02 · Introductory

Electric potential and voltage

Voltage is a difference between two points, not a property floating at one node without a reference.

80–105 minIntroductory

In this lesson

Define potential difference as energy per unit charge.
Read AoN voltage arrows correctly: head = positive reference.
Write VAB = VA − VB consistently.
Relate voltage difference to energy change for a charge.

Starting question

If someone says “this node is at 5 V,” what information is missing?

Voltage is not an isolated property of one point. It is a comparison between two electrical states: two terminals, two nodes, or one node and a declared reference.

02.1Potential difference is energy change per unit charge

If a charge q experiences a change ΔU in electric potential energy while moving between two points, the corresponding potential difference is

ΔV = ΔU/q

ΔU = qΔV

Thus one volt is one joule per coulomb:

1 V = 1 J/C

This definition is more precise than the popular phrase “voltage is electrical pressure.” The pressure analogy can help intuition in some situations, but energy per charge is the quantity the equations actually mean.

02.2Voltage requires two references and an orientation

Figure 2.1 — A voltage always relates two referencesThe arrow runs from B to A; its head is the positive reference, so VAB = VA − VB.

AoN defines the voltage arrow from the negative reference toward the positive reference. The arrow B→A therefore defines

V_AB = V_A − V_B

Reversing the reference reverses the sign:

V_BA = −V_AB

A negative answer does not invalidate the reference. It tells us that the actual ordering of the two potentials is opposite to the positive direction we chose.

02.3Ground is a coordinate choice, not necessarily Earth

A circuit drawing often declares one node to be 0 V and calls it GND. A node label V_A then means V_A−V_GND.

If we choose a different reference, every node-potential number may shift by the same constant, but physical voltage differences between pairs of nodes do not change.

02.4Worked example: signs and energy together

Let V_A=2 V and V_B=7 V relative to the chosen reference. Then

V_AB = 2 − 7 = −5 V

For q=+3 µC moving from B to A,

ΔU = qV_AB = 3 µC × (−5 V) = −15 µJ

A positive charge moving from the higher-potential point B to the lower-potential point A loses electric potential energy. The sign agrees with the qualitative picture.

What if q were −3 µC?

The same voltage difference would give ΔU=+15 µJ because the charge sign reverses the energy change.

02.5What voltage is not

Voltage is not “amount of charge.”
Voltage is not current; a voltage can exist with no steady current.
A node voltage is incomplete until its reference is known.
In this course we use the electrostatic/lumped notion of potential difference. Time-varying electromagnetic fields require a more general treatment introduced in Volume II.

02.6Practice for mastery

If V_A=−2 V and V_B=+3 V, find V_AB and V_BA.
A +4 µC charge gains 20 µJ of potential energy. What potential difference did it traverse?
Every node-potential number is shifted upward by 100 V after choosing a new reference. What happens to V_AB?
Explain why “a node has 5 V” is incomplete language unless the reference is understood.

Solutions

1. VAB=−5 V and VBA=+5 V.

2. ΔV=5 V.

3. Nothing; the same offset cancels in the difference.

4. Voltage is defined between two references; a node label silently assumes a second node such as GND.

Voltage gives energy change per charge. The next chapter asks how rapidly charge crosses a chosen section: that rate is current.