Electric Charge, Electric Potential, and Voltage

E01: Electric Charge, Electric Potential, and Voltage

Foundations of Electricity and Electrical Measurement, mini-lecture 1.

A voltmeter has two leads. It compares two points and displays a number in volts. To understand that simple measurement, we need three distinct ideas: Electric Charge, electric potential energy, and electric potential. The distinction is important because the voltage between two points is not the same as the energy possessed by one particular charge.

The central relationship is

(1)

where ΔU is the change of electrostatic potential energy of a test charge q moved between two points, and ΔV is the potential difference between those points. This relationship applies to a fixed test charge in a prescribed electrostatic field.

Learning objectives

After this lecture, the reader should be able to:

1. distinguish positive and negative charge and convert coulombs into numbers of elementary charges;
2. explain why electric potential energy depends on the chosen charge, while electric potential describes the electric environment;
3. calculate changes in energy from voltage, including the sign;
4. derive the potential difference from the electric work integral;
5. predict the direction of electric-field force on positive and negative charges;
6. explain why a voltmeter has two leads and why a voltage requires a reference point;
7. interpret battery terminal voltage and separate it from an ideal source model.

1 Electric charge

Electric charge is a property of matter that determines its electric interactions. There are two signs, positive and negative. Protons carry positive elementary charge, electrons carry negative elementary charge, and neutrons are electrically neutral.

The elementary charge is defined exactly in the SI system as

(2)

An electron has charge −e and a proton has charge +e.

For a macroscopic object whose net charge is due to an imbalance of elementary charges, we may write

(3)

where n is an integer that may be positive or negative. This form describes the net charge of ordinary objects made from charged particles. The coulomb, written C, is the SI unit of charge.

An object with charge −3e has three more electrons than needed for an electrically neutral object of the same composition. A positive charge need not mean that protons moved: a solid can become positively charged by losing electrons.

Charge conservation

Electric charge is conserved. In a closed system the algebraic total charge cannot change merely because charge moves from one object to another. Rubbing, contact, and electrostatic induction redistribute charge; they do not create net charge. E02 will connect this principle to the current at a circuit junction.

How charge interacts

For two stationary point charges separated by distance r in vacuum, Coulomb’s law gives the force magnitude

(4)

Like charges repel and opposite charges attract. This force can do work when charges move. The work provides the connection to electric potential energy.

2 From mechanical work to electric potential energy

Recall the mechanical definition of work:

(5)

For an electrostatic field, the electric force is conservative. We can therefore define an electric potential energy U such that

(6)

This is the same sign convention used with gravitational potential energy: if a conservative force does positive work on a particle, its associated potential energy decreases.

The magnitude and sign of the energy change depend on which charge is moved. For example, a charge twice as large experiences twice the force at the same point in a prescribed field and therefore has twice the change in potential energy between two fixed locations.

3 Electric potential: energy per unit charge

To describe the electric environment independently of the charge we use to probe it, we define an electric potential difference by

(7)

The ratio is independent of the small test charge q, provided that the test charge does not appreciably disturb the charges creating the field.

The electric potential V at a point is potential energy per unit charge, relative to a chosen reference:

(8)

when the same zero of potential energy is used consistently. More generally, it is the potential difference that is directly meaningful:

(9)

The SI unit follows immediately:

(10)

A potential difference of one volt means a change in electrostatic potential energy of one joule per coulomb of charge moved between the two locations. It does not mean that every particle acquires one joule of energy.

The sign matters

Suppose V _B − V _A = +5 V. Moving a positive +2 μC test charge from A to B changes its potential energy by

ΔU	= qΔV,	(11)
	= 2 × 10−6 C(5 J∕C),	(12)
	= +1.0 × 10−5 J.	(13)

The energy has increased by 10 μJ. For a negative −2 μC charge taking the same path in the same fixed field,

(14)

The potential difference has not changed. The sign of the moving charge changes the sign of its energy change.

4 Electric fields and voltage

An electric field is defined through the force on a small positive test charge:

(15)

For a stationary test charge in an electrostatic field,

(16)

Combining this with dU = −dW_electric and dU = q dV yields

(17)

Cancel the nonzero test charge:

(18)

After integration,

(19)

This is the fundamental connection between an electrostatic field and a voltage difference.

Along a displacement parallel to the electric field, potential decreases. A positive test charge accelerates in the field direction if no other forces oppose it. A negative test charge feels a force opposite the field and may move toward higher potential while its potential energy decreases.

Uniform-field example: parallel plates

Consider two large, oppositely charged parallel plates separated by d. Far from their edges, the field can be treated approximately as uniform:

(20)

If A is on the left and B on the right, with separation d in the field direction, then

(21)

As an example, for E = 200 V∕m and d = 0.050 m,

(22)

A positive 2 nC charge moving from A to B changes its potential energy by

ΔU	= qΔV,	(23)
	= (2 × 10−9)(−10) J,	(24)
	= −2.0 × 10−8 J.	(25)

The field does +20 nJ of work on the charge.

Scope of the electrostatic formula

The line integral above is path independent when the electric field is electrostatic. In more general situations, changing magnetic flux can produce electric fields for which the integral around a closed path is not zero. A single-valued scalar potential alone then does not describe the entire electric field. We will revisit that distinction in the induction and electromagnetic-wave laboratories.

5 Potential near a point charge

Outside a stationary point charge Q in vacuum, Coulomb’s law gives the radial electric field

(26)

Choose V (∞) = 0. From the potential integral,

V (r) − V (∞)	= −∫ ∞r ds,	(27)
	= .	(28)

Thus

(29)

This is a simple example where the absolute numerical value of V becomes meaningful only after we state a chosen reference. For an extended circuit with a selected reference node, we usually set that node to 0 V instead of using infinity.

6 Voltage means a difference between two points

The expression voltage at point A is shorthand. A voltage must be relative to another point or reference. Define

(30)

A voltmeter with its red lead connected to A and its black lead connected to B ideally displays V _AB.

Suppose V _A = +4.0 V and V _B = +1.5 V relative to a chosen reference. Then

(31)

Swapping the voltmeter leads produces

(32)

The circuit has not changed; only the order of the comparison changed.

Reference potential and circuit ground

We may add the same constant C to all potentials without changing any potential difference:

(33)

That freedom is why one node is often designated ground or 0 V in a circuit diagram.

A reference node is not automatically a physical connection to Earth. Protective Earth, chassis ground, and signal reference are different concepts, even though some instruments connect them internally. The practical wiring and safety implications are covered in E05, E13, and EDL01.

7 Batteries and terminal voltage

A battery uses chemical processes to separate charge and maintain a potential difference between its terminals. The source provides energy per unit charge to charges moving through it via nonelectrostatic forces. This energy per unit charge is the battery’s electromotive force, often represented by ℰ and measured in volts. Despite its name, electromotive force is not a mechanical force measured in newtons.

For a simple battery model with internal resistance r, delivering a positive current I to an external load,

(34)

When I ≈ 0 and the battery is approximately at equilibrium, the terminal voltage is often close to ℰ. Under load, the terminal voltage can be smaller. Real batteries may have more complicated behavior, but the simple model is useful for our first measurements.

An ideal voltage source maintains its specified terminal voltage regardless of current. A real battery and a real function generator are not perfect ideal sources. E08 develops their circuit models.

What the voltmeter measures

A digital multimeter set to DC voltage mode estimates the potential difference between its input terminals. It is normally connected in parallel with the two points being compared. A practical voltmeter has finite input resistance, so it draws a small current and can alter the potential difference it is trying to measure. E09 and E10 quantify that measurement loading effect.

8 The electronvolt: a unit of energy

A particle with charge magnitude e that undergoes a potential difference of magnitude 1 V has an energy change of magnitude

(35)

We define the electronvolt:

(36)

An electronvolt is a unit of energy, not a unit of voltage. It is especially useful when the energies of individual electrons or ions are much smaller than a joule.

If an electron moves from A to B and V _B − V _A = +3 V,

(37)

If its potential energy decrease becomes kinetic energy without losses, its kinetic energy increases by 3 eV.

9 A simple EDL01 measurement

Use a battery-operated digital multimeter and a small, low-voltage battery. Never use the current input for this voltage measurement.

1. Set the multimeter to DC voltage mode. Place the black lead in COM and the red lead in the marked voltage input.
2. Touch the red probe to the positive battery terminal and the black probe to the negative terminal. Record the reading and its polarity.
3. Reverse the leads without changing the battery. Record the reading.
4. Connect two cells in series using an appropriate holder. Predict the voltage across both cells from the individual terminal voltages, then measure it.
5. Choose a reference terminal as 0 V and label the potentials at the other terminals. Check differences by direct measurement.

Safety. Use only low-voltage, current-limited or small dry-cell sources. Do not short a battery, connect a multimeter’s current input across a source, or connect a grounded oscilloscope lead without checking its ground connections. Avoid mains wiring and energized Power equipment in this introductory laboratory.

10 Worked example: energy and polarity

A charge of q = −4.0 μC moves from point A to B, where

(38)

Find the change in the charge’s electrostatic potential energy and the work done by the electric force.

Step 1: Determine the potential difference.

(39)

Step 2: Apply the energy relation.

ΔU	= qΔV,	(40)
	= (−4.0 × 10−6)(−3.0) J,	(41)
	= +12 μJ.	(42)

Step 3: Relate work to potential energy.

(43)

The field does negative work along the stated displacement and the negative charge gains electrostatic potential energy. No statement about the charge’s actual speed follows without additional information about other forces and its initial motion.

11 Common misconceptions

• Voltage is not charge. Charge is measured in coulombs; voltage is energy per unit charge in joules per coulomb.
• Voltage is not energy. A 10 V difference changes the energy of a 1 C charge by 10 J, but changes the energy of a 2 C charge by 20 J.
• Higher potential does not always mean higher potential energy. For a negative test charge, increasing potential lowers its potential energy.
• Zero volts does not mean zero electric field. A point or node may be labeled 0 V while experiencing a nonzero field.
• Ground is not necessarily Earth. Many isolated battery circuits use an arbitrary local voltage reference.
• A voltmeter is not an ideal observer. Its finite input resistance can load the circuit and change the displayed voltage.
• EMF is not a force. It represents the work supplied per unit charge by an energy source in the relevant source model.

12 Concept checks

1. If the red and black probes of an ideal voltmeter are reversed, what happens to the displayed reading?
2. Two different charges move between the same points in the same unperturbed electrostatic field. Which quantity is the same for both: ΔV or ΔU?
3. Can an electron lose potential energy while moving toward a point of higher electric potential?
4. Is a circuit node labeled 0 V necessarily connected to physical Earth?
5. In what direction does the electrostatic potential change when moving along an electric field line?

Answers

1. The ideal reading changes sign but not magnitude.
2. The potential difference is the same; the energy change scales with charge and its sign.
3. Yes. Since q = −e, ΔU = qΔV is negative when ΔV > 0.
4. No. It may simply be the chosen local reference.
5. It decreases, since dV = −E ⋅ dr.

For more numerical practice, see E01E1, Worked Problems in Charge, Potential, and Voltage.

13 Summary and next step

Electric charge determines how matter responds to an electric field. Electrostatic potential energy describes the energy of a particular charge in a prescribed field. Electric potential describes the field’s energy landscape per unit charge.

The defining equations are

(44)

and

(45)

A voltmeter compares two points, so choosing a reference and reading the polarity correctly are central measurement skills. In E02 we introduce electric current and show how conservation of charge constrains current flow through a circuit.

References and learning connections

References