Physics

Chapter 7: Electric Potential

Imron Rosyadi

7.1 Electric Potential Energy

Learning Objectives

By the end of this section, you will be able to:

• Define the work done by an electric force.
• Define electric potential energy.
• Apply work and potential energy in systems with electric charges.

Today, we’re diving into Chapter 7 on Electric Potential. We’ll start by understanding electric potential energy, how it’s defined by the work done by electric forces, and how to apply these concepts to systems of charges.

Electric Potential Energy: Analogy

When a free positive charge q is accelerated by an electric field, it gains kinetic energy.

This is similar to an object accelerated by a gravitational field.

A charge accelerated by an electric field is analogous to a mass going down a hill.

Key Concept:

• Potential energy decreases as kinetic energy increases.
• \(-\Delta U = \Delta K\)
• Work done by a conservative force: \(W = -\Delta U\)

Note

The electrostatic (Coulomb) force is a conservative force. Work done by conservative forces is independent of the path taken.

Think of it like a ball rolling down a hill. As it goes down, its gravitational potential energy decreases, and its kinetic energy increases. Similarly, a positive charge “falls” down an “electrical hill,” converting electric potential energy into kinetic energy. The key here is that the electrostatic force, like gravity, is conservative, meaning the work done doesn’t depend on the path.

Defining Electric Potential Energy

Consider a fixed positive charge q at the origin.

Move a test charge Q towards q from \(P_1\) to \(P_2\).

Displacement of “test” charge Q in the presence of fixed “source” charge q.

The applied force \(\vec{F}\) balances the electric force \(\vec{F}_e\):

\(\vec{F} = -\vec{F}_e = -k_e \frac{qQ}{r^2} \hat{r}\)

Work Done by Applied Force:

\(W_{12} = \int_{P_1}^{P_2} \vec{F} \cdot d\vec{l}\)

The work done by the applied force changes the electric potential energy of Q.

Electric Potential Energy (\(U\)):

• Units: Joules (J)
• If conservative force does negative work, system gains potential energy.
• If conservative force does positive work, system loses potential energy.
• \(\Delta U = -W\)

Important

The work done by the applied force when bringing like charges closer is positive, increasing the system’s potential energy (\(\Delta U > 0\)).

To formally define electric potential energy, imagine moving a test charge Q in the presence of another charge q. We apply an external force to precisely counteract the electric force. The work done by this external force defines the change in electric potential energy. If the applied force does positive work, the system gains potential energy.

Electric Potential Energy of Point Charges

The work done moving Q from \(r_1\) to \(r_2\) is:

\[ W_{12} = k_e q Q \left[ \frac{1}{r_1} - \frac{1}{r_2} \right] \]

This result depends only on the endpoints, confirming the conservative nature of the electric force.

Defining Absolute Potential Energy:

• Take reference point \(r_{ref} = \infty\), where \(U_{ref} = 0\).
• The electric potential energy U of a system of two point charges q and Q separated by distance r is:

\[ U(r) = \frac{k_e q Q}{r} \quad (\text{zero reference at } r = \infty) \]

Note

This formula is symmetrical for q and Q, representing the potential energy of the two-charge system.

By integrating the force, we find a simple expression for the work done, which depends only on the initial and final positions. This path independence is a hallmark of conservative forces. To define a specific potential energy value, we set a reference point, usually infinity, where the interaction is zero. This gives us the standard formula for the potential energy of two point charges.

Example: Kinetic Energy of a Charged Particle

A +3.0-nC charge Q is initially at rest 10 cm from a +5.0-nC charge q fixed at the origin. Q is accelerated away from q, reaching 15 cm.

The charge Q is repelled by q, thus having work done on it and gaining kinetic energy.

1. What is the work done by the electric field between \(r_1\) and \(r_2\)?
2. How much kinetic energy does Q have at \(r_2\)?

Let’s apply these ideas to an example. We have two positive charges, one fixed and one free to move. Since they are both positive, they will repel.

Example: Kinetic Energy of a Charged Particle (Solution)

Strategy:

Calculate work using \(W_{12} = k_e q Q \left[ \frac{1}{r_1} - \frac{1}{r_2} \right]\).

Since Q starts from rest, the work done by the electric field is equal to its final kinetic energy.

Solution:

1. \(W_{12} = (8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2)(5.0 \times 10^{-9} \text{ C})(3.0 \times 10^{-9} \text{ C}) \left[ \frac{1}{0.10 \text{ m}} - \frac{1}{0.15 \text{ m}} \right]\)

   \(W_{12} = 4.5 \times 10^{-7} \text{ J}\)

2. The kinetic energy at \(r_2\) is \(K_f = W_{12} = 4.5 \times 10^{-7} \text{ J}\).

Tip

The electric field does positive work on Q, so Q gains kinetic energy.

We use the formula for work derived from the potential energy. Plugging in the values, we find the work done by the electric field. Since the charge started from rest, this work is entirely converted into kinetic energy.

Systems of Multiple Point Charges

The total electric potential energy of a system with multiple charges is the sum of the potential energies of all unique pairs of charges.

\[ U_{total} = \frac{k_e}{2} \sum_{i \ne j} \frac{q_i q_j}{r_{ij}} \]

• The factor of \(1/2\) prevents double-counting each pair.
• Potential energy is positive for like charges, negative for opposite charges.
• This relates to the work needed to assemble the system from infinity.

Important

Bringing like charges closer requires positive work (energy input), increasing potential energy.

Bringing opposite charges closer involves negative work (energy released by the system), decreasing potential energy.

Just like forces, potential energy is additive. For a system of multiple charges, we calculate the potential energy for every unique pair of charges and sum them up. The factor of 1/2 ensures we don’t count the interaction between two charges twice. The sign of the potential energy tells us about the nature of the interaction.

7.2 Electric Potential and Potential Difference

Learning Objectives

By the end of this section, you will be able to:

• Define electric potential, voltage, and potential difference.
• Define the electron-volt.
• Calculate electric potential and potential difference from potential energy and electric field.
• Describe systems in which the electron-volt is a useful unit.
• Apply conservation of energy to electric systems.

Now, let’s move from potential energy to electric potential. We’ll define key terms like voltage and the electron-volt, learn how to calculate potential from energy and field, and see how the conservation of energy applies to electric systems.

Electric Potential and Voltage

Electric potential V (or simply potential) is defined as the electric potential energy per unit charge.

\[ V = \frac{U}{q} \]

The change in potential energy $ U $ is crucial, so we focus on potential difference $ V $ between two points:

\[ \Delta V = V_B - V_A = \frac{\Delta U}{q} \]

Electric Potential Difference (\(V_B - V_A\)):

• Change in potential energy of a charge q moved from A to B, divided by the charge.
• Units: Joules per Coulomb (J/C), named Volt (V) after Alessandro Volta.
• \(1 \text{ V} = 1 \text{ J/C}\)

Note

Voltage is the common name for electric potential difference.

Always refers to the potential difference between two points.

While potential energy depends on the magnitude of the charge, electric potential is a property of the electric field itself, independent of the test charge. It’s essentially “potential energy per unit charge.” The change in potential, or potential difference, is what we commonly call voltage. This scalar quantity is super useful, especially when compared to dealing with vector electric fields directly.

Potential Difference vs. Electrical Potential Energy

Potential Difference ($ V $):

• Energy per unit charge (\(\text{J/C}\) or \(\text{V}\)).
• \(\Delta V = \frac{\Delta U}{q}\)
• Independent of the test charge q.
• A property of the electric field.

Electrical Potential Energy ($ U $):

• Total energy (\(\text{J}\)).
• \(\Delta U = q \Delta V\)
• Depends on both the charge q and the potential difference \(\Delta V\).
• The energy stored or released when a charge moves between two points.

Analogy:

Imagine two batteries, one for a motorcycle and one for a car, both 12-V.

• Both have the same voltage (\(\Delta V = 12 \text{ V}\)).
• The car battery stores much more energy (\(\Delta U\)) because it can move a larger amount of charge (q).

A battery moves negative charge from its negative terminal through a headlight to its positive terminal.

It’s crucial to distinguish these two. Voltage is like the “potential” for work, while potential energy is the actual work done for a specific charge. A 12-V motorcycle battery and a 12-V car battery have the same voltage, but the car battery can deliver much more total energy because it moves a greater quantity of charge.

The Electron-Volt (eV)

On a submicroscopic scale, an energy unit called the electron-volt (eV) is more convenient.

Definition:

• The energy gained by a fundamental charge (e.g., an electron or proton) accelerated through a potential difference of 1 Volt.
• \(1 \text{ eV} = (1.60 \times 10^{-19} \text{ C})(1 \text{ V}) = 1.60 \times 10^{-19} \text{ J}\)

A typical electron gun accelerates electrons using a potential difference between two separated metal plates.

Applications:

• Chemical valence energies (\(\approx 5 \text{ eV}\) to break organic molecules).
• Molecular and nuclear binding energies (nuclear decay energies \(\approx 1 \text{ MeV}\)).
• Particle accelerators.

Tip

An electron accelerated through \(X\) Volts gains \(X\) electron-volts of energy. A doubly charged ion (\(+2e\)) accelerated through \(Y\) Volts gains \(2Y\) electron-volts of energy.

For atomic and nuclear scales, the joule is too large. This is where the electron-volt comes in. It’s the energy a single electron gains when accelerated by 1 volt. This unit simplifies calculations dramatically for particle physics and chemistry, where energies are often expressed in eV, keV, or MeV.

Conservation of Energy in Electric Systems

For conservative forces, like the electrostatic force, the total mechanical energy of a system is conserved.

\[ K + U = \text{constant} \] or \[ K_i + U_i = K_f + U_f \]

• A loss in potential energy (\(U\)) for a charged particle results in an increase in its kinetic energy (\(K\)).

Example: Calculate the final speed of a free electron accelerated from rest through a potential difference of 100 V.

Solution:

\(K_i = 0\), \(U_i = qV\), \(K_f = \frac{1}{2}mv^2\), \(U_f = 0\)

\(qV = \frac{1}{2}mv^2\)

\(v = \sqrt{\frac{2qV}{m}} = \sqrt{\frac{2(-1.60 \times 10^{-19} \text{ C})(-100 \text{ J/C})}{9.11 \times 10^{-31} \text{ kg}}} = 5.93 \times 10^6 \text{ m/s}\)

Note

The negative charge of the electron and negative voltage change result in a positive kinetic energy.

Just as in classical mechanics, the principle of conservation of energy holds. Any loss in electric potential energy is converted into other forms, often kinetic energy. This allows us to calculate things like the final speed of a particle accelerated through a known voltage, as shown in the example of an electron accelerated by 100 V.

Voltage and Electric Field Relationship

The general relationship between potential difference and electric field is:

\[ \Delta V = V_B - V_A = - \int_A^B \vec{E} \cdot d\vec{l} \]

• The potential difference is negative when displacement is in the same direction as the field.
• The electric field points toward lower electric potential.

For a Uniform Electric Field (e.g., parallel plates):

\[ E = - \frac{\Delta V}{\Delta s} \quad \text{or} \quad \Delta V = -E \Delta s \]

• Units: \(1 \text{ N/C} = 1 \text{ V/m}\)

Potential of a Point Charge (\(q\)) at distance r:

\[ V(r) = \frac{k_e q}{r} \]

Tip

This equation is for the electric potential at a point r relative to a zero reference at infinity.

Voltage and electric field are intimately linked. The electric field is essentially the negative gradient of the potential. This means that the electric field always points in the direction of decreasing potential. For a uniform field, this relationship simplifies to E = -Delta V / Delta s. This also gives us a clear understanding of the units and how to calculate potential from a point charge.

Problem-Solving Strategy: Electrostatics

1. Examine the situation: Is static electricity involved? What are the charges, forces, and fields?
2. Identify the system: What charges are involved, their locations, and types?
3. Identify unknowns: What quantities need to be determined? (e.g., force, field, potential, energy). Drawing a free-body diagram can be helpful for forces.
4. Identify knowns: List given values and inferred information. Distinguish between force (\(\vec{F}\)) and electric field (\(\vec{E}\)).
5. Solve: Use appropriate equations or methods (e.g., Coulomb’s Law, potential energy formula, integral for potential/field).
6. Examine the answer: Is it reasonable? Are the units correct? Are the numerical values plausible?

To effectively tackle electrostatics problems, follow this systematic strategy. It helps ensure you understand the problem, select the right tools, execute the solution, and verify your results. Always double-check your units and the physical reasonableness of your answers.

7.3 Calculations of Electric Potential

Learning Objectives

By the end of this section, you will be able to:

• Calculate the potential due to a point charge.
• Calculate the potential of a system of multiple point charges.
• Describe an electric dipole and define dipole moment.
• Calculate the potential of a continuous charge distribution.

In this section, we’ll build on our understanding of electric potential by learning how to calculate it for various charge configurations: individual point charges, systems of multiple point charges, electric dipoles, and continuous charge distributions.

Potential Due to Multiple Point Charges

The electric potential V at a point P due to a system of N point charges is the algebraic sum of the individual potentials:

\[ V_P = \sum_{i=1}^N V_i = k_e \sum_{i=1}^N \frac{q_i}{r_i} \]

• \(r_i\) is the distance from charge \(q_i\) to point P.
• Electric potential is a scalar quantity, so addition is straightforward (no vectors).
• This is consistent with the superposition principle.

Note

This formula assumes zero potential at infinitely far away.

Example: What Voltage Is Produced by a Small Charge on a Metal Sphere? A 1-cm-diameter metal sphere has a –3.00-nC static charge. What is the voltage 5.00 cm away from its center?

Solution:

\(V = \frac{k_e q}{r} = \frac{(8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2)(-3.00 \times 10^{-9} \text{ C})}{0.0500 \text{ m}} = -539 \text{ V}\)

Just like electric fields, electric potentials obey the superposition principle. To find the total potential at a point from multiple charges, you simply add the potentials from each charge algebraically. This is much simpler than vector addition for electric fields. The example shows how to calculate the potential from a charged sphere, which behaves like a point charge externally.

The Electric Dipole

An electric dipole consists of two equal but opposite charges (\(+q\) and \(-q\)) separated by a fixed distance d.

A general diagram of an electric dipole, and the notation for the distances from the individual charges to a point P in space.

Electric Dipole Moment (\(\vec{p}\)):

• A vector quantity defined as: \(\vec{p} = q\vec{d}\)
• Points from the negative charge to the positive charge.
• Magnitude: \(qd\).

Potential of a Dipole at Distant Points (\(r \gg d\)):

\[ V_P = \frac{k_e q d \cos\theta}{r^2} \quad \text{or} \quad V_P = \frac{k_e \vec{p} \cdot \hat{r}}{r^2} \]

• Where \(\theta\) is the angle between \(\vec{p}\) and \(\vec{r}\).

Electric dipoles are fundamental for modeling molecular interactions. They consist of two equal and opposite charges separated by a distance. We define an electric dipole moment, a vector quantity, which simplifies calculations, especially for points far from the dipole. The potential then depends on the angle relative to the dipole moment.

Potential of Continuous Charge Distributions

For continuous charge distributions, we sum the potentials from infinitesimal charge elements \(dq\) using integration:

\[ V_P = k_e \int \frac{dq}{r} \]

Where r is the distance from each infinitesimal charge element \(dq\) to point P.

Infinitesimal Charge Elements (\(dq\)):

• Line charge (1D): \(dq = \lambda dl\) (\(\lambda\) = linear charge density)
• Surface charge (2D): \(dq = \sigma dA\) (\(\sigma\) = surface charge density)
• Volume charge (3D): \(dq = \rho dV\) (\(\rho\) = volume charge density)

When charges aren’t discrete points but are spread continuously, we use calculus. We divide the distribution into tiny charge elements, \(dq\), and integrate their contributions to the potential. The specific form of \(dq\) depends on whether the charge is distributed along a line, over a surface, or throughout a volume.

Example: Potential of a Line of Charge

Find the electric potential of a uniformly charged, nonconducting wire of length L and linear density \(\lambda\) at a point P on the perpendicular bisector (x-axis).

We want to calculate the electric potential due to a line of charge.

Solution:

Choose origin at center of wire, y-axis along wire.

\(dq = \lambda dy\)

Distance to P is \(r = \sqrt{x^2 + y^2}\)

\(V_P = k_e \int_{-L/2}^{L/2} \frac{\lambda dy}{\sqrt{x^2 + y^2}} = k_e \lambda \left[ \ln(y + \sqrt{y^2 + x^2}) \right]_{-L/2}^{L/2}\)

\(V_P = k_e \lambda \ln \left[ \frac{L/2 + \sqrt{(L/2)^2 + x^2}}{-L/2 + \sqrt{(-L/2)^2 + x^2}} \right]\)

Important

Note the simplification from using scalar potential compared to vector electric field calculations. The potential approaches zero as \(x \to \infty\).

Let’s apply the integration method. For a uniformly charged wire, we consider a small segment \(dy\) and integrate its potential contribution. The resulting logarithmic expression, while a bit complex, is still easier to derive than the corresponding electric field vector.

7.4 Determining Field from Potential

Learning Objectives

By the end of this section, you will be able to:

• Explain how to calculate the electric field in a system from the given potential.
• Calculate the electric field in a given direction from a given potential.
• Calculate the electric field throughout space from a given potential.

Now that we know how to calculate potential, we’ll learn the reverse: how to determine the electric field from a known potential. This involves taking derivatives, which transforms a scalar potential back into a vector electric field.

Electric Field as the Gradient of Potential

The electric field \(\vec{E}\) points in the direction of decreasing potential \(V\).

The magnitude of \(\vec{E}\) equals the rate of decrease of \(V\) with distance.

Relationship between Voltage and Uniform Electric Field:

\[ E = -\frac{\Delta V}{\Delta s} \]

• \(\Delta s\) is the distance over which potential changes by \(\Delta V\).
• The minus sign indicates \(\vec{E}\) points towards decreasing \(V\).

For Continually Changing Potentials:

Electric field components in Cartesian coordinates:

\(E_x = -\frac{\partial V}{\partial x}\)

\(E_y = -\frac{\partial V}{\partial y}\)

\(E_z = -\frac{\partial V}{\partial z}\)

This leads to the gradient operator (del operator):

\[ \vec{E} = -\nabla V \]

The electric field is the negative gradient of the electric potential. This means two things: the electric field always points “downhill” (towards lower potential), and its strength is proportional to how rapidly the potential changes with distance. We can calculate the components of the electric field by taking partial derivatives of the potential with respect to each spatial coordinate.

Del Operator in Different Coordinate Systems

The del operator (\(\nabla\)) allows us to calculate the electric field from the potential in a single step, using the appropriate coordinate system.

Cartesian Coordinates:

\[ \nabla = \hat{i} \frac{\partial}{\partial x} + \hat{j} \frac{\partial}{\partial y} + \hat{k} \frac{\partial}{\partial z} \]

Cylindrical Coordinates:

\[ \nabla = \hat{r} \frac{\partial}{\partial r} + \hat{\phi} \frac{1}{r} \frac{\partial}{\partial \phi} + \hat{z} \frac{\partial}{\partial z} \]

Spherical Coordinates:

\[ \nabla = \hat{r} \frac{\partial}{\partial r} + \hat{\theta} \frac{1}{r} \frac{\partial}{\partial \theta} + \hat{\phi} \frac{1}{r \sin\theta} \frac{\partial}{\partial \phi} \]

Tip

Choose the coordinate system that best matches the symmetry of your potential function.

Depending on the symmetry of the problem, using the appropriate coordinate system for the del operator can greatly simplify calculations. For example, if your potential has spherical symmetry, using the spherical del operator is much more efficient than using Cartesian coordinates.

Example: Electric Field of a Point Charge

Calculate the electric field of a point charge from its potential \(V = k_e q/r\).

Strategy:

Since the potential has spherical symmetry, use the spherical del operator:

\(\vec{E} = -\nabla V = -\left( \hat{r} \frac{\partial}{\partial r} + \hat{\theta} \frac{1}{r} \frac{\partial}{\partial \theta} + \hat{\phi} \frac{1}{r \sin\theta} \frac{\partial}{\partial \phi} \right) \left( \frac{k_e q}{r} \right)\)

Solution:

\(\vec{E} = -k_e q \left( \hat{r} \frac{\partial}{\partial r}\left(\frac{1}{r}\right) + \hat{\theta} \frac{1}{r} \frac{\partial}{\partial \theta}\left(\frac{1}{r}\right) + \hat{\phi} \frac{1}{r \sin\theta} \frac{\partial}{\partial \phi}\left(\frac{1}{r}\right) \right)\)

• Only the \(\frac{\partial}{\partial r}\) term is non-zero because \(V\) only depends on \(r\).

\(\frac{\partial}{\partial r}\left(\frac{1}{r}\right) = -\frac{1}{r^2}\)

\(\vec{E} = -k_e q \left( \hat{r} \left(-\frac{1}{r^2}\right) + \hat{\theta} (0) + \hat{\phi} (0) \right)\)

\(\vec{E} = \frac{k_e q}{r^2} \hat{r}\)

Important

This matches Coulomb’s Law, demonstrating consistency between potential and field. The field points radially outward from a positive charge, towards decreasing potential.

Let’s see this in action for a point charge. The potential \(k_e q/r\) has spherical symmetry, so we use the spherical del operator. All terms except the radial derivative are zero, and the result precisely matches Coulomb’s law for the electric field of a point charge. This confirms that our definitions and methods are consistent.

7.5 Equipotential Surfaces and Conductors

Learning Objectives

By the end of this section, you will be able to:

• Define equipotential surfaces and equipotential lines.
• Explain the relationship between equipotential lines and electric field lines.
• Map equipotential lines for one or two point charges.
• Describe the potential of a conductor.
• Compare and contrast equipotential lines and elevation lines on topographic maps.

We’ll now visualize electric potential using equipotential surfaces and lines. We’ll explore their relationship with electric field lines, see how they look for different charge configurations, and understand the special properties of conductors as equipotential surfaces.

Equipotential Surfaces and Lines

Equipotential Surfaces/Lines:

• Locations where the electric potential is constant.
• In 3D: Equipotential surfaces.
• In 2D: Equipotential lines.

An isolated point charge Q with its electric field lines in red and equipotential lines in black.

Key Properties:

• Perpendicular to Electric Field Lines: No work is done moving a charge along an equipotential, because \(\Delta V = 0\). Since \(W = q E d \cos\theta = 0\) (and \(q, E, d \ne 0\)), \(\cos\theta\) must be 0, so \(\theta = 90^\circ\).
• Conductors are Equipotential Surfaces (in static situations): If there were a potential difference across a conductor’s surface, charges would flow.
• Grounding: Connecting a conductor to Earth fixes its potential at zero volts.

Equipotential surfaces are like contour lines on a topographic map – they connect points of equal elevation, but here, it’s equal electric potential. Crucially, electric field lines are always perpendicular to these surfaces. This means no work is done moving a charge along an equipotential line. Conductors, in static equilibrium, are always equipotential surfaces, a principle leveraged in grounding for safety.

Equipotential Maps for Charge Configurations

For a Point Charge:

• Equipotential surfaces are concentric spheres centered on the charge.
• Potential decreases with distance (\(V = k_e q/r\)).

For Two Equal and Opposite Charges (Dipole):

Electric field lines (red) and equipotential lines (black) for two equal but opposite charges.

• Equipotentials are complex closed loops.
• The line equidistant from both charges has zero potential.

For Parallel Plates:

The electric field and equipotential lines between two metal plates.

• Equipotentials are evenly spaced, parallel planes between the plates.

The shape of equipotential surfaces tells us a lot about the electric field. For a single point charge, they’re spheres. For a dipole, they’re more complex loops. For parallel plates, they’re simple parallel planes, reflecting the uniform electric field between them. These maps provide a visual understanding of the potential landscape.

Distribution of Charges on Conductors

The electric field inside a conductor in electrostatic equilibrium is zero.

The entire volume of a conductor is at the same potential.

Charge Distribution on Irregular Conductors:

\[ \sigma_1 R_1 = \sigma_2 R_2 \]

• Surface charge density (\(\sigma\)) is inversely proportional to the radius of curvature (R) at the surface.
• Charges tend to accumulate at sharp points (smaller R).
• This means the electric field is strongest at sharp points.

The surface charge density and the electric field of a conductor are greater at regions with smaller radii of curvature.

Applications:

• Lightning Rods: Sharp points on rods create large electric fields, allowing charge to neutralize gradually before a lightning strike occurs.

An important consequence of conductors being equipotential surfaces is how charges distribute themselves. On irregularly shaped conductors, charge density is highest at points of smallest curvature, leading to stronger electric fields there. This principle is used in lightning rods to safely discharge atmospheric electricity.

7.6 Applications of Electrostatics

Learning Objectives

By the end of this section, you will be able to:

• Describe some of the many practical applications of electrostatics, including several printing technologies.
• Relate these applications to Newton’s second law and the electric force.

Finally, let’s look at some real-world applications of electrostatics. From large-scale generators to everyday devices like printers and air purifiers, electrostatic principles are at work all around us.

Van de Graaff Generator

Schematic of Van de Graaff generator.

How it works:

• A motor-driven insulating belt continuously transports charge to a large conducting sphere.
• Charge rapidly moves to the outer surface of the sphere.
• Pointed conductors spray charge onto the belt and collect it from the belt.

Purpose:

• Generates very high static voltages (millions of volts).
• Used for physics demonstrations and nuclear physics research.

Note

High electric fields can ionize surrounding air, limiting maximum achievable voltage.

The Van de Graaff generator is a classic example of applying electrostatic principles. An insulating belt mechanically transports charge, accumulating it on a large metal sphere. The charge moves to the outer surface, building up immense voltages. It’s a powerful demonstration of static electricity and has historical importance in nuclear research.

Xerography (Photocopying)

Xerography is a dry copying process based on electrostatics.

Key Stages:

1. Charging the drum: A photoconducting drum (insulator in dark, conductor in light) is uniformly charged.
2. Image formation: Light projected from the original image discharges areas of the drum corresponding to white areas; dark areas retain charge.
3. Toner application: Negatively charged toner particles are attracted to the positively charged (dark) areas of the drum.
4. Transfer to paper: A highly positive-charged paper pulls the toner from the drum.
5. Fusing: Heat and pressure rollers melt the toner onto the paper, creating a permanent copy.

Xerography, the process behind photocopiers, is a clever application of electrostatics and photoconductivity. A charged drum selectively loses charge where light hits, creating an electrostatic image. Charged toner then adheres to these charged areas, which is then transferred and fused onto paper.

Laser Printers

In a laser printer, a laser beam is scanned across a photoconducting drum, leaving a positively charged image.

• Similar to xerography, but uses a precisely controlled laser beam to “write” the image onto the photoconducting drum.
• The laser selectively discharges areas, creating an electrostatic image with high precision.
• Allows for very high-quality text and graphics from computer output.

Laser printers refine the xerographic process by using a laser to “draw” the image onto the drum. The laser’s precision enables exceptionally high-quality printing, making it a cornerstone of modern document production.

Ink Jet Printers and Electrostatic Painting

Ink Jet Printers:

The nozzle of an ink-jet printer produces small ink droplets, which are sprayed with electrostatic charge.

• Tiny ink droplets are given an electrostatic charge.
• Charged deflection plates guide the droplets with high precision to form images on paper.
• Can produce full-color images by mixing primary colored inks.

Electrostatic Painting:

• Paint droplets are electrostatically charged.
• Mutual repulsion spreads the paint evenly.
• The object to be painted is oppositely charged, attracting the paint.

Advantages:

• Reaches hard-to-get places.
• Applies an even coat with minimal waste.
• Excellent for irregularly shaped objects.

Inkjet printers and electrostatic painting also rely on charge manipulation. Inkjet printers precisely steer charged ink droplets using electric fields. Electrostatic painting uses charged paint droplets that repel each other for even coating and are attracted to the oppositely charged object, providing efficient and uniform coverage, especially for complex shapes.

Smoke Precipitators and Electrostatic Air Cleaning

Schematic of an electrostatic precipitator. Air is passed through grids of opposite charge.

How it works:

1. Airborne particles (smoke, dust, pollen) are given an electrostatic charge (usually positive) by passing through a charging grid.
2. The charged particles then pass through an oppositely charged collecting grid.
3. The collecting grid attracts and retains the charged particles, cleaning the air.

Applications:

• Industrial: Removes over 99% of particles from factory emissions (e.g., coal and oil burning).
• Residential: Home air purifiers (often integrated with HVAC systems) remove pollutants, irritants, and allergens.

Finally, electrostatic precipitators are vital for air quality. They charge airborne particles, then use an oppositely charged grid to attract and remove them. This technology is used in large industrial settings to clean factory emissions and in smaller home units to purify indoor air, effectively removing pollutants and allergens.

Key Takeaways

Electric Potential Energy (\(U\)):

• Work done to move a charge in an electric field.
• For point charges \(q_1, q_2\): \(U = k_e \frac{q_1 q_2}{r}\).
• Conservative force: work is path-independent.

Electric Potential (\(V\)) / Voltage (\(\Delta V\)):

• Potential energy per unit charge: \(V = U/q\).
• Potential difference: \(\Delta V = \Delta U/q\).
• Units: Volts (V), where \(1 \text{ V} = 1 \text{ J/C}\).

Electron-Volt (eV):

• Energy gained by a fundamental charge accelerated through 1 V.
• \(1 \text{ eV} = 1.60 \times 10^{-19} \text{ J}\).
• Useful for atomic/molecular energy scales.

Relationship between \(\vec{E}\) and \(V\):

• \(\vec{E} = -\nabla V\). The electric field points in the direction of decreasing potential.
• For uniform field: \(E = -\frac{\Delta V}{\Delta s}\).

Equipotential Surfaces:

• Regions of constant electric potential.
• Always perpendicular to electric field lines.
• Conductors in electrostatic equilibrium are equipotential surfaces.
• Charge density is higher at sharp points on conductors.

Applications of Electrostatics:

• Van de Graaff generators, xerography (photocopiers, laser printers), inkjet printers, electrostatic painting, smoke precipitators, air cleaners.

To summarize, electric potential energy is about the work done on charges. Electric potential, or voltage, is potential energy per unit charge, a scalar quantity. The electron-volt is a convenient energy unit for microscopic scales. The electric field is the negative gradient of the potential, always pointing “downhill.” Equipotential surfaces are constant potential regions, always perpendicular to field lines, and conductors are equipotentials. These principles underpin many technologies, from generators to printers and air purifiers.

Key Equations

Equation	Description
\(W = -\Delta U\)	Work done by conservative force
\(U(r) = k_e \frac{qQ}{r}\)	Electric potential energy of two point charges
\(V = \frac{U}{q}\)	Electric potential
\(\Delta V = V_B - V_A = \frac{\Delta U}{q}\)	Electric potential difference (voltage)
\(1 \text{ eV} = 1.60 \times 10^{-19} \text{ J}\)	Electron-volt to Joules conversion
\(\Delta V = -\int_A^B \vec{E} \cdot d\vec{l}\)	Potential difference from electric field
\(V(r) = \frac{k_e q}{r}\)	Potential of a point charge
\(V_P = k_e \sum_{i=1}^N \frac{q_i}{r_i}\)	Potential of multiple point charges
\(\vec{p} = q\vec{d}\)	Electric dipole moment
\(V_P = k_e \frac{\vec{p} \cdot \hat{r}}{r^2}\)	Potential of a dipole at distant points
\(\vec{E} = -\nabla V\)	Electric field from potential
\(E = -\frac{\Delta V}{\Delta s}\)	Uniform electric field from potential difference
\(\sigma_1 R_1 = \sigma_2 R_2\)	Charge distribution on connected conductors

Here’s a summary of the key equations from this chapter. Make sure you understand what each symbol represents and when to use each equation. These are your primary tools for solving problems related to electric potential.

Key Terms

Term	Definition
Conservative force	A force for which the work done on a particle moving between two points is independent of the path taken.
Electric potential energy	The potential energy of a charged particle due to its position in an electric field.
Electric potential (V)	The electric potential energy per unit charge at a given point in an electric field. Often called voltage.
Potential difference	The difference in electric potential between two points, equal to the work done per unit charge in moving a charge between the points.
Volt (V)	The SI unit of electric potential and potential difference, defined as one joule per coulomb (J/C).
Electron-volt (eV)	An energy unit equal to the work done on an electron (or proton) accelerated through a potential difference of 1 volt.
Electric dipole	A pair of equal and opposite electric charges separated by a small distance.
Electric dipole moment (\(\vec{p}\))	A vector quantity that measures the strength and orientation of an electric dipole, defined as \(q\vec{d}\).
Equipotential surface/line	A surface or line over which the electric potential is constant. No work is done moving a charge along an equipotential.
Grounding	The process of connecting an object to the Earth to establish electrical contact, setting its potential to zero.
Xerography	A dry copying process based on electrostatics, used in photocopiers and laser printers.
Photoconductor	A material that is an electrical insulator in the dark but becomes a conductor when exposed to light.

And finally, a glossary of important terms. Understanding these definitions is fundamental to grasping the concepts in this chapter. Review them to solidify your knowledge.