Electric PotentialElectric Potential

Chapter 23Chapter 23

Electric potential energyElectric potential energy

When an electrostatic force acts When an electrostatic force acts between two or more charged between two or more charged particles within a system of particles within a system of particles, we can assign an particles, we can assign an electric electric potential energypotential energy UU to the system. If to the system. If the system changes its configuration the system changes its configuration from an initial state from an initial state ii to a different to a different final state final state ff, the electrostatic force , the electrostatic force does work does work WW on the particles: on the particles:

Electric potentialElectric potential

The potential energy per unit charge, The potential energy per unit charge, which can be symbolized as which can be symbolized as UU//qq, is , is independent of the charge independent of the charge qq of the of the particle we happen to use and is particle we happen to use and is characteristic only of the electric fieldcharacteristic only of the electric field we are investigating. The potential we are investigating. The potential energy per unit charge at a point in an energy per unit charge at a point in an electric field is called the electric field is called the electric electric potentialpotential VV (or simply the (or simply the potentialpotential) ) at that point. Thus, at that point. Thus,

Potential differencePotential difference

The The electric potential differenceelectric potential difference ΔΔVV between any two points between any two points ii and and ff in in an electric field is equal to the an electric field is equal to the difference in potential energy per difference in potential energy per unit charge between the two points: unit charge between the two points:

When +2.0 C of charge moves at When +2.0 C of charge moves at constant speed from a point with constant speed from a point with

zero potential to a point with zero potential to a point with potential +6.0 V, the amount of potential +6.0 V, the amount of

work done is work done is A.A. 2 J. 2 J.

B.B. 3 J. 3 J.

C.C. 6 J. 6 J.

D.D. 12 J. 12 J.

E.E. 24 J. 24 J.

Equipotential surfacesEquipotential surfaces

Adjacent points that have the same Adjacent points that have the same electric potential form an electric potential form an equipotential surfaceequipotential surface

Fig. 24-3

The concept of difference in The concept of difference in electric potential is most electric potential is most closely associated with closely associated with

A.A. the mechanical force on an electron. the mechanical force on an electron. B.B. the number of atoms in one gram-the number of atoms in one gram-

atom. atom. C.C. the charge on one electron. the charge on one electron. D.D. the resistance of a certain specified the resistance of a certain specified

column of mercury. column of mercury. E.E. the work per unit quantity of electric the work per unit quantity of electric

charge. charge.

Charges Charges QQ and and qq ( (QQ ≠≠ qq), separated by a distance ), separated by a distance dd, produce a potential , produce a potential VVPP = 0 at point P. This = 0 at point P. This

means that means that A.A. no force is acting on a test charge placed at point P.no force is acting on a test charge placed at point P.

B.B. QQ and and qq must have the same sign. must have the same sign.

C.C. the electric field must be zero at point P. the electric field must be zero at point P.

D.D. the net work in bringing the net work in bringing QQ to distance to distance dd from from qq is is zero. zero.

E.E. the net work needed to bring a charge from infinity to the net work needed to bring a charge from infinity to point P is zero. point P is zero.

VV from from E E

VV of a point charge of a point charge

23-1923-19

A point particle has a charge equal to A point particle has a charge equal to +2.00 +2.00 μμC and is fixed at the origin. (C and is fixed at the origin. (aa) ) What is the electric potential What is the electric potential VV at a at a point 4.00 m from the origin assuming point 4.00 m from the origin assuming that that VV = 0 at infinity? ( = 0 at infinity? (bb) How much ) How much work must be done to bring a second work must be done to bring a second point particle that has a charge of point particle that has a charge of +3.00 +3.00 μμC from infinity to a distance of C from infinity to a distance of 4.00 m from the +2.00-4.00 m from the +2.00-μμC charge? C charge?

23-2123-21 A uniform electric field has a magnitude 2.00 A uniform electric field has a magnitude 2.00

kN/C and points in the +kN/C and points in the +xx direction. ( direction. (aa) What is ) What is the electric potential difference between the the electric potential difference between the xx = 0.00 m plane and the = 0.00 m plane and the xx = 4.00 m plane? A = 4.00 m plane? A point particle that has a charge of +3.00 point particle that has a charge of +3.00 μμC is C is released from rest at the origin. (released from rest at the origin. (bb) What is the ) What is the change in the electric potential energy of the change in the electric potential energy of the particle as it travels from the particle as it travels from the xx = 0.00 m plane = 0.00 m plane to the to the xx = 4.00 m plane? ( = 4.00 m plane? (cc) What is the kinetic ) What is the kinetic energy of the particle when it arrives at the energy of the particle when it arrives at the xx = = 4.00 m plane? (4.00 m plane? (dd) Find the expression for the ) Find the expression for the electric potential electric potential VV((xx) if its value is chosen to ) if its value is chosen to be zero at be zero at xx = 0. = 0.

VV of a group of charges of a group of charges

V obeys the principle of V obeys the principle of superpositionsuperposition

Q

200V 00V a b c

Q

Two equal positive charges are placed in Two equal positive charges are placed in an external electric field. The an external electric field. The

equipotential lines shown are at 100 V equipotential lines shown are at 100 V intervals. The potential for line intervals. The potential for line cc is is

A.A. 100 V.100 V.

B.B. 100 V.100 V.

C.C. 200 V.200 V.

D.D. 200 V.200 V.

E.E. zerozero

22-2622-26

Four point charges, each having a Four point charges, each having a magnitude of 2.00 magnitude of 2.00 μμC, are fixed at the C, are fixed at the corners of a square whose edges are 4.00 corners of a square whose edges are 4.00 m long. Find the electric potential at the m long. Find the electric potential at the center of the square if (center of the square if (aa) all the charges ) all the charges are positive, (are positive, (bb) three of the charges are ) three of the charges are positive and one charge is negative, and positive and one charge is negative, and ((cc) two charges are positive and two ) two charges are positive and two charges are negative. (Assume the charges are negative. (Assume the potential is zero very far from all charges.) potential is zero very far from all charges.)

23-3123-31

Two identical positively charged Two identical positively charged point particles are fixed on the point particles are fixed on the xx axis axis at at xx = + = +aa and and xx = – = –aa (a) Write an (a) Write an expression for the electric potential expression for the electric potential VV((xx) as a function of ) as a function of xx for all points for all points on the on the xx axis. (b) Sketch axis. (b) Sketch VV((xx) versus ) versus xx for all points on the for all points on the xx axis. axis.

In what direction can you move In what direction can you move relative to an electric field so relative to an electric field so

that the electric potential does that the electric potential does not change?not change?

A.A. parallel to the electric fieldparallel to the electric field

B.B. perpendicular to the electric perpendicular to the electric field field

In what direction can you move In what direction can you move relative to an electric field so relative to an electric field so

that the electric potential that the electric potential increases at the greatest rate?increases at the greatest rate?A.A. in the direction of the electric fieldin the direction of the electric field

B.B. opposite to the direction of the opposite to the direction of the electric fieldelectric field

C.C. perpendicular to the electric field perpendicular to the electric field

Charges +Charges +QQ and – and –QQ are arranged at the corners of a are arranged at the corners of a square as shown. When the magnitude of the electric square as shown. When the magnitude of the electric

field field EE and the electric potential and the electric potential VV are determined at P, are determined at P, the center of the square, we find that the center of the square, we find that

A.A. EE ≠≠ 0 and 0 and VV > 0. > 0.

B.B. EE = 0 and = 0 and VV = 0. = 0.

C.C. EE = 0 and = 0 and VV > 0. > 0.

D.D. EE ≠≠ 0 and 0 and VV < 0. < 0.

E.E. None of these is None of these is correct.correct.

VV of a charge distribution of a charge distribution

23-4423-44

A conducting spherical shell of inner A conducting spherical shell of inner radius radius bb and outer radius and outer radius cc is concentric is concentric with a small metal sphere of radius with a small metal sphere of radius a < b.a < b. The metal sphere has a positive charge The metal sphere has a positive charge Q.Q. The total charge on the conducting The total charge on the conducting spherical shell is –spherical shell is –Q.Q. (Assume the (Assume the potential is zero very far from all potential is zero very far from all charges.) (charges.) (aa) What is the electric potential ) What is the electric potential of the spherical shell? (of the spherical shell? (bb) What is the ) What is the electric potential of the metal sphere? electric potential of the metal sphere?

23-5123-51

A rod of length A rod of length LL has a total charge has a total charge Q,Q, uniformly distributed along its length. uniformly distributed along its length. The rod lies along the The rod lies along the yy axis with its axis with its center at the origin. (center at the origin. (aa) Find an ) Find an expression for the electric potential as expression for the electric potential as a function of position along the a function of position along the xx axis. axis. ((bb) Show that the result obtained in ) Show that the result obtained in Part (Part (aa) reduces to ) reduces to V = kQ/|x|V = kQ/|x| for for |x| |x| >> L.>> L. Explain why this result is Explain why this result is expected. expected.

EE from from VV

The electric potential in a region of space is The electric potential in a region of space is given by given by VV = 2 = 2xyxy + 3 + 3yy22 in units of V. The in units of V. The

electric field, in V/m, in this region is electric field, in V/m, in this region is

above. theof None E.

ˆ)62(ˆ2 D.

)ˆ3ˆ(2 C.

)ˆ3ˆ(2 B.

ˆ)62(ˆ2 A.

jyxiy

jiy

jiy

jyxiy

23-7323-73

Two positive point charges each Two positive point charges each have a charge of +have a charge of +qq and are fixed on and are fixed on the the yy axis at axis at yy = + = +aa and and yy = – = –a.a. ( (aa) ) Find the electric potential at any Find the electric potential at any point on the point on the xx axis. ( axis. (bb) Use your ) Use your result in Part (result in Part (aa) to find the electric ) to find the electric field at any point on the field at any point on the xx axis. axis.

23-8023-80

A charge of +2.00 nC is uniformly A charge of +2.00 nC is uniformly distributed on a ring of radius 10.0 distributed on a ring of radius 10.0 cm that lies in the cm that lies in the xx = 0 plane and is = 0 plane and is centered at the origin. A point centered at the origin. A point charge of +1.00 nC is initially charge of +1.00 nC is initially located on the located on the xx axis at axis at xx = 50.0 cm. = 50.0 cm. Find the work required to move the Find the work required to move the point charge to the origin. point charge to the origin.

If the potential If the potential VV of an array of charges of an array of charges versus the distance from the charges is versus the distance from the charges is as shown in graph 1, which graph A, B, as shown in graph 1, which graph A, B, C, D, or E shows the electric field C, D, or E shows the electric field EE as a as a

function of distance function of distance rr? ?

23-3823-38

The electric potential due to a particular The electric potential due to a particular charge distribution is measured at many charge distribution is measured at many points along the points along the xx axis. A plot of the data axis. A plot of the data is shown in is shown in Figure 23-34. At what location . At what location (or locations) is the (or locations) is the xx component of the component of the electric field equal to zero? At this electric field equal to zero? At this location (or these locations) is the location (or these locations) is the potential also equal to zero? Explain your potential also equal to zero? Explain your answer. answer.

VV of a system of charges of a system of charges

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

23-4123-41

An infinite line charge of linear An infinite line charge of linear charge density +1.50 charge density +1.50 μμC/m lies on C/m lies on the the zz axis. Find the electric potential axis. Find the electric potential at distances from the line charge of at distances from the line charge of ((aa) 2.00 m, () 2.00 m, (bb) 4.00 m, and () 4.00 m, and (cc) 12.0 ) 12.0 m. Assume that we choose m. Assume that we choose VV = 0 at a = 0 at a distance of 2.50 m from the line of distance of 2.50 m from the line of charge. charge.

VV of an isolated of an isolated conductorconductor

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

VV surfaces surfaces

Which of the following Which of the following statements regarding potential statements regarding potential

is true? is true?

A.A. The units of potential are N/C. The units of potential are N/C.

B.B. Potential is a vector quantity. Potential is a vector quantity.

C.C. Equipotential surfaces are at right Equipotential surfaces are at right angles to lines of electric force. angles to lines of electric force.

D.D. Potential differences can be measured Potential differences can be measured directly with a ballistic galvanometer. directly with a ballistic galvanometer.

E.E. Equipotential surfaces for an isolated Equipotential surfaces for an isolated point charge are cubes concentric with point charge are cubes concentric with the charge. the charge.

The work required to The work required to bring a positively bring a positively charged body from charged body from very far away is very far away is greatest for which greatest for which point?point?

Three charges are brought from infinity Three charges are brought from infinity and placed at the corner of an and placed at the corner of an

equilateral triangle. Which of the equilateral triangle. Which of the following statements is following statements is truetrue??

A.A. The work required to assemble the charges is The work required to assemble the charges is always positive.always positive.

B.B. The electrostatic potential energy of the The electrostatic potential energy of the system is always positive.system is always positive.

C.C. The electrostatic potential energy does not The electrostatic potential energy does not depend on the order the charges are placed at depend on the order the charges are placed at the corners. the corners.

D.D. The work required to assemble the charges The work required to assemble the charges depends on which charge is placed at which depends on which charge is placed at which corner.corner.

E.E. The electrostatic potential energy depends on The electrostatic potential energy depends on which charge is placed at which corner. which charge is placed at which corner.

Electrical potential Electrical potential energyenergy

The electrostatic potential energy of The electrostatic potential energy of a system of point charges is the a system of point charges is the work needed to bring the charges work needed to bring the charges from an infinite separation to their from an infinite separation to their final positions. final positions.

Which of the points shown in Which of the points shown in the diagram are at the same the diagram are at the same

potential? potential?

A.A. 2 and 5 2 and 5

B.B. 2, 3, and 2, 3, and 5 5

C.C. 1 and 4 1 and 4

D.D. 1 and 5 1 and 5

E.E. 2 and 4 2 and 4

The electric field for a charge distribution is The electric field for a charge distribution is E E = 0 for = 0 for

rr < 1 m, and for . < 1 m, and for .

A. 4000 V.

B. 2000 V.

C. 1000 V.

D. Zero.

E. Cannot be determined precisely.

E (

10 V

/m)

3

r (m)0 1 2 3 4

0

1

2

3

4

rr

E ˆmV4000

2

m 1r

Use the reference point Use the reference point VV = 0 as = 0 as rr infinity. The infinity. The potential for potential for rr < 1 m is < 1 m is

The figure depicts a uniform The figure depicts a uniform electric field. Along which electric field. Along which

direction is the increase in the direction is the increase in the electric potential a maximum? electric potential a maximum?

Which of the curves on the graph Which of the curves on the graph represents the electrostatic potential represents the electrostatic potential

energy of a small negative charge energy of a small negative charge plotted as a function of its distance from plotted as a function of its distance from

a positive point charge? a positive point charge?

23-6623-66

Point charges Point charges q1, q2,q1, q2, and and qq3 are fixed at 3 are fixed at the vertices of an equilateral triangle the vertices of an equilateral triangle whose sides are 2.50 m long. Find the whose sides are 2.50 m long. Find the electrostatic potential energy of this electrostatic potential energy of this system of charges for the following charge system of charges for the following charge values: (values: (aa) ) q1 = q2 = qq1 = q2 = q3 = +4.20 3 = +4.20 μμC, (C, (bb) ) q1 q1 = q= q2 = +4.20 2 = +4.20 μμC and C and qq3 = –4.20 3 = –4.20 μμC; C; and (and (cc) ) q1 = qq1 = q2 = –4.20 2 = –4.20 μμC and C and qq3 = +4.20 3 = +4.20 μμC. (Assume the potential energy is zero C. (Assume the potential energy is zero when the charges are very far from each when the charges are very far from each other.) other.)

23-7923-79 A positive point charge +A positive point charge +QQ is located on the is located on the xx

axis at axis at xx = – = –aa. (. (aa) How much work is required to ) How much work is required to bring an identical point charge from infinity to bring an identical point charge from infinity to the point on the the point on the xx axis at axis at xx = + = +a?a? ( (bb) With the ) With the two identical point charges in place at two identical point charges in place at xx = – = –aa and and xx = + = +a,a, how much work is required to bring a how much work is required to bring a third point charge –third point charge –QQ from infinity to the origin? from infinity to the origin? ((cc) How much work is required to move the ) How much work is required to move the charge –charge –QQ from the origin to the point on the from the origin to the point on the xx axis at axis at xx = 2 = 2aa along the semicircular path shown along the semicircular path shown ((Figure 23-35)? )?

23-8423-84

A metal sphere centered at the origin has a A metal sphere centered at the origin has a surface charge density that has a surface charge density that has a magnitude of 24.6 nC/m2 and a radius less magnitude of 24.6 nC/m2 and a radius less than 2.00 m. A distance of 2.00 m from the than 2.00 m. A distance of 2.00 m from the origin, the electric potential is 500 V and origin, the electric potential is 500 V and the electric field strength is 250 V/m. the electric field strength is 250 V/m. (Assume the potential is zero very far from (Assume the potential is zero very far from the sphere.) (the sphere.) (aa) What is the radius of the ) What is the radius of the metal sphere? (metal sphere? (bb) What is the sign of the ) What is the sign of the charge on the sphere? Explain your charge on the sphere? Explain your answer. answer.

23-9123-91

Show that the total work needed to Show that the total work needed to assemble a uniformly charged assemble a uniformly charged sphere that has a total charge of sphere that has a total charge of QQ and radius and radius RR is given by 3 is given by 3QQ2/(20 2/(20 εε00RR). Energy conservation tells us ). Energy conservation tells us that this result is the same as the that this result is the same as the resulting electrostatic potential resulting electrostatic potential energy of the sphere. energy of the sphere.