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+Q

UB

ExamplePositive 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

+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

U= QoV=Qo(–Q/d+Q/d)=0

Electric Potential on Perpendicular Bisector of DipoleYou 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

- 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

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!

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)

Electric Field & Potential: Example Simple Relationship!

(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?

Electric Field & Potential: Example Simple Relationship!

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 Simple Relationship!

- 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.

An uncharged conductor: Simple Relationship!

A uniform electric field:

An uncharged conductor in the initially uniform electric field:

Conductors change the field around them!“Sharp”conductors Simple Relationship!

- 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: Simple Relationship!

- 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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