over a path.

Newcomers often forget that potential, field, and charge distribution are three interchangable ways of thinking about the same underlying physical situation. While voltage has a priviliged place because it's how most circuit systems are analyzed as Systems of Equations, it's often possible to gain a lot of intuition by remembering that voltage differences imply electric fields, and those electric fields imply forces on other charges within the system.

The study of electronics is essentially the study of how charges move around in a system, and just as the first step in solving any kinematics problem is to look at the forces on each object, a useful step in electronics is to think about the forces on those charges. Those forces are just the (negative) gradient of the voltage.

In mechanics and kinematics, forces on an object lead to accelerations and velocities of those objects.

In electronics, forces on charges lead to acceleration and velocities of those charges, which are described by currents. An electric current is just a particular way of describing the average flow rate of electric charges, as covered in Electrons in Motion.

### Energy Interpretation

Work, in the mechanical sense, means:

$$\text{Work} = \vec{\text{Force}} \cdot \vec{\text{Distance}}$$

A force applied to an object moved over a distance is work, in units of energy.

In a region of space where the electric field is $\vec{E}$, let's calculate how much work is needed to move a charge $Q$ over some small path segment $\vec{\Delta x}$:

\begin{align} \Delta \text{Work} & = \vec{\text{Force}} \cdot \vec{\text{Distance}} \\ & = \left( \text{Charge} \cdot \vec{\text{Field}} \right) \cdot \vec{\text{Distance}} \\ & = Q \vec{E} \cdot \vec{\Delta x} \end{align}

Instead of looking at the total work for all the charges $Q$, let's consider the work per unit charge so that we have one convenient work value to consider whether we're discussing 1 electron or 1 trillion electrons:

$$\frac {\Delta \text{Work}} {\text{Charge}} = \vec{E} \cdot \vec{\Delta x}$$

This left-hand side is the definition of voltage or electric potential:

$$\Delta \frac {\text{Work}} {\text{Charge}} = \text{Electric Potential} = \text{Voltage}$$

$$\Delta V = \frac {\Delta W} {Q} = \vec{E} \cdot \vec{\Delta x}$$

Volts are measured in units of energy per unit charge, or:

$$\text{Volts} = \frac {\text{Joules}} {\text{Coulomb}}$$

### Voltage is Relative

Voltage is always relative. This means it is always defined as a difference between two locations.

Notice that so far we've only defined an equation for a change in potential: $\Delta V$.

There is no "absolute" work: a force only pushes a charge from one location to another, and is meaningless without both start and end locations.

You must specify both locations whenever you talk about a voltage. For example,

$$V_{A B} = \int _B ^A - \vec{E} \cdot \vec{dl} = \int _A ^B \vec{E} \cdot \vec{dl}$$

describes an integral over a path betewen two points B and A.

As we'll discuss in a later section on Kirchhoff's Voltage Law and Kirchhoff's Current Law, the variable $V_{A B}$ would be described as "the voltage at A with respect to B," or "the voltage difference from B to A." This labeling and its directionality is confusing enough that we discuss it again in the Labeling Voltages, Currents, and Nodes section.

In the Ground section we'll discuss why sometimes we might write $V_A$, but that this is just a bookkeeping shortcut to more conveniently write voltages with a globally shared, implied reference point.

### Path Dependence (Sometimes)

In the Electrons at Rest section we described the idea that the electric potential doesn't actually exist if there are time-varying magnetic fields.

In that case, the voltage "from B to A" would actually depend on which path we took. In a non-conservative field, we could make a closed loop and possibly gain or lose voltage. This would make a voltage be path dependent.

As discussed in that section, while true, this greatly complicates our analysis, and 99% of the time we are OK to assume that the electric field is conservative and so a voltage potential function exists, so voltages are not path dependent.

These three statements:

1. There are no time-varying magnetic fields.
2. The electric field is conservative.
3. The voltage from B to A is independent of which path we take.

are three identical ways to state the same assumption. This assumption is a key one in the Lumped Element Model and Kirchhoff's Voltage Law. We make this incorrect assumption work in practice by modeling the behavior of these time-dependent magnetic fields into the behavior of components like inductors.

### Current

Current and voltage are the two primary variables of interest.

The equations we set up to solve a circuit will all be functions relating various voltages and currents.

Physically speaking, the two are related because electric fields cause charges to accelerate. The direction and magnitude of the average veolcity of this movement is simply called "current."

As discussed in the earlier Electrons in Motion section, current is defined in terms of the flow of positive charge. As our charge carrier particle is typically the negatively charged electron, we have to be careful about the sign of current.

### Through and Across

Current is always measured through a surface (such as one terminal of a component). Imagine that surface as an imaginary area slicing somewhere in your system, counting electric charges as they pass.

Voltage is always measured across two points. It is always a difference in potential energy (per unit charge) in moving from one point to the other.

The Kirchhoff's Voltage Law and Kirchhoff's Current Law section will explore how the implied meanings of the words "through" and "across" lead us to being able to write a complicated network as a System of Equations. The Labeling Voltages, Currents, and Nodes section will address the sometimes tricky bookkeeping to properly account for positive and negative signs.

### What's Next

In the next section, Ideal Sources, we'll discuss idealized voltage and current sources which can define particular fixed voltage differences and fixed current flows in branches of our circuits.