Chapter 25

Electric Potential

1

Electrical Potential Energy

2

When a test charge is placed in an electric field, it experiences a force.

�⃑� = 𝑞°𝐸 The force is conservative. 𝑑𝑠 : is an infinitesimal displacement vector that

is oriented tangent to a path through space.

The work done by a conservative force equals the decrease in potential energy.

This will define the electric potential energy .

Electric Potential Energy, cont.

3

The work done by the electric force on the a charge q is

The potential energy of the charge-field system is changed by:

For a finite displacement of the charge from A to B, the change in potential energy of the system is

Because the force is conservative, the line integral does not depend on the path taken by the charge.

Electric Potential

4

The potential energy per unit charge, U/qo, is the electric potential. The electric potential is:

The potential is a scalar quantity. Units: 1 Volt (V) ≡ 1 J/C = 1 N/C As a charged particle moves in an electric field, it will experience a change in

potential.

o

UVq

=

B

Ao

UV dq∆

∆ = = − ⋅∫ E s

Electric Potential, Cont.

5

We often take the value of the potential to be zero at some convenient point in the field. (usually at infinity)

Assume a charge moves in an electric field without any change in its kinetic energy. The work by the field is:

W = ̶ ΔU = ̶ q ΔV

Electron-Volts

6

Another unit of energy that is commonly used in atomic and nuclear physics is the electron-volt.

One electron-volt is defined as the energy a charge-field system gains or loses when a charge of magnitude e (an electron or a proton) is moved through a potential difference of 1 volt.

1 eV = 1.60 x 10-19 J

Potential Difference in a Uniform Field

7

If the electric field is uniform:

The displacement points from A to B and is parallel to the field lines.

The negative sign indicates that the electric potential at point B is lower than at point A.

Electric field lines always point in the direction of decreasing electric potential.

B B

B A A AV V V d E d Ed∆ = − = − ⋅ = − = −∫ ∫E s s

More About Directions

8

The charged particle gains kinetic energy and the potential energy of the charge-field system decreases by an equal amount.

Another example of Conservation of Energy

( )f iK U q V q V V∆ = −∆ = − ∆ = − −

Directions, cont.

9

If a positive charge is released from rest in an electric field, it accelerates in the direction of the field; then:

The electric potential difference (ΔV) decreases,

The electric potential energy (ΔU) decreases,

The kinetic energy (ΔK) increases.

If a negative charge is released from rest in an electric field, it accelerates in a direction opposite the direction of the field:

The electric potential difference (ΔV) increases,

The electric potential energy (ΔU) decreases,

The kinetic energy (ΔK) increases.

Equipotentials

10

Point B is at a lower potential than point A. Points B and C are at the same potential.

The name equipotential surface is given to any surface consisting of a continuous distribution of points having the same electric potential.

All points in a plane perpendicular to a uniform electric field are at the same electric potential.

cosB AC A

V V V Es θV V Ed

∆ = − = −

= − = −

Electric Potential due to a Point Charges

11

Electric Potential due to a Point Charges , cont.

12

⇒ 𝑉𝑝 = 𝐾𝑒𝑞𝑅

Electric Potential due to a Point Charges , cont.

13

The potential difference

between points A and B will be

:

A positive point charge produces a field directed radially outward

Electric Potential due to a Point Charges, final

14

The electric potential is independent of the path between points A and B. It is customary to choose a reference potential of V = 0 at rA = ∞. The potential due to a point charge at some point r is

The potential due to a group of point charges is

V = Ke�qirii

V = Keqr

Potential Energy of Multiple Charges

15

Potential Energy of Multiple Charges, cont.

16

The potential energy of the system is:

If the two charges are the same sign, U is positive and work must be done to bring the charges together

If the two charges have opposite signs, U is negative and work is done to keep the charges apart.

𝑈 = 𝐾𝑒𝑞1𝑞2𝑟12

Potential Energy of Multiple Charges, cont.

17

If there are more than two charges, then find U for each pair of charges and add them.

For three charges:

𝑈12 = 𝐾𝑒𝑞1𝑞2𝑟12

𝑈13 = 𝐾𝑒𝑞1𝑞3𝑟13

𝑈23 = 𝐾𝑒𝑞2𝑞3𝑟23

Potential Energy of Multiple Charges, final

18

𝑈 = 𝐾𝑒 𝑞1𝑞2𝑟12 +𝑞1𝑞3𝑟13

+ 𝑞2𝑞3𝑟23

Since U is a scalar, then

Obtaining the Value of the Electric Field from the Electric Potential

19

Recall that:

which tells us how to find ΔV if the electric field 𝐸 is known.

What if the situation is reversed?

Assume, to start, that the field has only an x component. Then:

∆𝑉 = ∫ 𝑑𝑉 = −𝑉𝑓𝑉𝑖 ∫ 𝐸� .𝑑�̅�𝑓𝑖

⇒ 𝑑𝑉 = −𝐸� .𝑑�̅�

𝐸𝑥 = −𝑑𝑉𝑑𝑥

Obtaining the Value of E from V, cont.

20

Given V (x, y, z) you can find Ex, Ey and Ez as partial derivatives:

Equipotential surfaces must always be perpendicular to the electric field lines passing through them. Because:

The motion through a displacement ds along an equipotential surface:

𝑑𝑉 = 0 𝐸� .𝑑�̅� = 0

E and ds (and Equipotential surface) must be perpendicular.

𝐸𝑥 = −𝜕𝑉𝜕𝑥 , 𝐸𝑦 = −𝜕𝑉𝜕𝑦 , 𝐸𝑧 = −𝜕𝑉𝜕𝑧

E and V for an Infinite Sheet of Charge

21

The equipotential lines are the dashed blue lines.

The electric field lines are the brown lines.

The equipotential lines are everywhere perpendicular to the field lines.

E and V for a Point Charge

22

The equipotential lines are the dashed blue lines.

The electric field lines are the brown lines.

The electric field is radial. Er = - dV / dr The equipotential lines are

everywhere perpendicular to the field lines.

Electric Potential Due to Continuous Charge Distributions

23

Consider a small charge element dq Treat it as a point charge.

The potential at some point due to this

charge element is

𝑑𝑉 = Ke𝑑𝑞𝑟

To find the total potential, you need to integrate to include the contributions from all the elements.

𝑉 = 𝐾𝑒 �𝑑𝑞𝑟

Example 25.5 Electric Potential Due to a Uniformly Charged Ring

24

Answer: The potential is given by

Find an expression for the electric potential at a point P located on the

perpendicular central axis of a

uniformly charged ring of radius a

and total charge Q.

𝑉 = 𝐾𝑒 � 𝑑𝑞𝑟 =𝐾𝑒𝑄𝑎2+𝑥2

𝑄

0

Example 25.5 Electric Potential Due to a Uniformly Charged Ring, solution.

25

Example 25.6 Electric Potential Due to a Uniformly Charged Disk

26

A uniformly charged disk has radius R and surface charge density σ. Find the electric potential at a point P along the perpendicular central axis of the disk.

Answer: The potential is given by

𝑉 = 2𝜋𝑘𝑒𝜎� 𝑟𝑑𝑟𝑥2+𝑟2𝑅

0

= 2π𝐾𝑒𝜎 𝑅2 + 𝑥2 − 𝑥

Example 25.6 Electric Potential Due to a Uniformly Charged Disk

27

2πr dr

𝑉 = 2𝜋𝐾𝑒𝜎� 𝑟𝑑𝑟𝑟2+𝑥2𝑅

0

𝜎=𝑄𝐴=𝑑𝑞𝑑𝐴

Example 25.7 Electric Potential Due to a Finite Line of Charge

28

A rod of length L, located along the x axis has a total charge Q and a uniform linear charge density λ. Find the electric potential at a point P located on the y axis a distance a from the origin.

𝑉 = 𝐾𝑒λ∫ 𝑑𝑥𝑥2+𝑎2

ℓ0

𝑉 = 𝐾𝑒λln ℓ+ 𝑎2+ℓ2𝑎

Answer: The potential is given by

Example 25.7 Electric Potential Due to a Finite Line of Charge

29

λ = 𝑄𝐿 =𝑑𝑞𝑑𝑥

Properties of a Conductor in Electrostatic Equilibrium

30

When there is NO net motion of charge within a conductor, the conductor is said to be in

electrostatic equilibrium, it has the following

properties:

1) The electric field is Z E R O everywhere

inside the conductor. Whether the conductor

is solid or hollow

2) If the conductor is isolated and carries a

charge, the charge resides on its surface.

Properties of a Conductor in Electrostatic Equilibrium

31

3) The electric field at a point just outside a

charged conductor is perpendicular to the

surface and has a magnitude of: 𝐸 = 𝜎𝜀°

.

4) On an irregularly shaped conductor, the

surface charge density is greatest at locations

where the radius of curvature is the smallest.

𝐸 =𝜎𝜀°

Potential Due to a Charged Conductor

32

The surface of any charged conductor in electrostatic equilibrium is an equipotential surface.

Consider two points on the surface of the charged conductor as shown.( A and B )

. 𝐸 is always perpendicular to the displacement 𝑑𝑠 . Therefore,

𝑉𝐵 − 𝑉𝐴 = 𝐸.𝑑𝑠 = 0

⇒ 𝑉𝐵 = 𝑉𝐴

𝑑𝑠

V Due to a Charged Conductor, cont.

33

Because the electric field is zero inside the conductor, consider a third point C inside the

conductor, the potential difference:

We conclude that: the electric potential is constant everywhere inside the conductor and

equal to the value at the surface.

𝑉𝐵 − 𝑉𝐴 = 𝐸.𝑑𝑠 = 0

⇒ 𝑉𝐵 = 𝑉𝐴=𝑉𝐶

C

Irregularly Shaped Objects

34

The charge density is high where the radius of curvature is small.

And low where the radius of curvature is large

The electric field is large near the convex points having small radii of curvature and reaches very high values at sharp points.

Potential of a Charged Isolated Conducting Sphere (OR) Conducting Spherical Shell

35

R = Radius Q = Charge r = distance from the center

Field Due to a Plane of Charge

36

Recall that for a Charged Disk, the electric field is :

IF: X

Field Due to a Plane of Charge

37

Find the electric field due to an infinite plane of positive charge with uniform surface charge density σ.

.

Note, this does not depend on r. Therefore, the field is uniform everywhere.

The total charge in the surface is: qin=σA. Applying Gauss’s law:

Field Due to two Planes of Charge

38

Suppose two infinite planes of charge are parallel to each other, one positively charged

and the other negatively charged. The surface charge densities of both planes are of the same magnitude. What does the electric field look like in this situation?

.

E+

E-

Enet

Example

39

In a certain region, the electric potential due to a charge distribution is given by the equation V(x,y,z) = 3x2y2 + yz3 - 2z3x, where x, y, and z are measured in meters and V is in volts. Calculate the electric field vector at the position (x,y,z) = (1, 1, 1).

Example 25.6

40

Chapter 25

41

End of Part 01

Chapter 25Electrical Potential EnergyElectric Potential Energy, cont.Electric PotentialElectric Potential, Cont.Electron-VoltsPotential Difference in a Uniform FieldMore About DirectionsDirections, cont.EquipotentialsElectric Potential due to a Point ChargesElectric Potential due to a Point Charges , cont.Electric Potential due to a Point Charges , cont.Electric Potential due to a Point Charges, finalPotential Energy of Multiple ChargesPotential Energy of Multiple Charges, cont.Potential Energy of Multiple Charges, cont.Potential Energy of Multiple Charges, finalObtaining the Value of the Electric Field� from the Electric PotentialObtaining the Value of E from V, cont.E and V for an Infinite Sheet of ChargeE and V for a Point ChargeElectric Potential Due to� Continuous Charge DistributionsExample 25.5 Electric Potential Due to a Uniformly Charged RingExample 25.5 Electric Potential Due to a Uniformly Charged Ring, solution.Example 25.6 Electric Potential Due to a Uniformly Charged DiskExample 25.6 Electric Potential Due to a Uniformly Charged DiskExample 25.7 Electric Potential Due to a Finite Line of ChargeExample 25.7 Electric Potential Due to a Finite Line of ChargeProperties of a Conductor in �Electrostatic EquilibriumProperties of a Conductor in Electrostatic EquilibriumPotential Due to a Charged ConductorV Due to a Charged Conductor, cont.Irregularly Shaped ObjectsPotential of a Charged Isolated Conducting Sphere�(OR) Conducting Spherical ShellField Due to a Plane of ChargeField Due to a Plane of ChargeField Due to two Planes of ChargeExampleExample 25.6Chapter 25