Chapter 22 Electric Potential (Voltage)

Question 29.6 Work and Electric Potential II

P 1

2

3

4

P 1

2

3

4

Question 29.5 Work and Electric Potential I

Electric potential energy •  Recall how a conservative force is related to the

potential energy associated with that force:

•  The electric potential energy is the potential energy due to the electric force, which can be expressed in terms of the electric field.

•  If location A is chosen to be the zero point, then the electric potential at location B (which we now call r) is given by

Potential energy of particle is a scalar function of space.

Consider uniform electric field (say inside a parallel capacitor)

If a proton is taken from location B to location C, how does its potential energy change?

1.  it decreases 2.  it increases 3.  it doesn’t change

Suppose a proton is released from rest just below the top (positive) plate of an parallel plate capacitor with an electric field strength E = 100 N/C. If the distance between the plates is d = 3 mm, how fast is it moving when it hits the bottom (negative) plate?

Electric Potential (Voltage) •  Electric potential, or voltage, is defined as the electric

potential energy per unit charge a test particle would have if it were located at a position

•  Potential energy deals with the energy of a particle. Voltage deals with all locations in space (no particle needs to be there).

•  Analogous to how a particle experiences a force, but an electric field can exist at any point in space.

Potential Difference (Voltage Difference) •  Voltage difference is defined as

•  Because the electrostatic field is conservative, it doesn’t matter what path is taken between those points.

•  In a uniform field, the potential difference becomes

Clicker Question In a parallel plate capacitor, the electric field is uniform and is directed from the positive plate to the negative plate. An electron goes from location A to location C. Which statement is true?

A)  The electron goes from a high voltage to a lower voltage.

B)  The electron goes from a low voltage to a higher voltage.

C)  The voltage is the same at both locations.

Millikan’s Oil Drop Experiment •  Charged oil droplets made to levitate inside capacitor

•  Measure voltage difference across plates •  Release and measure terminal velocity (which gives droplet

radius/mass) •  Determine net charge on droplet.

Clicker question •  The figure shows three straight paths AB of the same

length, each in a different electric field. Which one of the three has the largest magnitude of a voltage difference between the two points?

A.  (a) B.  (b) C.  (c)

Voltage of a point charge •  The point-charge field varies with

position, so we need to integrate:

•  Taking the zero of potential at infinity and letting gives

VB = V (�r) =kq

r

Rutherford scattering. A helium nucleus of mass 4 mp is emitted with an initial speed of v0 = 4.9 x 105 m/s towards a gold nucleus of charge q2 = 79 e. What is the minimum distance between the two particles (assume the gold nucleus doesn’t move)?

Example

Voltage due to a charge distribution •  If the electric field of the charge distribution is known,

the voltage can be found by integration. •  Alternatively, the voltage can be found by summing

point-charge potentials: •  For discrete point charges,

•  For a continuous charge distribution,

V (P ) = −�

�E · d�r = −� �

�

i

�Ei

�· d�r = −

�

i

��Ei · d�r =

�

i

Vi(P )

Clicker Question Two identical positive charges of charge Q are a distance d

apart. What is the voltage at the midway point between the charges?

a)  k Q/d b)  2 k Q/d c)  4 k Q/d d)  8 k Q/d e)  0

Clicker question Location P is equidistant from the two charges of an electric

dipole. The voltage at P is

a)  positive b)  zero c)  negative

CT 29.12c

At which labeled point is voltage highest?

A

B

C

D

E

Potential of charged sphere •  Outside sphere, electric field is identical to that of a point

charge.

•  What is V for (r<R)?

Maximum voltage of a Van de Graaff generator.

•  Molecules in air get ionized for electric fields greater than roughly Emax = 3 x 106 V/m. What is the maximum voltage of a charged sphere of radius R=0.2 m?

Voltage due to a charged ring •  For a uniformly charged ring of

total charge Q, integration gives the potential on the ring axis:

•  Very hard integral in general! If P is on x axis, then r is independent of θ.

V =�

k dq

rdq = λadθ

V (x, y, z) =� 2π

0

kλa dθ

r(θ, x, y, z)

•  Integrating the potentials of charged rings gives the potential of a uniformly charged disk:

•  This result reduces to the infinite-sheet potential close to the disk, and the point-charge potential far from the disk.

V (x) =�

kdQ

r=

� a

0k

λ2πr dr√x2 + r2

Equipotentials •  An equipotential is a surface on which the potential

(voltage) is constant. •  In two-dimensional drawings, we

represent equipotentials by curves similar to the contours of height on a map.

•  The electric field is always perpendicular to the equipotentials. (∆V = − �E · ∆�s = 0)

Conductors •  There’s no electric field inside a conductor in

electrostatic equilibrium. •  And even at the surface there’s no field

component parallel to the surface. •  Therefore it takes no work to move charge

inside or on the surface of a conductor in electrostatic equilibrium.

A conductor in electrostatic equilibrium is an equipotential. •  The electric field must be perpendicular to the

surface of a conductor (in electrostatic equilibrium

•  For a very small displacement, •  Suppose

Then

Can do the same thing in other direction:

The derivatives here are partial derivatives, expressing the variation with respect to one variable alone.

Determining E from V •  Voltage can be determined if electric field is known •  Can electric field be determined if voltage is known?

∆V = − �E · ∆�r

�E · ∆�r = Ex ∆x

Ex = −∆V

∆x= −∂V

∂x

�E = −∇V = −�

∂V

∂xi +

∂V

∂yj +

∂V

∂zk

�

(gradient of V)

∆�r = ∆x i

•  For which region is the magnitude of the electric field the highest?

80 V

Distance (cm)1 2 3 4 5 6 7 8 9 10

Dist

ance

(cm

)

4

65

7

8

9

3

2

1

A

B

200 V180 V

160 V

140 V120 V

D

C100 V

1.  A 2.  B 3.  C 4.  D

CT 29.13b

What is the approximate magnitude of the electric field at point A? (Each equipotential line is 2 m from the nearest-neighbor equipotential.)

A

0V -1.4V

-2.1V -1.8V

A) 0.1 Volts/m B) 0.2 Volts/m C) 1.6 Volts/m D) 0.7 Volts/m

E) None of these

Example: Electric field along axis of charged Ring:

•  Recall that the voltage due to a charged ring is:

Use this to determine E(x):