physics 2113 lecture: 14 fri 25 sep electric potential iii physics 2113 jonathan dowling

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Physics 2113 Physics 2113 Lecture: 14 FRI 25 SEP Lecture: 14 FRI 25 SEP Electric Potential III Electric Potential III Physics 2113 Jonathan Dowling

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Page 1: Physics 2113 Lecture: 14 FRI 25 SEP Electric Potential III Physics 2113 Jonathan Dowling

Physics 2113 Physics 2113 Lecture: 14 FRI 25 SEPLecture: 14 FRI 25 SEP

Electric Potential IIIElectric Potential III

Physics 2113

Jonathan Dowling

Page 2: Physics 2113 Lecture: 14 FRI 25 SEP Electric Potential III Physics 2113 Jonathan Dowling

Conservative Forces, Work, and Potential Energy

Work Done (W) is Integral of Force (F)

Potential Energy (U) is Negative of Work Done

Hence Force is Negative Derivative of Potential Energy

Page 3: Physics 2113 Lecture: 14 FRI 25 SEP Electric Potential III Physics 2113 Jonathan Dowling

Coulomb’s Law for Point Charge

P1P2

q1q2P1P2

q2

Force [N]= Newton

Potential Energy [J]=Joule

Potential Voltage [J/C]=[V]=Volt

ElectricField [N/C]=[V/m]

Page 4: Physics 2113 Lecture: 14 FRI 25 SEP Electric Potential III Physics 2113 Jonathan Dowling

Potential At Center of Ring of Potential At Center of Ring of ChargeCharge

• Divide the charge distribution into

differential elements

• Write down an expression for

potential voltalge from a typical

element — treat as point charge

• Integrate!

• Simple example: circular rod of

radius r, total charge Q; find V at

center.

dq

r

Same result holds along z-axis!

Page 5: Physics 2113 Lecture: 14 FRI 25 SEP Electric Potential III Physics 2113 Jonathan Dowling

Potential Along Axis of Ring of ChargePotential Along Axis of Ring of Charge• Divide the charge distribution into

differential elements

• Write down an expression for

potential voltalge from a typical

element — treat as point charge

• Integrate!dq

R’

dq

Page 6: Physics 2113 Lecture: 14 FRI 25 SEP Electric Potential III Physics 2113 Jonathan Dowling

PotentialPotential Voltage of Continuous Charge Voltage of Continuous Charge Distribution: Return of the Rod!Distribution: Return of the Rod!

• Uniformly charged rod• Total charge +Q• Length L• What is V at position P?

P

r = x

L a

dq=λdx

Units: [Nm2/C2][C/m]=[Nm/C]=[J/C]=[V] dq

Page 7: Physics 2113 Lecture: 14 FRI 25 SEP Electric Potential III Physics 2113 Jonathan Dowling

PotentialPotential Voltage of Continuous Charge Voltage of Continuous Charge Distribution: Return of the Rod!Distribution: Return of the Rod!

• Uniformly charged rod• Total charge +Q• Length L• What is V at position P?• What about a>>L?

P

r = x

L a

dq=λdx

In this limit we recover V=kq/r for point charge!

Page 8: Physics 2113 Lecture: 14 FRI 25 SEP Electric Potential III Physics 2113 Jonathan Dowling

PotentialPotential Voltage of Continuous Charge Voltage of Continuous Charge

• Uniformly charged rod• Total charge +Q• Length L• What is V at position P?

Page 9: Physics 2113 Lecture: 14 FRI 25 SEP Electric Potential III Physics 2113 Jonathan Dowling

PotentialPotential Voltage of Continuous Charge Voltage of Continuous Charge

• What about d>>L?

We recover formulaV=kq/r for a point chargeAgain!

Page 10: Physics 2113 Lecture: 14 FRI 25 SEP Electric Potential III Physics 2113 Jonathan Dowling

Electric Field & Potential: Electric Field & Potential: A Simple Relationship! A Simple Relationship!

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!

Focus only on a simple case: electric field that points along +x axis but whose magnitude varies with x.

Note: • MINUS sign!• Units for E: VOLTS/METER (V/m)

Note: • MINUS sign!• Units for E: VOLTS/METER (V/m)

Page 11: Physics 2113 Lecture: 14 FRI 25 SEP Electric Potential III Physics 2113 Jonathan Dowling

E from V: Example E from V: Example

• Uniformly charged rod

• Total charge +Q• Length L• We Found V at P!• Find E from V?

Px

L a

dq

Units: √ Electric Field!

Page 12: Physics 2113 Lecture: 14 FRI 25 SEP Electric Potential III Physics 2113 Jonathan Dowling

Electric Field & Potential: ICPPElectric Field & Potential: ICPP

• Hollow metal spherical shell of radius R has a charge on the shell +q

• Which of the following is magnitude of the electric potential voltage V as a function of distance r from center of sphere?

V

rr=R

(a) V

rr=R

(b)

R

Hint: Inside sphere there are no charges so E=0.But E=dV/dr=0. What can V be?

Page 13: Physics 2113 Lecture: 14 FRI 25 SEP Electric Potential III Physics 2113 Jonathan Dowling

Δx

(a) Since Δx is the same, only |ΔV| matters!

|ΔV1| =200, |ΔV2| =220, |ΔV3| =200

|E2| > |E3| = |E1|

The bigger the voltage drop the stronger the field.

(b) = 3

(c) F = qE = ma

accelerate leftward

Page 14: Physics 2113 Lecture: 14 FRI 25 SEP Electric Potential III Physics 2113 Jonathan Dowling

Equipotentials and ConductorsEquipotentials and Conductors• Conducting surfaces are

EQUIPOTENTIALs

• At surface of conductor, E is normal (perpendicular) 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.

EV

Page 15: Physics 2113 Lecture: 14 FRI 25 SEP Electric Potential III Physics 2113 Jonathan Dowling

Conductors Change the Field Conductors Change the Field Around Them!Around Them!

An Uncharged Conductor:

A Uniform Electric Field:

An Uncharged Conductor in the Initially Uniform Electric Field:

Page 16: Physics 2113 Lecture: 14 FRI 25 SEP Electric Potential III Physics 2113 Jonathan Dowling

Sharp ConductorsSharp Conductors• Charge density is higher at

conductor surfaces that have small radius of curvature

• E = σε 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)

Page 17: Physics 2113 Lecture: 14 FRI 25 SEP Electric Potential III Physics 2113 Jonathan Dowling

Sharp Conductors: Lightning RodsSharp Conductors: Lightning Rods

Page 18: Physics 2113 Lecture: 14 FRI 25 SEP Electric Potential III Physics 2113 Jonathan Dowling

Ben Franklin Invents the Lightning Rod!

Page 19: Physics 2113 Lecture: 14 FRI 25 SEP Electric Potential III Physics 2113 Jonathan Dowling

LIGHTNING SAFE CROUCHIf caught out of doors during an approaching storm and your skin tingles or hair tries to stand on end, immediately do the "LIGHTNING SAFE CROUCH ”. Squat low to the ground on the balls of your feet, with your feet close together. Place your hands on your knees, with your head between them. Be the smallest target possible, and minimize your contact with the ground.

Page 20: Physics 2113 Lecture: 14 FRI 25 SEP Electric Potential III Physics 2113 Jonathan Dowling

WE’LL BOTH BE DEAD IN

SECONDS!

WE’LL BOTH BE DEAD IN

SECONDS!

QUICK — TAKE A SELFIE!

QUICK — TAKE A SELFIE!

Page 21: Physics 2113 Lecture: 14 FRI 25 SEP Electric Potential III Physics 2113 Jonathan Dowling

Summary:Summary:• Electric potential: work needed to bring +1C from infinity; units = V = Volt• 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 : dV = kdq/r

• Electric field and electric potential: E= -dV/dx• Electric potential energy: work used to build the system, charge by

charge. Use W = U = 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.

Page 22: Physics 2113 Lecture: 14 FRI 25 SEP Electric Potential III Physics 2113 Jonathan Dowling
Page 23: Physics 2113 Lecture: 14 FRI 25 SEP Electric Potential III Physics 2113 Jonathan Dowling