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Physics 2102 Jonathan Dowling. Physics 2102 Lecture 6. Electric Potential II. –Q. +Q. U B. Example. Positive and negative charges of equal magnitude Q are held in a circle of radius R. 1. What is the electric potential at the center of each circle? V A = V B = V C =

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physics 2102 lecture 6

Physics 2102

Jonathan Dowling

Physics 2102 Lecture 6

Electric Potential II

example

–Q

+Q

UB

Example

Positive and negative charges of equal magnitude Q are held in a circle of radius R.

1. What is the electric potential at the center of each circle?

  • VA =
  • VB =
  • VC =

2. Draw an arrow representing the approximate direction of the electric field at the center of each circle.

3. Which system has the highest electricpotential energy?

A

B

C

electric potential of a dipole on axis

a

+Q

-Q

r

Electric Potential of a Dipole (on axis)

What is V at a point at an axial distance r away from the midpoint of a dipole (on side of positive charge)?

Far away, when r >> a:

V

electric potential on perpendicular bisector of dipole

d

U= QoV=Qo(–Q/d+Q/d)=0 

Electric Potential on Perpendicular Bisector of Dipole

You bring a charge of Qo = –3C from infinity to a point P on the perpendicular bisector of a dipole as shown. Is the work that you do:

  • Positive?
  • Negative?
  • Zero?

a

+Q

-Q

P

-3C

continuous charge distributions
Continuous Charge Distributions
  • Divide the charge distribution into differential elements
  • Write down an expression for potential from a typical element — treat as point charge
  • Integrate!
  • Simple example: circular rod of radius r, total charge Q; find V at center.

r

dq

potential of continuous charge distribution example

x

dx

a

L

Potential of Continuous Charge Distribution: Example
  • Uniformly charged rod
  • Total charge Q
  • Length L
  • What is V at position P shown?

P

electric field potential a simple relationship
Electric Field & Potential: A Simple Relationship!

Focus only on a simple case:

electric field that points along +x axis but whose magnitude varies with x.

Notice the following:

  • Point charge:
    • E = kQ/r2
    • V = kQ/r
  • Dipole (far away):
    • E ~ kp/r3
    • V ~ kp/r2
  • E is given by a DERIVATIVE of V!
  • Of course!
  • Note:
  • MINUS sign!
  • Units for E -- VOLTS/METER (V/m)
slide8

Electric Field & Potential: Example

(b)

V

V

(a)

r

r

r=R

r=R

V

(c)

r

r=R

+q

  • Hollow metal sphere of radius R has a charge +q
  • Which of the following is the electric potential V as a function of distance r from center of sphere?
slide9

Electric Field & Potential: Example

V

E

  • Outside the sphere:
  • Replace by point charge!
  • Inside the sphere:
  • E =0 (Gauss’ Law)
  • E = –dV/dr = 0 IFF V=constant

+q

equipotentials and conductors
Equipotentials and Conductors
  • Conducting surfaces are EQUIPOTENTIALs
  • At surface of conductor, E is normal to surface
  • Hence, no work needed to move a charge from one point on a conductor surface to another
  • Equipotentials are normal to E, so they follow the shape of the conductor near the surface.
conductors change the field around them

An uncharged conductor:

A uniform electric field:

An uncharged conductor in the initially uniform electric field:

Conductors change the field around them!
sharp conductors
“Sharp”conductors
  • Charge density is higher at conductor surfaces that have small radius of curvature
  • E = s/e0 for a conductor, hence STRONGER electric fields at sharply curved surfaces!
  • Used for attracting or getting rid of charge:
    • lightning rods
    • Van de Graaf -- metal brush transfers charge from rubber belt
    • Mars pathfinder mission -- tungsten points used to get rid of accumulated charge on rover (electric breakdown on Mars occurs at ~100 V/m)

(NASA)

summary
Summary:
  • Electric potential: work needed to bring +1C from infinity; units = V
  • Electric potential uniquely defined for every point in space -- independent of path!
  • Electric potential is a scalar -- add contributions from individual point charges
  • We calculated the electric potential produced by a single charge: V=kq/r, and by continuous charge distributions : V= kdq/r
  • Electric field and electric potential: E= -dV/dx
  • Electric potential energy: work used to build the system, charge by charge. Use W=qV for each charge.
  • Conductors: the charges move to make their surface equipotentials.
  • Charge density and electric field are higher on sharp points of conductors.
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