Ch. 23 Electric Potential. Chapter Overview ► Review Work and Potential Energy ► Define...

Ch. 23 Ch. 23 Electric PotentialElectric Potential

Chapter OverviewChapter Overview► Review Work and Potential EnergyReview Work and Potential Energy►Define Potential DifferenceDefine Potential Difference► Compute the Potential Difference from the Compute the Potential Difference from the

Electric FieldElectric Field► Find the electric potential due to point chargesFind the electric potential due to point charges► Calculate the electric potential for a continuous Calculate the electric potential for a continuous

charge distributioncharge distribution► Compute the Electric Field from the PotentialCompute the Electric Field from the Potential►Describe Properties of Equipotential SurfacesDescribe Properties of Equipotential Surfaces

Review of WorkReview of Work►A force of 20 N is applied at an angle A force of 20 N is applied at an angle

of 20of 20° above the horizontal ° above the horizontal to a block to a block sitting on a frictionless horizontal sitting on a frictionless horizontal surface. a) Sketch the situation. b) surface. a) Sketch the situation. b) If If the force moves the block 2.5 m, how the force moves the block 2.5 m, how much work was done?much work was done?

W = Fd cosW = Fd cosθθ= 20 N x 2.5 m x cos 20°= 20 N x 2.5 m x cos 20°= 47 J= 47 J

dFW

Electrostatic Potential EnergyElectrostatic Potential Energy►Work done by an electrostatic force isWork done by an electrostatic force is►dW = dW = FF∙∙ddrr►Electrostatic Force is a “conservative” Electrostatic Force is a “conservative”

forceforce►dW = -dUdW = -dUelectrostaticelectrostatic

A -2.0 A -2.0 µC charge is placed in a µC charge is placed in a uniform electric field of 500 uniform electric field of 500 N/CN/Ca) Sketch the situation. b) a) Sketch the situation. b) What is the work done by the What is the work done by the electric field on the charge if it electric field on the charge if it moves 5.5 m? c) What is the moves 5.5 m? c) What is the change in electrostatic change in electrostatic potential energy of the potential energy of the charge? charge?

How would your answer for How would your answer for the the ΔΔUUelecelec change if the change if the charge moved was twice as charge moved was twice as big? big? (CT)(CT)

1 2 3 4

0% 0%0%0%

1.1. There would be no changeThere would be no change2.2. It would be twice as bigIt would be twice as big3.3. It would be ½ as bigIt would be ½ as big4.4. It cannot be determinedIt cannot be determined

11 22 33 44 55

How would your answer for How would your answer for the the ΔΔUUelecelec change if the charge change if the charge moved was half as big? moved was half as big? (CT)(CT)

1 2 3 4

0% 0%0%0%

1.1. There would be no changeThere would be no change2.2. It would be twice as bigIt would be twice as big3.3. It would be ½ as bigIt would be ½ as big4.4. It cannot be determinedIt cannot be determined

11 22 33 44 55

What type of relationship is What type of relationship is there between the charge and there between the charge and the the ΔΔUUelecelec? ? (GR)(GR)

Definition of the Potential Definition of the Potential DifferenceDifference

►Since the Since the ΔΔUUelecelec is proportional to the is proportional to the charge, we define a quantity which is charge, we define a quantity which is independent of the charge by dividing independent of the charge by dividing the potential energy by the charge.the potential energy by the charge.

Definition of the Potential Definition of the Potential DifferenceDifference

► If a charge goes from point A to If a charge goes from point A to point B and has a change in point B and has a change in electric potential energy of electric potential energy of dUdUelecelec, , then the potential difference then the potential difference between A and B isbetween A and B is

0qdUdV elec

Definition of the Potential Definition of the Potential DifferenceDifference

► Remember that E = F/q for a test Remember that E = F/q for a test chargecharge

► so so

ldEldqF

qdWdV

00

Definition of Potential Definition of Potential DifferenceDifference

► If we add up the small changes in potential If we add up the small changes in potential then we can obtain the finite potential then we can obtain the finite potential difference between the points in space A difference between the points in space A and Band B

ldEV

AB

Definition of the Potential Definition of the Potential DifferenceDifference

►SI Units?SI Units?

►This combination of units is called the This combination of units is called the volt, Vvolt, V

► 1 V = 1 J/C1 V = 1 J/C►volt is named after Alessandro Volta volt is named after Alessandro Volta

who invented the batterywho invented the battery

Ex. Two points in space are in Ex. Two points in space are in a region of uniform electric a region of uniform electric field of magnitude 550 N/C field of magnitude 550 N/C with the field pointing in the with the field pointing in the direction from one point to the direction from one point to the other. a) Sketch the situation other. a) Sketch the situation and depict the electric field. b) and depict the electric field. b) Find the potential difference Find the potential difference between the two points.between the two points.

Connection between Connection between Electrostatic Potential Energy Electrostatic Potential Energy

and Potential Differenceand Potential Difference►Knowing the potential difference Knowing the potential difference

between two point tells you the between two point tells you the change in electrostatic potential change in electrostatic potential energy between those two pointsenergy between those two points

►ΔΔUUelecelec = q = q00ΔΔV = -WV = -Welecelec

Ex. 2.0 J of work are done Ex. 2.0 J of work are done when a charge of 5.0 pC when a charge of 5.0 pC moves from point A to point moves from point A to point B. a) Sketch the situation. B. a) Sketch the situation. b) Find the change in Ub) Find the change in Uelecelec of of the charge. c) Find the the charge. c) Find the potential difference between potential difference between points A and B. points A and B.

Ex. A battery maintains a Ex. A battery maintains a constant potential difference constant potential difference between two pieces of metal between two pieces of metal of 6.0 V. A charge of -2.0 nC of 6.0 V. A charge of -2.0 nC moves between the plates. moves between the plates. a) Sketch the situation b) a) Sketch the situation b) What is the change of What is the change of electric potential energy for electric potential energy for the charge. the charge.

Ex. Two parallel plates of metal are Ex. Two parallel plates of metal are placed in a vacuum chamber. A placed in a vacuum chamber. A battery is used to place a potential battery is used to place a potential difference of 400 V between the difference of 400 V between the plates. An electron is released plates. An electron is released from rest at the plate with the from rest at the plate with the lower potential. a) Sketch the lower potential. a) Sketch the situation. b) What will happen to situation. b) What will happen to the electron and why? c) What will the electron and why? c) What will be the speed of the electron once it be the speed of the electron once it has crossed the gap between the has crossed the gap between the plates?plates?

Energy ConservationEnergy Conservation►The electrostatic force is a The electrostatic force is a

conservative force, so work done by it conservative force, so work done by it is stored as a potential energyis stored as a potential energy

►Review: The Work – Kinetic Energy Review: The Work – Kinetic Energy Theorem WTheorem Wnetnet = = ΔΔK where K = ½ mvK where K = ½ mv22

Energy ConservationEnergy Conservation► If no non-conservative work is done If no non-conservative work is done

then mechanical energy is conservedthen mechanical energy is conserved►E = K + UE = K + U

► If WIf Wncnc = 0 then = 0 then ΔΔE = 0E = 0

The Electric Potential of a Point The Electric Potential of a Point ChargeCharge

Find the Potential difference between Find the Potential difference between the points A and B along the same the points A and B along the same radial line outwards from the positive radial line outwards from the positive point charge Qpoint charge Q

Q A B

b

a

b

a

b

a

r

r

r

r

r

r

drdrrrV

r

ldEV

22

2

rkQˆˆ

rkQ

so dr,r̂ dl case in this and

ˆrkQE

chargepoint afor but

ab rkQ

rkQ

rkQ V

get weintegral out the Carrying

b

a

r

r

Suppose the points A and B were at Suppose the points A and B were at the same radii but in different the same radii but in different directions as shown. Would the directions as shown. Would the answer to the potential difference answer to the potential difference change?change?

Q A

1 2 3

0% 0%0%

1.1. YesYes2.2. NoNo3.3. Depends on the Depends on the

exact positionexact position

B

11 22 33 44 55

The Electric Potential of a Point The Electric Potential of a Point ChargeCharge

► The potential difference is given by the The potential difference is given by the difference ofdifference of

► The answer depends only on the radius The answer depends only on the radius of each point and not the direction of each point and not the direction

ab rkQ

rkQV

The Electric Potential of a Point The Electric Potential of a Point ChargeCharge

►We can conclude, if a point charge of We can conclude, if a point charge of charge Q is a distance r from a point P, charge Q is a distance r from a point P, then the electric potential at the point P is then the electric potential at the point P is given bygiven by

rQkV

At what distance from the At what distance from the charge Q is the electric charge Q is the electric potential 0? potential 0? (TPS)(TPS)

1 2 3 4

0% 0%0%0%

1.1. r = 0 (at the charge)r = 0 (at the charge)2.2. r = r = ∞ (very far from the ∞ (very far from the

chargecharge3.3. The electric potential is The electric potential is

never 0never 04.4. Cannot be determinedCannot be determined

11 22 33 44 55

At what distance from the At what distance from the charge Q is the electric charge Q is the electric

potential 0? potential 0? (TPS)(TPS)

1 2 3 4

0% 0%0%0%

1.1. r = 0 (at the charge)r = 0 (at the charge)2.2. r = r = ∞ (very far from ∞ (very far from

the chargethe charge3.3. The electric potential The electric potential

is never 0is never 04.4. Cannot be determinedCannot be determined

11 22 33 44 55

The Electric Potential of a Point The Electric Potential of a Point ChargeCharge

► We can conclude, if a point charge of charge We can conclude, if a point charge of charge Q is a distance r from a point P, then the Q is a distance r from a point P, then the electric potential at the point P is given byelectric potential at the point P is given by

► The electric potential of a point charge has The electric potential of a point charge has built in that the potential is 0 when r = built in that the potential is 0 when r = ∞∞

rQkV

Ex. A point P is 5.0 cm away Ex. A point P is 5.0 cm away from a small charged object from a small charged object with charge 2.0 with charge 2.0 μμC. a) C. a) Sketch the situation. b) Sketch the situation. b) Find Find the electric potential at the the electric potential at the point P c) What work would point P c) What work would be done to place a -2.0 be done to place a -2.0 μμC at C at the point P from very far the point P from very far away. d) What does the away. d) What does the work?work?

SuperpositionSuperposition►The electric potential due to several The electric potential due to several

charges is simply the sum of the charges is simply the sum of the potentials of the individual chargespotentials of the individual charges

►The electric potential is a scalar. No The electric potential is a scalar. No need to take into account direction, need to take into account direction, but do need to include signs in but do need to include signs in calculationcalculation

Ex. a) Find the electric potential at the point Ex. a) Find the electric potential at the point P. b) How much work would it require to bring P. b) How much work would it require to bring a 7.5 pC charge to point P from very far away? a 7.5 pC charge to point P from very far away? c) Does the field do work or is work done on c) Does the field do work or is work done on the field?the field?

-5.0 pc 3.0 PC

The Potential of a Continuous The Potential of a Continuous Distribution of ChargeDistribution of Charge

►Suppose we a continuous distribution Suppose we a continuous distribution of chargeof charge

►We can divide the object into small We can divide the object into small pieces that we treat like point objectspieces that we treat like point objects

++++++++++++++++++++

P

ri

dqi

The Potential of a Continuous The Potential of a Continuous Distribution of ChargeDistribution of Charge

►To find the total potential at P, we add To find the total potential at P, we add up all the little piecesup all the little pieces

rdqkV

Ex. Find the potential on the Ex. Find the potential on the axis of a uniform positively axis of a uniform positively charged ring of charge at a charged ring of charge at a point P a distance x from the point P a distance x from the center of the ring.center of the ring.

The Potential of a Continuous The Potential of a Continuous Distribution of ChargeDistribution of Charge

►Each small piece of charge dqEach small piece of charge dqii contributes a small piece to the contributes a small piece to the potential dVpotential dVii given by given by

rdqkdV i

i

The Potential of a Continuous The Potential of a Continuous Distribution of ChargeDistribution of Charge

► To find the total potential at the point PTo find the total potential at the point P

rdqkV

Determining the Potential from Determining the Potential from the Fieldthe Field

►We can in principle find the potential We can in principle find the potential for any distribution of charge usingfor any distribution of charge using

►However if we know the electric field, it However if we know the electric field, it can be easier to use the definition of can be easier to use the definition of the potential difference to find the the potential difference to find the potentialpotential

rdqkV

ldEV

Ex. A very long, insulating, Ex. A very long, insulating, solid cylinder of radius is solid cylinder of radius is uniformly charged with a uniformly charged with a positive charge. a) Sketch the positive charge. a) Sketch the cylinder and discuss what the cylinder and discuss what the symmetry tells you about the E-symmetry tells you about the E-field. b) Use Gauss’s Law to field. b) Use Gauss’s Law to find the Electric Field for r < R find the Electric Field for r < R and R > r. c) Use the electric and R > r. c) Use the electric field to find the potential for r < field to find the potential for r < R and r > RR and r > R

Determining E from VDetermining E from V► If you know the potential in a region of If you know the potential in a region of

space, you can determine the electric fieldspace, you can determine the electric field► SinceSince

►We can find E from V by taking a We can find E from V by taking a derivativederivative

► In 1D we can write In 1D we can write

ldEV

dxdVEx

Ex. A potential along the x-axis is Ex. A potential along the x-axis is given by V(x) = kQ/|x|. Find the given by V(x) = kQ/|x|. Find the electric field as a function of x. If electric field as a function of x. If Q is positive, what is the direction Q is positive, what is the direction of the electric field in the +x and of the electric field in the +x and –x directions?–x directions?

Ex. Two parallel metal plates Ex. Two parallel metal plates are separated by 2.0 mm. The are separated by 2.0 mm. The plates are connected to the plates are connected to the opposite terminals of a 6.0 V opposite terminals of a 6.0 V battery. a) Sketch the battery. a) Sketch the situation. b) Find the average situation. b) Find the average electric field in the region electric field in the region between the plates. c) Indicate between the plates. c) Indicate the direction of the electric field the direction of the electric field in the region between the in the region between the platesplates

A region of space has a A region of space has a constant positive electric constant positive electric potential. What can you say potential. What can you say about the electric field in that about the electric field in that region?region?

1 2 3 4

1.1. It is constant and It is constant and positivepositive

2.2. It is constant and It is constant and negativenegative

3.3. It is 0It is 04.4. Cannot be determinedCannot be determined11 22 33 44 55 66 77 88 99 1010 1111 1212 1313 1414 1515 1616 1717 1818 1919 2020

2121 2222 2323 2424 2525 2626 2727 2828 2929 3030 3131 3232 3333 3434 3535 3636 3737 3838 3939 4040

4141 4242 4343 4444 4545 4646 4747 4848 4949 5050

A region of space has an A region of space has an electric field of o. What can electric field of o. What can you say about the electric you say about the electric potential in that region?potential in that region?

1 2 3 4

1.1. It is constant and It is constant and positivepositive

2.2. It is constant and It is constant and negativenegative

3.3. It is 0It is 04.4. Cannot be determinedCannot be determined11 22 33 44 55 66 77 88 99 1010 1111 1212 1313 1414 1515 1616 1717 1818 1919 2020

2121 2222 2323 2424 2525 2626 2727 2828 2929 3030 3131 3232 3333 3434 3535 3636 3737 3838 3939 4040

4141 4242 4343 4444 4545 4646 4747 4848 4949 5050

Equipotential SurfacesEquipotential Surfaces►A surface at a constant electric potential is A surface at a constant electric potential is

an equipotential surfacean equipotential surface►Electric field lines are perpendicular to Electric field lines are perpendicular to

equipotential surfaces and point form equipotential surfaces and point form higher ot lower potentialhigher ot lower potential

►Two conductors in contact at electrostatic Two conductors in contact at electrostatic equilibrium will be equipotentialequilibrium will be equipotential

►Electric Field is higher near more sharply Electric Field is higher near more sharply curved points on an equipotential surfacecurved points on an equipotential surface

The figure shows a set of equi- The figure shows a set of equi- potential surfaces measured by a potential surfaces measured by a student. Find the average electric student. Find the average electric field and indicate the direction of field and indicate the direction of the electric field in each region.the electric field in each region.

Equipotential SurfacesEquipotential Surfaces►An equipotential surface is one in An equipotential surface is one in

which the potential difference between which the potential difference between any two points on the surface is 0any two points on the surface is 0

► Is a conductor in electrostatic Is a conductor in electrostatic equilibrium an equipotential surface?equilibrium an equipotential surface?

Dielectric BeakdownDielectric Beakdown