Physics 272: Electricity and Magnetism

Mark Palenik

Thursday June 14th

Topics

• Finish up dipoles

• Polarization of materials

• Conductors

• Charging/discharging

iclicker (not credit)

• An atom is placed in a UNIFORM electric field pointing to the right and polarizes. What is the net force on the atom?

a) To the left

b) To the right

c) No net force

𝐸

Force on a dipole (clicker, no credit)

• Net force on a dipole = Force on positive charge + Force on negative charge

• 𝐹 𝑛𝑒𝑡 = 𝐹 𝑝𝑜𝑠+𝐹 𝑛𝑒𝑔 = 𝑞𝐸𝑝𝑜𝑠 − q𝐸𝑛𝑒𝑔 = 𝑞 𝐸𝑝𝑜𝑠 − 𝐸𝑛𝑒𝑔

• Assume we have a permanent dipole (not induced by an external electric field) In which situation could this force be non-zero?

a) It is placed in a uniform field

b) A charge is placed along the perpendicular axis

c) A charge is placed along the parallel axis

d) Either b) or c)

𝐸

+

+

Force on a point dipole

• We said𝐹 𝑛𝑒𝑡 = 𝑞 𝐸𝑝𝑜𝑠 − 𝐸𝑛𝑒𝑔

• For a point dipole, P (dipole moment) is important, q is meaningless

• Charges are distance s apart

• 𝐹 𝑛𝑒𝑡 = lim𝑠→0

𝑃

𝑠𝐸𝑝𝑜𝑠 − 𝐸𝑛𝑒𝑔 = lim

𝑠→0

𝑃

𝑠𝐸(x + 𝑠) − 𝐸(𝑥) =

𝑃 lim𝑠→0

𝐸(x+s)−𝐸(𝑥)

𝑠

What does this look like? (think calculus)

It’s a derivative! 𝑑𝑦

𝑑𝑥=

𝑦 𝑥+ℎ −𝑦(𝑥)

ℎ

- +

x

s

x+s

Dipoles respond to E field derivatives

• Force on a dipole = Dipole moment * Derivitive of E field along dipole axis.

• E.g. if dipole is along x axis

• 𝐹 𝑛𝑒𝑡 = 𝑃𝑑𝐸

𝑑𝑥

• More formally we could write

• 𝐹 𝑛𝑒𝑡 = 𝑃 ∙ 𝛻𝐸

Tensor

Polarization of materials

• Recall: Electric field lines push on positive charges and pull negative charges

• Neutral atoms will polarize under the influence of electric fields

• Materials are made of atoms, so materials will polarize

– We call this INDUCED polarization

• Conductors and insulators behave differently

Electric fields and materials

• Plastic is an insulator: charges are not free to flow

• Metals are conductors

𝐸

𝐸

𝐸

Plastic sphere: each atom becomes polarized, since the electrons cannot flow

Metal sphere, electrons can flow. Sphere polarizes. Protons DO NOT move

E=0

Polarization of a plastic sphere

• Draw the E field lines from the positive charge and the polarization of this plastic sphere.

– Remember two things: 1) electrons are pushed opposite to the direction of field lines

– 2) The field lines are NOT horizontal

+ Plastic Sphere

Non-external electric field

• The external electric field acting on a plastic sphere is drawn below.

• Draw the electric field produced by the induced dipoles below

𝐸

The field of the induced dipoles inside the sphere opposes the applied field. As we will see, conductors are a special case.

Note the field outside the sphere does not oppose the applied E

Net field

• Superposition still applies

• One part of the field comes from the external, applied field

• Another comes from the polarization field

• 𝐸𝑛𝑒𝑡 = 𝐸𝑒𝑥𝑡 + 𝐸𝑝𝑜𝑙

• 𝐸𝑝𝑜𝑙 is induced by the external field. This

happens in both conductors and insulators

Clicker Question

+

x

Plastic Block

-

A

B D

C

What is the direction of the electric field at X due to the dipole?

Clicker Question

+

x

Plastic Block

-

A

B D

C

What is the direction of the electric field at X due to the plastic

box?

Clicker Question

+

x

Plastic Block

-

A

B D

C

What is the direction of the NET electric field at X?

Electric fields and metals

• Electrons are free to move inside of metals (nuclei are still fixed, i.e. the atoms don’t move)

• What happens if I apply an external field to a piece of metal?

• Will the electrons move? Will they ever stop moving? If so, when?

Electric field in a conductor

• At equilibrium, E inside any conductor is zero

• Equilibrium is defined as when charges are static

• If there is an E field, the charges will move

• Therefore, in equilibrium, E=0 throughout the conductor

• Charges arrange themselves so that they cancel the external E field

𝐸 Enet=0

Applied field points to the right. The external applied field and field produced by the sphere cancel each other inside.

The electric field produced by the sphere with this arrangement of charges points left

Reasons charges stop a) E pushes charges toward

equilibrium b) “Friction” like forces slow

electrons

iClicker question: non-spherical conductor

• Which best describes the NET electric field inside of this triangular conductor with a positive charge nearby

a) It points to the right, because of the positive charge

b) Non-zero because the conductor is not spherical, but not necessarily to the right

c) Zero, because the triangle is a conductor

d) It points toward the positive charge

+

Metal triangle

Charged conductor

• Where will excess charges go if I put them on a conductor?

– Keep in mind how charges affect each other.

• When is the sphere in equilibrium?

++ +++ ++

In inject several positive charges into the center of a metal sphere, so it is no longer neutral. Where do they go?

Charged conductor and external charge

• Qualitatively describe the net field of a charged, conducting sphere and a point charge

• Recall the field outside a uniformly charged sphere is the same as that of a point charge?

• Is the total field the same as the sum of the field of the individual objects?

– In what circumstances might this or might this not be true?

- Conducting sphere Point charge

Discharging by contact

On approach: body polarizes

On contact: charge redistributes over

larger surface

Charging by induction

What is a conductor?

• Charges in conductors can be mobile.

• Either electrons or ions can flow

• In metals, there is a well ordered crystal structure, many close energy levels: a sea of electrons that can flow

Gray is the electron sea. The red +’s are the positive cores

What is a conductor 2

• In ionic solutions, molecules have split into + and – ions, which can flow in solution. E.g. NaCl becomes Na+ and Cl-