Lecture 2. Electric Field, Dipoles

Outline:

Electric Field Electric Field Lines Electric Dipoles in External Electric Fields Dipoles in Nature

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Lecture 1 review:

Electrostatics: Coulomb’s Law + Superposition Principle Electric Charges

Iclicker Question

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Three point charges lie at the vertices of an equilateral triangle as shown. All three charges have the same magnitude, but Charges #1 and #2 are positive (+q) and Charge #3 is negative (–q).

The net electric force that Charges #2 and #3 exert on Charge #1 is in

A. the +x-direction. B. the –x-direction.

C. the +y-direction. D. the –y-direction.

E. none of the above

#1

#2

#3

+q

+q

–q

x

y

Iclicker Question

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Three point charges lie at the vertices of an equilateral triangle as shown. All three charges have the same magnitude, but Charges #1 and #2 are positive (+q) and Charge #3 is negative (–q).

The net electric force that Charges #2 and #3 exert on Charge #1 is in

A. the +x direction. B. the –x direction.

C. the +y direction. D. the –y direction.

E. none of the above

#1

#2

#3

+q

+q

–q x

y

𝐹 =1

4𝜋𝜖0𝑞1𝑞2

𝑟2 − 𝑟1 2

Iclicker Question

�⃗�1→2 �⃗�2→1

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Consider two point-like charges of the same mass: #1 has charge +q, and #2 has charge Q = +10q. You hang them from threads near each other. The angle between the thread supporting #1 and the vertical is 𝜶𝟏, the angle between the thread supporting #2 and the vertical is 𝜶𝟐. Choose the statement with which you agree: A. α1 > α2

B. α1 < α2

C. α1 = α2

D. You need to know the mass to answer the question.

#1 #2 𝛼1 𝛼2

𝐹 =1

4𝜋𝜖0𝑞1𝑞2

𝑟2 − 𝑟1 2

Iclicker Question

�⃗�1→2 �⃗�2→1

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Consider two point-like charges of the same mass: #1 has charge +q, and #2 has charge Q = +10q. You hang them from threads near each other. The angle between the thread supporting #1 and the vertical is 𝛼1, the angle between the thread supporting #2 and the vertical is 𝛼2. Choose the statement with which you agree: A. α1 > α2

B. α1 < α2

C. α1 = α2

D. You need to know the mass to answer the question.

#1 #2 𝛼1 𝛼2

X-components: Newton’s 3d Law 𝐹1→2 = 𝐹2→1

Y-components: 𝑚1𝑔 = 𝑚2𝑔 = 𝑚𝑔

Concept of the Electromagnetic Field

Reasons to introduce electric (and magnetic) fields:

Coulomb’s Law suggests an instantaneous interaction: no matter how far apart the charges are, they instantly “know” if the location of another charge has been changed (“action-at-a-distance”). However, no information can travel faster than light. General cure for the “action-at-a-distance” problem: charges generate “fields”, and these fields act upon other charges. The field perturbation propagates in vacuum with the speed of light. Electrostatics (charges at rest): the field description of electrostatic interactions is equivalent to the description based on Coulomb’s Law. However, we will need to modify Coulomb’s Law in electrodynamics.

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The electromagnetic field has considerable objective reality, and in particular it possesses energy and momentum. It is by means of electric and magnetic fields (radiation heat exchange) that the Sun’s energy reaches us.

Electric Field in Electrostatics

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Electric field in electrostatics: electrostatic interaction between charges can be described by the model of the vector electrostatic field. Instead of “action-at-a-distance” between the charges, we can consider the interaction of a charge with the field created by all other charges at its location.

𝐸 𝑟 = �𝐸𝑖𝑖

𝑟

- the field at the location 𝑟 due to all

other charges

The superposition principle implies that the electric fields created by different charges do not interact with each other, the net field is just the vector superposition of the fields due to individual point charges:

�⃗�𝑛𝑛𝑛 = ��⃗�𝑖𝑖

- the force on a charge at the location 𝑟 due to all other charges

𝐹2→1 =1

4𝜋𝜖0𝑞1𝑞2

𝑟2 − 𝑟1 2 = 𝑞11

4𝜋𝜖0𝑞2

𝑟2 − 𝑟1 2 = 𝑞1𝐸2 𝑟1

the 𝐸 field due

to 𝑞2 at the location of 𝑞1

Electric Field of a Point Charge

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�⃗�𝑄→𝑞 𝑟 =1

4𝜋𝜖0𝑄𝑞𝑟 2 �̂�

Consider two charges: +Q (at the origin) and +q (at 𝑟).

The force exerted by Q on q:

The electric field due to Q at the location 𝑟: 𝐸𝑄 𝑟 ≡�⃗�𝑄→𝑞 𝑟

𝑞 =1

4𝜋𝜖0𝑄𝑟 2 �̂�

The E field due to Q at (⋅) 𝑟 is a vector that points along the vector of the force on a positive charge q placed at (⋅) 𝑟 (𝐸 is directed along �̂� for +Q, along −�̂� for –Q).

Units of the electric field: 𝑵𝑪

=−−−−−

A few numbers:

• At 1 m from a charge of 1C, the field would be 9×109 N/C (ordinary materials break down in such a strong field).

• The strength of the electric field near the Earth’s surface is 100-300 N/C (V/m).

𝑟 +𝑄

+𝑞 �⃗�𝑄→𝑞 𝑟

�⃗�𝑄→𝑞 𝑟 = 𝑞𝐸𝑄 𝑟

(later we’ll be also using an

equivalent unit, Volt/meter)

Visualization of the Field: Electric Field Lines (Curves)

To map the electrostatic field, we introduce the concept of the electric field lines:

- direction of the field vector is tangential to the field line (curve);

- intensity of the field at a given point is proportional to the local density of field lines.

This picture resembles a laminar flow of some incompressible fluid from positive charges (“source”) to negative charges (“sink”), though there is no real displacement of matter in space.

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For a point charge, 𝐸 ∝ 1𝑟2

. Thus, the density of lines

∝ 1𝑟2

. The area of a sphere centered at the charge ∝ 𝑟2. Thus, the total number of lines is fixed: they don’t “vanish into thin air”, must be terminated either at another (negative) charge or continue to infinity.

Iclicker Question

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Does this 2D plot do justice to a 3D electrostatic field?

Hint: the density of the field lines should be proportional to the intensity of the field .

A. Yes

B. No

C. Depends on the field strength and direction.

Iclicker Question

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Does this 2D plot do justice to a 3D electrostatic field?

Hint: the density of the field lines should be proportional to the intensity of the field .

A. Yes

B. No

C. Depends on the field strength and direction.

In 3D, the density of lines ∝ 1/r2.

In 2D, the density of lines ∝ 1/r.

Thus, this is just a cartoon, no true scientific value .

How to Draw the Electric Field Lines Convention: - the electric field lines originate on positive charges;

- terminate on negative charges.

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Field lines don’t form sharp bends (there is only one tangent line to a field “curve” at each point).

+q +q +2q -q +q -q

Iclicker Question Which field line configuration correctly represents an electrostatic field?

A

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B C

D E

Iclicker Question Which field line configuration correctly represents an electrostatic field?

A

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B C

D E

two tangent lines at (⋅) O

O

the line forms a closed loop

(OK in electrodynamics, forbidden in electrostatics)

lines form sharp corners two tangent lines at bends

straight lines belong to a solitary point charge, but the symmetry is broken

must be a charge at the intersection

Experiments on Field Visualization

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1. Charge separation by friction.

2. The girl acquires a charge (unevenly) distributed across her surface.

3. Like charges on individual hairs repel each other and force the hairs to stand away from each other and the girl’s head.

4. Girl’s hairs (roughly) follow the field lines.

Demonstration: Van de Graaff Generator

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Robert J. Van de Graaff 1901-1967

1) hollow metal sphere 2) upper collecting electrode 3) upper roller (for example an acrylic glass) 4) side of the belt with positive charges 5) opposite side of the belt with negative charges 6) lower roller (metal) 7) lower electrode (ground) 8) spherical device with negative charges, used to discharge the main sphere 9) spark produced by the difference of potentials

Electric Field of a Dipole

+q -q Dipoles: the second most important (after a

point charge) configuration of charges.

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2D plot of the field lines in the x-y plane 3D plot of the field intensity in the x-y plane

Dipoles in a Uniform External Electric Field

+q -q

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𝜏 = 2 × 𝐸𝑞𝑑2𝑠𝑠𝑠 𝜙 = 𝐸𝑞𝑑𝑠𝑠𝑠 𝜙 = 𝐸𝐸𝑠𝑠𝑠 𝜙

The net force on a dipole is zero; however, there is a non-zero torque:

𝜏 = �⃗� × 𝐸

Potential energy of a dipole in an electric field: 𝑈 𝜙 = −𝐸𝐸𝑝𝑝𝑠 𝜙

𝐸 – the dipole moment

In the vector form: (�⃗� directed from – to +)

Polar Water Molecules Polar = built-in dipole moment

Life on Earth very much depends on a large dipole moment of water molecules!

Large dipole moment → Hydrogen Bonding

As a result, it’s the most unusual liquid: it is much denser than expected and as a solid it is much lighter than expected when compared with its liquid form.

Anomalies: high freezing and melting point (due to this our planet is bathed in liquid water), large heat capacity, high thermal conductivity (high water content in organisms contribute to thermal regulation and prevent local temperature fluctuations), high latent heat of evaporation (resistance to dehydration and considerable evaporative cooling), excellent solvent due to its polarity, high dielectric constant, etc., etc.

http://www.lsbu.ac.uk/water/anmlies.html

H+

O2-

H+

3-dimensional bonding network: water looks like a "gel" consisting of a single, huge hydrogen-bonded cluster.

𝐸 ≅ 6 × 10−30 C⋅m

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�⃗�

𝑞∗ =𝐸𝑑

=6 × 10−30𝐶𝑚0.6 × 10−10𝑚

= 1 × 10−19𝐶 ≈ 0.6𝑒

𝑑 = 0.96A × 𝑝𝑝𝑠 52.20 = 0.6A = 0.6 × 10−10𝑚

- the effective charge on O

Microwave Ovens

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RF heating (or high-frequency heating) is the process in which a high-frequency alternating electric field (i.e. microwave electromagnetic radiation) heats a dielectric material. At higher frequencies, this heating is caused by molecular dipole rotation within the dielectric.

Water molecules feel the torque and align themselves in an electric field. As the field alternates, the molecules reverse direction (dipole rotation). Rotating molecules push, pull, and collide with other molecules (through electrical forces), converting the energy of the electric field into the thermal energy (heat).

Conclusion

Electric Field: math. tool and phys. reality

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Next time: Lecture 3. Electric Field Flux, Gauss’ Law. §§ 22.1 – 22.4.

Electric Dipoles