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### Chapter 24. Electric Potential

24.1. What is Physics?

24.2. Electric Potential Energy

24.3. Electric Potential

24.4. Equipotential Surfaces

24.5. Calculating the Potential from the Field

24.6. Potential Due to a Point Charge

24.7. Potential Due to a Group of Point Charges

24.8. Potential Due to an Electric Dipole

24.9. Potential Due to a Continuous Charge Distribution

24.10. Calculating the Field from the Potential

24.11. Electric Potential Energy of a System of Point Charges

24.12. Potential of a Charged Isolated Conductor

What is Physics?

- Gravitational force: F=Gm1m2/r2

- Electrostatic force: F=Gq1q2/r2

- One thing is in common: both of these forces are conservative

Electric Potential Energy

Electrostaticforce

Gravitational force

Note: Electric energy is one type of energy.

Reference Point of Electric Potential Energy

The reference point can be anywhere. For convenience, we usually set charged particles to be infinitely separated from one another to be zero potential energy

The potential energy U of the system at any point f is

where W∞ is the work done by the electric field on a charged particle as that particle moves in from infinity to point f.

Example 1

A proton, located at point A in an electric field, has an electric potential energy of UA = 3.20 × 10-19 J. The proton experiences an average electric force of 0.80 × 10-9 N, directed to the right. The proton then moves to point B, which is a distance of 1.00 × 10-10 m to the right of point A. What is the electric potential energy of the proton at point B ?

Electric Potential

The electric potential V at a given point is the electric potential energy U of a small test charge q0 situated at that point divided by the charge itself:

If we set at infinity as our reference potential energy,

SI Unit of Electric Potential: joule/coulomb=volt (V)

- Note:
- Both the electric potential energy U and the electric potential V are scalars.
- The electric potential energy U and the electric potential V are not the same. The electric potential energy is associated with a test charge, while electric potential is the property of the electric field and does not depend on the test charge.

The Electric Potential Difference

The electric potential difference between any two points i and f in an electric field.

- It is equal to the difference in potential energy per unit charge between the two points.
- the negative work done by the electric field on a unite charge as that particle moves in from point i to point f.

- Note:
- Only the differencesΔV and ΔU are measurable in terms of the work W.
- Theis ΔV property of the electric field and has nothing to do with a test charge
- The common name for electric potential difference is "voltage".

Notes Continue

- Electric field always points from higher electric potential to lower electric potential.
- A positive charge accelerates from a region of higher electric potential energy (or higher potential) toward a region of lower electric potential energy (or lower potential).
- A negative charge accelerates from a region of lower potential toward a region of higher potential.

Conceptual Example The Accelerations of Positive and Negative Charges

Three points, A, B, and C, are located along a horizontal line, as Figure 19.4 illustrates. A positive test charge is released from rest at A and accelerates toward B. Upon reaching B, the test charge continues to accelerate toward C. Assuming that only motion along the line is possible, what will a negative test charge do when it is released from rest at B?

Example 2Work, Electric Potential Energy, and Electric Potential

The work done by the electric force as the test charge (q0=+2.0×10–6 C) moves from A to B is WAB=+5.0×10–5 J. (a) Find the difference, ΔU=UB–UA, in the electric potential energies of the charge between these points. (b) Determine the potential difference, ΔV=VB–VA, between the points.

Example 3Operating a Headlight

Determine the number of particles, each carrying a charge of 1.60×10–19 C (the magnitude of the charge on an electron), that pass between the terminals of a 12-V car battery when a 60.0-W headlight burns for one hour.

Example 4Electric Field and Electric Potential

Two identical point charges (+2.4×10–9 C) are fixed in place, separated by 0.50 m. (see Figure 19.32). Find the electric field and the electric potential at the midpoint of the line between the charges qA and qB.

Equipotential Surfaces

An equipotential surface is a surface on which the electric potential is the same everywhere.

Relation of Equipotential Surfaces and the Electric Field

- The net electric force does no work as a charge moves on an equipotential surface.
- The electric field created by any charge or group of charges is everywhere perpendicular to the associated equipotential surfaces and points in the direction of decreasing potential.

What will happen if the electric field E is not perpendicular to the equipotential surface?

Check Your Understanding

The drawing shows a cross-sectional view of two spherical equipotential surfaces and two electric field lines that are perpendicular to these surfaces. When an electron moves from point A to point B (against the electric field), the electric force does +3.2×10–19 J of work. What are the electric potential differences (a) VB–VA, (b) VC–VB, and (c) VC–VA?

Calculating the Field fromthe Potential

The potential gradient gives the component of the electric field along the displacement Δs

Check Your Understanding

The sketch below shows cross sections of equipotential surfaces between two charged conductors that are shown in solid black. Various points on the equipotential surfaces near the conductors are labeled A, B, C, ..., I. At which of the labeled points will the electric field have the greatest magnitude

Example 4

The metal contacts of an electric wall socket are about 1.0 cm apart and are maintained at a potential difference of 120 V. What is the average electric field strength between the contacts? What is the direction of the electric field if the left contact is the higher potential? The lower potential? Treat the potential difference between the contacts as being constant in time.

Sample Problem

The electric potential at any point on the central axis of a uniformly charged disk is given by Eq. 24-37 ,

Starting with this expression, derive an expression for the electric field at any point on the axis of the disk.

Potential of a Charged Isolated Conductor

An excess charge placed on an isolated conductor will distribute itself on the surface of that conductor so that all points of the conductor—whether on the surface or inside—come to the same potential. This is true even if the conductor has an internal cavity and even if that cavity contains a net charge.

Isolated Conductor in an External Electric Field

- The free conduction electrons distribute themselves on the surface in such a way that the electric field they produce at interior points cancels the external electric field that would otherwise be there.
- The electron distribution causes the net electric field at all points on the surface to be perpendicular to the surface.

Sample Problem

(a) Figure 24-5a shows two points i and f in a uniform electric field E . The points lie on the same electric field line (not shown) and are separated by a distance d. Find the potential difference ΔV by moving a positive test charge q0 from i to f along the path shown, which is parallel to the field direction. (b) Now find the potential difference ΔV by moving the positive test charge q0 from i to f along the path icf shown in Fig. 24-5b.

Potential Due to a Point Charge

A zero reference potential is at infinity

Potential Due to a Group of Point Charges

The potential at a point due to any number of point charges can be found by simply finding the potential at the point due to each alone and adding the potentials: Vtot=V1+V2+∙∙∙+VN

Sample Problem

- (a) In Fig. 24-9a, 12 electrons (of charge −e) are equally spaced and fixed around a circle of radius R. Relative to V=0 at infinity, what are the electric potential and electric field at the center C of the circle due to these electrons?
- (b) If the electrons are moved along the circle until they are nonuniformly spaced over a 120° arc (Fig. 24-9b), what then is the potential at C? How does the electric field at C change (if at all)?

Line of Charge

In Fig. 24-12a, a thin nonconducting rod of length L has a positive charge of uniform linear density λ . Let us determine the electric potential V due to the rod at point P, a perpendicular distance d from the left end of the rod.

Charged Disk

In Section 22.7 , we calculated the magnitude of the electric field at points on the central axis of a plastic disk of radius R that has a uniform charge density σ on one surface. Here we derive an expression for V(z), the electric potential at any point on the central axis.

Electric Potential Energy of a System of Point Charges

The electric potential energy of a system of fixed point charges is equal to the work that must be done by an external agent to assemble the system, bringing each charge in from an infinite distance.

Sample Problem

Figure 24-16 shows three point charges held in fixed positions by forces that are not shown. What is the electric potential energy U of this system of charges? Assume that d=12 cm and that

Conceptual Questions

1. The drawing shows three possibilities for the potentials at two points, A and B. In each case, the same positive charge is moved from A to B. In which case, if any, is the most work done on the positive charge by the electric force? Account for your answer.

2. The electric field at a single location is zero. Does this fact necessarily mean that the electric potential at the same place is zero? Use a spot on the line between two identical point charges as an example to support your reasoning.

3. The potential is constant throughout a given region of space. Is the electric field zero or nonzero in this region? Explain.

4. In a region of space where the electric field is constant everywhere, is the potential constant everywhere? Account for your answer.

5. A positive test charge is placed in an electric field. In what direction should the charge be moved relative to the field, such that the charge experiences a constant electric potential? Explain.

6. The potential at a point in space has a certain value, which is not zero. Is the electric potential energy the same for every charge that is placed at that point? Give your reasoning.

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