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FOR SCIENTISTS AND ENGINEERS

A STRATEGIC APPROACH

4/E

PHYSICS

RANDALL D. KNIGHT

Chapter 25 Lecture

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Chapter 25 The Electric Potential

IN THIS CHAPTER, you will learn to use the electric potential and electric potential energy.

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Chapter 25 Preview

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Chapter 25 Preview

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Chapter 25 Preview

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Chapter 25 Preview

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Chapter 25 Preview

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Chapter 25 Preview

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Chapter 25 Reading Questions

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The electric potential energy of a system of two point charges is proportional to

  1. The distance between the two charges.
  2. The square of the distance between the two charges.
  3. The inverse of the distance between the two charges.
  4. The inverse of the square of the distance between the two charges.

Reading Question 25.1

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The electric potential energy of a system of two point charges is proportional to

  1. The distance between the two charges.
  2. The square of the distance between the two charges.
  3. The inverse of the distance between the two charges.
  4. The inverse of the square of the distance between the two charges.

Reading Question 25.1

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What are the units of potential difference?

  1. Amperes
  2. Potentiometers
  3. Farads
  4. Volts
  5. Henrys

Reading Question 25.2

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What are the units of potential difference?

  1. Amperes
  2. Potentiometers
  3. Farads
  4. Volts
  5. Henrys

Reading Question 25.2

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New units of the electric field were introduced in this chapter. They are

  1. V/C
  2. N/C
  3. V/m
  4. J/m2
  5. Ω/m

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Reading Question 25.3

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New units of the electric field were introduced in this chapter. They are

Reading Question 25.3

  1. V/C
  2. N/C
  3. V/m
  4. J/m2
  5. Ω/m

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Which of the following statements about equipotential surfaces is true?

  1. Tangent lines to equipotential surfaces are always parallel to the electric field vectors.
  2. Equipotential surfaces are surfaces that have the same value of potential energy at every point.
  3. Equipotential surfaces are surfaces that have the same value of potential at every point.
  4. Equipotential surfaces are always parallel planes.
  5. Equipotential surfaces are real physical surfaces that exist in space.

Reading Question 25.4

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Which of the following statements about equipotential surfaces is true?

  1. Tangent lines to equipotential surfaces are always parallel to the electric field vectors.
  2. Equipotential surfaces are surfaces that have the same value of potential energy at every point.
  3. Equipotential surfaces are surfaces that have the same value of potential at every point.
  4. Equipotential surfaces are always parallel planes.
  5. Equipotential surfaces are real physical surfaces that exist in space.

Reading Question 25.4

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The electric potential inside a capacitor

  1. Is constant.
  2. Increases linearly from the negative to the positive plate.
  3. Decreases linearly from the negative to the positive plate.
  4. Decreases inversely with distance from the negative plate.
  5. Decreases inversely with the square of the distance from the negative plate.

Reading Question 25.5

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The electric potential inside a capacitor

  1. Is constant.
  2. Increases linearly from the negative to the positive plate.
  3. Decreases linearly from the negative to the positive plate.
  4. Decreases inversely with distance from the negative plate.
  5. Decreases inversely with the square of the distance from the negative plate.

Reading Question 25.5

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Chapter 25 Content, Examples, and QuickCheck Questions

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Energy

  • The kinetic energy of a system, K, is the sum of the kinetic energies Ki = 1/2mivi2 of all the particles in the system.
  • The potential energy of a system, U, is the interaction energy of the system.
  • The change in potential energy, ΔU, is –1 times the work done by the interaction forces:
  • If all of the forces involved are conservative forces (such as gravity or the electric force) then the total energy K + U is conserved; it does not change with time.

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Work Done by a Constant Force

  • Recall that the work done by a constant force depends on the angle θ between the force F and the displacement Δr:

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A Gravitational Analogy

  • Every conservative force is associated with a potential energy.
  • In the case of gravity, the work done is

Wgrav = mgyi – mgyf

  • The change in gravitational potential energy is

ΔUgrav = – Wgrav

where

Ugrav = U0 + mgy

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Two rocks have equal mass. Which has more gravitational potential energy?

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  1. Rock A
  2. Rock B
  3. They have the same potential energy.
  4. Both have zero potential energy.

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Two rocks have equal mass. Which has more gravitational potential energy?

QuickCheck 25.1

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  1. Rock A
  2. Rock B
  3. They have the same potential energy.
  4. Both have zero potential energy.

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Increasing PE

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A Uniform Electric Field

  • A positive charge q inside a capacitor speeds up as it “falls” toward the negative plate.
  • There is a constant force �F = qE in the direction of the displacement.
  • The work done is

Welec = qEsi – qEsf

  • The change in electric potential energy is

ΔUelec = –Welec

where

Uelec = U0 + qEs

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Two positive charges are equal. Which has more electric potential energy?

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  1. Charge A
  2. Charge B
  3. They have the same potential energy.
  4. Both have zero potential energy.

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Two positive charges are equal. Which has more electric potential energy?

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  1. Charge A
  2. Charge B
  3. They have the same potential energy.
  4. Both have zero potential energy.

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Increasing PE

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Electric Potential Energy in a Uniform Field

  • A positive charge inside a capacitor speeds up and gains kinetic energy as it “falls” toward the negative plate.
  • The charge is losing potential energy as it gains kinetic energy.

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  • For a positive charge, U decreases and K increases as the charge moves toward the negative plate.
  • A positive charge moving opposite the field direction is going “uphill,” slowing as it transforms kinetic energy into electric potential energy.

Electric Potential Energy in a Uniform Field

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  • A negative charged particle has negative potential energy.
  • U increases (becomes less negative) as the negative charge moves toward the negative plate.
  • A negative charge moving in the field direction is going “uphill,” transforming K → U as it slows.

Electric Potential Energy in a Uniform Field

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  • The figure shows the energy diagram for a positively charged particle in a uniform electric field.
  • The potential energy increases linearly with distance, but the total mechanical energy Emech is fixed.

Electric Potential Energy in a Uniform Field

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Two negative charges are equal. Which has more electric potential energy?

QuickCheck 25.3

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  1. Charge A
  2. Charge B
  3. They have the same potential energy.
  4. Both have zero potential energy.

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Two negative charges are equal. Which has more electric potential energy?

QuickCheck 25.3

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  1. Charge A
  2. Charge B
  3. They have the same potential energy.
  4. Both have zero potential energy.

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Increasing PE for negative charge

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A positive charge moves as shown. Its kinetic energy

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  1. Increases.
  2. Remains constant.
  3. Decreases.

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A positive charge moves as shown. Its kinetic energy

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  1. Increases.
  2. Remains constant.
  3. Decreases.

Increasing PE

Decreasing KE

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The Potential Energy of Two Point Charges

  • The change in electric potential energy of the system is ΔUelec = –Welec if
  • Consider two like charges q1 and q2.
  • The electric field of q1 pushes q2 as it moves from xi to xf.
  • The work done is

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The Potential Energy of Point Charges

  • Consider two point charges, q1 and q2, separated by a distance r. The electric potential energy is
  • This is explicitly the energy of the system, not the energy of just q1 or q2.

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  • Note that the potential energy of two charged particles approaches zero as r → ∞.

The Potential Energy of Two Point Charges

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  • Two like charges approach each other.
  • Their total energy is Emech > 0.
  • They gradually slow down until the distance separating them is rmin.
  • This is the distance of closest approach.

The Potential Energy of Two Point Charges

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  • Two opposite charges are shot apart from one another with equal and opposite momenta.
  • Their total energy is �Emech < 0.
  • They gradually slow down until the distance separating them is rmax.
  • This is their maximum separation.

The Potential Energy of Two Point Charges

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A positive and a negative charge are released from rest in vacuum. They move toward each other. As they do,

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QuickCheck 25.5

  1. A positive potential energy becomes more positive.
  2. A positive potential energy becomes less positive.
  3. A negative potential energy becomes more negative.
  4. A negative potential energy becomes less negative.
  5. A positive potential energy becomes a negative potential energy.

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A positive and a negative charge are released from rest in vacuum. They move toward each other. As they do,

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QuickCheck 25.5

  1. A positive potential energy becomes more positive.
  2. A positive potential energy becomes less positive.
  3. A negative potential energy becomes more negative.
  4. A negative potential energy becomes less negative.
  5. A positive potential energy becomes a negative �potential energy.

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The Electric Force Is a Conservative Force

  • Any path away from q1 can be approximated using circular arcs and radial lines.
  • All the work is done along the radial line segments, which is equivalent to a straight line from i to f.
  • Therefore the work done by the electric force depends only on initial and final position, not the path followed.

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Example 25.2 Approaching a Charged Sphere

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Example 25.2 Approaching a Charged Sphere

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Example 25.2 Approaching a Charged Sphere

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Example 25.3 Escape Speed

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Example 25.3 Escape Speed

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Example 25.3 Escape Speed

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Example 25.3 Escape Speed

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The Potential Energy of Multiple Point Charges

  • Consider more than two point charges. The potential energy is the sum of the potential energies due to all pairs of charges:

where rij is the distance between qi and qj.

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  • The summation contains the i < j restriction to ensure that each pair of charges is counted only once.

The Potential Energy of Multiple Point Charges

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Example 25.4 Launching an Electron

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Example 25.4 Launching an Electron

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Example 25.4 Launching an Electron

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Example 25.4 Launching an Electron

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The Potential Energy of a Dipole

  • The change in electric potential energy of the system is ΔUelec = –Welec if
  • Consider a dipole in a uniform electric field.
  • The forces F+ and F– exert a torque on the dipole.
  • The work done is

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The Potential Energy of a Dipole

  • The potential energy of a dipole is ϕ = 0º minimum at where the dipole is aligned with the electric field.
  • A frictionless dipole with mechanical energy Emech will oscillate back and forth between turning points on either side of ϕ = 0º.

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Example 25.5 Rotating a Molecule

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Example 25.5 Rotating a Molecule

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The Electric Potential

  • We define the electric potential V (or, for brevity, just the potential) as
  • The unit of electric potential is the joule per coulomb, which is called the volt V:

This battery is a source of electric potential. The electric potential difference between the + and – sides is 1.5 V.

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  • Test charge q is used as a probe to determine the electric potential, but the value of V is independent of q.
  • The electric potential, like the electric field, is a property of the source charges.

The Electric Potential

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Using the Electric Potential

  • As a charged particle moves through a changing electric potential, energy is conserved:

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A proton is released from rest at the dot. Afterward, the proton

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QuickCheck 25.6

  1. Remains at the dot.
  2. Moves upward with steady �speed.
  3. Moves upward with an increasing speed.
  4. Moves downward with a steady speed.
  5. Moves downward with an increasing speed.

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A proton is released from rest at the dot. Afterward, the proton

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QuickCheck 25.6

  1. Remains at the dot.
  2. Moves upward with steady �speed.
  3. Moves upward with an increasing speed.
  4. Moves downward with a steady speed.
  5. Moves downward with an increasing speed.

Decreasing PE

Increasing KE

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If a positive charge is released from rest, it moves in the direction of

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QuickCheck 25.7

  1. A stronger electric field.
  2. A weaker electric field.
  3. Higher electric potential.
  4. Lower electric potential.
  5. Both B and D.

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If a positive charge is released from rest, it moves in the direction of

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QuickCheck 25.7

  1. A stronger electric field.
  2. A weaker electric field.
  3. Higher electric potential.
  4. Lower electric potential.
  5. Both B and D.

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Problem-Solving Strategy: Conservation of Energy in Charge Interactions

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Problem-Solving Strategy: Conservation of Energy in Charge Interactions

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Example 25.6 Moving Through a Potential Difference

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Example 25.6 Moving Through a Potential Difference

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Example 25.6 Moving Through a Potential Difference

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Example 25.6 Moving Through a Potential Difference

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Example 25.6 Moving Through a Potential Difference

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The Electric Field Inside a Parallel-Plate Capacitor

  • This is a review of Chapter 23.

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The Electric Potential Inside a Parallel-Plate Capacitor

  • The electric potential inside a parallel-plate capacitor is

where s is the distance from the negative electrode.

  • The potential difference ΔVC, or “voltage” between the two capacitor plates is

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Units of Electric Field

  • If we know a capacitor’s voltage ΔV and the distance between the plates d, then the electric field strength within the capacitor is
  • This implies that the units of electric field are volts per meter, or V/m.
  • Previously, we have been using electric field units of newtons per coulomb.
  • In fact, as you can show as a homework problem, these units are equivalent to each other:

1 N/C = 1 V/m

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The Electric Potential Inside a Parallel-Plate Capacitor

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The Electric Potential Inside a Parallel-Plate Capacitor

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Two protons, one after the other, are launched from point 1 with the same speed. They follow the two trajectories shown. The protons’ speeds at points 2 and 3 are related by

QuickCheck 25.8

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  1. v2 > v3
  2. v2 = v3
  3. v2 < v3
  4. Not enough information to compare their speeds.

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Two protons, one after the other, are launched from point 1 with the same speed. They follow the two trajectories shown. The protons’ speeds at points 2 and 3 are related by

QuickCheck 25.8

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  1. v2 > v3
  2. v2 = v3
  3. v2 < v3
  4. Not enough information to compare their speeds.

Energy conservation

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The Parallel-Plate Capacitor

  • The figure shows the contour lines of the electric potential and the electric field vectors inside a parallel-plate capacitor.
  • The electric field vectors are perpendicular to the equipotential surfaces.
  • The electric field points in the direction of decreasing potential.

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The Zero Point of Electric Potential

  • Where you choose V = 0 is arbitrary. The three contour maps below represent the same physical situation.

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The Electric Potential of a Point Charge

  • Let q in the figure be the source charge, and let a second charge q', a distance r away, probe the electric potential of q.
  • The potential energy of the two point charges is

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  • The electric potential due to a point charge q is
  • The potential extends through all of space, showing the influence of charge q, but it weakens with distance as 1/r.
  • This expression for V assumes that we have chosen V = 0 to be at r = ∞.

The Electric Potential of a Point Charge

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What is the ratio VB/VA of the electric potentials at the two points?

QuickCheck 25.9

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  1. 9
  2. 3
  3. 1/3
  4. 1/9
  5. Undefined without knowing the charge

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What is the ratio VB/VA of the electric potentials at the two points?

QuickCheck 25.9

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  1. 9
  2. 3
  3. 1/3
  4. 1/9
  5. Undefined without knowing the charge

Potential of a point charge decreases inversely with distance.

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The Electric Potential of a Point Charge

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The Electric Potential of a Point Charge

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The Electric Potential of a Charged Sphere

  • Outside a uniformly charged sphere of radius R, the electric potential is identical to that of a point charge Q at the center:

where r ≥ R.

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  • If the potential at the surface V0 is known, then the potential at r ≥ R is

A plasma ball consists of a small metal ball inside a hollow glass sphere filled with low-pressure neon gas. The high voltage of the ball creates “lightning bolts” between the ball and the glass sphere.

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An electron follows the trajectory shown from i to f. At point f,

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QuickCheck 25.10

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  1. vf > vi
  2. vf = vi
  3. vf < vi
  4. Not enough information to compare the � speeds at these points.

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An electron follows the trajectory shown from i to f. At point f,

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QuickCheck 25.10

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  1. vf > vi
  2. vf = vi
  3. vf < vi
  4. Not enough information to compare the � speeds at these points.

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Increasing PE (becoming less negative) so decreasing KE

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Example 25.8 A Proton and a Charged Sphere

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Example 25.8 A Proton and a Charged Sphere

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Example 25.8 A Proton and a Charged Sphere

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Example 25.8 A Proton and a Charged Sphere

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  • The electric potential V at a point in space is the sum of the potentials due to each charge:

where ri is the distance from charge qi to the point in space where the potential is being calculated.

  • The electric potential, like the electric field, obeys the principle of superposition.

The Electric Potential of Many Charges

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The Electric Potential of an Electric Dipole

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  • Electrical activity within the body can be monitored by measuring equipotential lines on the skin.
  • The equipotentials near the heart are a slightly distorted but recognizable electric dipole.

The Electric Potential of a Human Heart

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At the midpoint between these two equal but opposite charges,

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QuickCheck 25.11

  1. E = 0; V = 0
  2. E = 0; V > 0
  3. E = 0; V < 0
  4. E points right; V = 0
  5. E points left; V = 0

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At the midpoint between these two equal but opposite charges,

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QuickCheck 25.11

  1. E = 0; V = 0
  2. E = 0; V > 0
  3. E = 0; V < 0
  4. E points right; V = 0
  5. E points left; V = 0

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At which point or points is the electric potential zero?

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QuickCheck 25.12

A.

B.

C.

D.

E. More than one of these.

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At which point or points is the electric potential zero?

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QuickCheck 25.12

A.

B.

C.

D.

E. More than one of these.

V = 0

V = 0

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Example 25.9 The Potential of Two Charges

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Example 25.9 The Potential of Two Charges

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Example 25.9 The Potential of Two Charges

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Problem-Solving Strategy: The Electric Potential of a Continuous Distribution of Charge

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Problem-Solving Strategy: The Electric Potential of a Continuous Distribution of Charge

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Example 25.10 The Potential of a Ring of Charge

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Example 25.10 The Potential of a Ring of Charge

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Example 25.10 The Potential of a Ring of Charge

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Example 25.10 The Potential of a Ring of Charge

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Example 25.10 The Potential of a Ring of Charge

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Chapter 25 Summary Slides

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General Principles

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General Principles

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Applications

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Applications

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