Chapter 23: Electric Potential

Covers the concept of electric potential, which is a scalar quantity related to electric fields and potential energy in electrostatics.

A point charge, Q, creates an electric field, E→, in space. A positive point charge, +q, is then moved with a constant velocity by an external force from infinity to a distance d from Q. The electric potential energy between +q and Q when they are separated by d is equal to:

• the work done by the external force on +q during its displacement from ∞ to d

• the negative of the work done by the electric field on +q during its displacement from ∞ to d

Two analogous situations are shown in the figure: a positive charge that moves a distance d in the direction of a constant electric field, E→, and a mass m that moves a distance h vertically downward near the surface of the Earth. The change in gravitational potential energy of the mass during the displacement h→ is −mgh. Drag and drop the analogous gravitational counterparts against the corresponding electrical quantities.

• q ←→ m

• E ←→ g

• d ←→ h

A particle with a small positive charge, +Q, is moved from point A to point B in a region of space containing an unknown electric field. If the electrical potential energy of the charged particle at point B is greater than the electrical potential energy of the charged particle at point A, which of the following statements about the electric potential at point A and at point B is true?

V_B > V_A

Three charged particles are arranged in a vertical line, as shown. Suppose now a fourth particle, having a charge of +Q, is initially placed at point A and then is moved to point B. How much work was done by the electric field on +Q during its displacement from A to B? (Note: A and B are directly to the right of the middle charge.)

–kqQ/d

Equipotential lines for a certain electric field are shown. Rank, in descending order, according to the change in electrical potential energy, ΔU, that occurs when a positive point charge is moved between the points indicated. Put the path for which the most positive ΔU occurs at the top of the list and the path for which the most negative ΔU occurs at the bottom of the list.

1. J to B
2. J to H
3. A to B
4. B to E
5. A to I

By convention, the electrical potential energy of a point charge is equal to _______ when the charge is infinitely far from any other charges. The electrical potential energy of a point charge, q, located at a point P is equal to the negative of the work done by the electric field as the charge is displaced from ________ to point P.

• zero

• infinity

A point charge, q, is moved from point A to point B through a constant electric field, E→, whose direction is shown in the figure. The work done on the charge by the electric field, W_el, during this displacement, d→, is equal to: (Note that x and y are the components of d→.)

• qEd cosθ

• qEy

Suppose an electric field is created in some region of space by an unknown distribution of charges. In order for an electrical potential to exist at an arbitrary point in this region of space, there must be a charge located at that point.

False

The diagram shows equipotential surfaces in some region of space. Which of the directions (labeled I through VIII in the figure) is possible for the direction of the electric field at point A?

VIII

A positive point charge +Q is initially placed at point A, where the electrical potential is V_A, and then is moved to point B, where the electrical potential is V_B. If numerical values are given for +Q, V_A and V_B, which of the following quantities could be calculated?

• The work done by the electric field on +Q as it moves from A to B.

• The change in electrical potential energy of +Q due to its displacement from A to B.

• The electrical potential energy of +Q at point B.

Three point charges are positioned as shown. Suppose the –Q charge now moves upward in a straight vertical line to the position marked by the × in the figure. Which of the following statements is true?

• An external force, in addition to the electric force, is needed to move –Q to position ×.

• The electrical potential energy of the system of three charges increases when –Q moves to position ×.

• The work done by the electric field on –Q during its move to position ×is negative.

Five statements are given below that refer to the equipotential lines shown in the figure. Note: W_e (path #) means "the work done by the electric field, E→, in moving a point charge, q, along the path specified," and ΔU (path #) means "the change in electrical potential energy of q along the path specified." Identify the statements that are true.

• W_e (path 1) = 0

• The electric field corresponding to these equipotentials is constant.

A positive point charge +Q is initially placed at point A, where the electrical potential is VA, and then is moved to point B, where the electrical potential is VB. If numerical values are given for +Q, VA and VB, which of the following quantities could be calculated?

• The work done by the electric field on +Q as it moves from A to B.

• The change in electrical potential energy of +Q due to its displacement from A to B.

• The electrical potential energy of +Q at point B.

A particle with a small positive charge, +Q, is moved from point A to point B in a region of space containing an unknown electric field. If the electrical potential energy of the charged particle at point B is greater than the electrical potential energy of the charged particle at point A, which of the following statements about the electric potential at point A and at point B is true?

VB > VA

Three charged particles are arranged in a vertical line, as shown. Suppose now a fourth particle, having a charge of +Q, is initially placed at point A and then is moved to point B. How much work was done by the electric field on +Qduring its displacement from A to B? Note: A and B are directly to the right of the middle charge.

-kqQ/d

A set of concentric circles both close and far from the charge distribution

A single point charge

Two separate sets of concentric circles that eventually merge into a single set of concentric ovals

Two charges of the same sign

Two separate sets of concentric circles that eventually become two separate oval shapes

Two charges of opposite signs

The electrical potential 15 cm from a single point charge is 10 V. What is the electrical potential 20 cm from the point charge?

7.5 V

A constant electric field, E→, exists in the region shown. Which of the following expressions correctly describes the difference in electrical potential, ∆VAB, between points A and B shown in the figure to the right? (Note: ∆VAB = VB−VA.)

-Ed*cos(theta)

Four point charges of equal magnitude, but different signs, are arranged in a straight line. Six points, labeled A - F, are shown in the figure. At which points is the electric potential zero?

B, A

Equipotential lines are shown in the figure. The lines are equally spaced at 2.0 cm intervals. What is the magnitude and direction of the electric field at point A?

500 V/m to the left