The document discusses electric fields and electric dipoles. It defines the electric field as a vector field generated by electric charges that acts upon other charges. Electric field lines are introduced to visualize electric fields, with higher density of lines indicating stronger fields. Dipoles, such as water molecules, have a built-in electric polarity due to unequal charge distribution. When placed in an external electric field, dipoles experience a torque attempting to align them with the field but do not experience a net force. Microwave ovens work by using an oscillating electric field to cause the rotation of polar water molecules in food, generating heat through molecular collisions.

1. 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

2. 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.

3. 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 𝑟⃗ = � 𝐸𝑖
𝑖
𝑟⃗
- the field at the
location 𝑟⃗ 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:
𝐹⃗ 𝑛𝑛𝑛 = � 𝐹⃗𝑖
𝑖
- the force on a charge
at the location 𝑟⃗ due
to all other charges
𝐹2→1 =
1
4𝜋𝜖0
𝑞1 𝑞2
𝑟⃗2 − 𝑟⃗1
2
= 𝑞1
1
4𝜋𝜖0
𝑞2
𝑟⃗2 − 𝑟⃗1
2
= 𝑞1 𝐸2 𝑟⃗1
the 𝐸 field due
to 𝑞2 at the
location of 𝑞1

4. Electric Field of a Point Charge
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𝐹⃗ 𝑄→𝑞 𝑟⃗ =
1
4𝜋𝜖0
𝑄𝑄
𝑟⃗ 2 𝑟̂
Consider two charges: +Q (at the origin) and +q (at 𝑟⃗).
The force exerted by Q on q:
The electric field due to Q at the location 𝑟⃗: 𝐸 𝑄 𝑟⃗ ≡
𝐹⃗ 𝑄→𝑞 𝑟⃗
𝑞
=
1
4𝜋𝜖0
𝑄
𝑟⃗ 2 𝑟̂
The E field due to Q at (⋅) 𝑟⃗ is a vector that points along the vector of the force on a
positive charge q placed at (⋅) 𝑟⃗ (𝐸 is directed along 𝑟̂ for +Q, along −𝑟̂ for –Q).
Units of the electric field:
𝑵
𝑪
=−−−− −
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).
𝑟⃗
+𝑄
+𝑞
𝐹⃗ 𝑄→𝑞 𝑟⃗
𝐹⃗ 𝑄→𝑞 𝑟⃗ = 𝑞𝐸 𝑄 𝑟⃗
(later we’ll be
also using an
equivalent unit,
Volt/meter)

5. 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 ∝
1
𝑟2. Thus, the density of lines
∝
1
𝑟2. 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.

6. 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

7. 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.

8. 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

9. 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

10. Dipoles in a Uniform External Electric Field
+q -q
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𝜏 = 2 × 𝐸𝐸
𝑑
2
𝑠𝑠𝑠 𝜙 = 𝐸𝐸𝑑𝑠𝑠𝑠 𝜙 = 𝐸𝐸𝐸𝐸𝐸 𝜙
The net force on a dipole is zero; however, there is
a non-zero torque:
𝜏⃗ = 𝑝⃗ × 𝐸
Potential energy of a dipole in an electric field: 𝑈 𝜙 = −𝑝𝑝𝑝𝑝𝑝 𝜙
𝑝 – the dipole moment
In the vector form: (𝑝⃗ directed from – to +)

11. 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.
𝑝 ≅ 6 × 10−30 C⋅m
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𝑝⃗
𝑞∗ =
𝑝
𝑑
=
6 × 10−30
𝐶𝐶
0.6 × 10−10 𝑚
= 1 × 10−19 𝐶 ≈ 0.6𝑒
𝑑 = 0.96A × 𝑐𝑐𝑐 52.20 = 0.6A = 0.6 × 10−10 𝑚
- the effective
charge on O

12. 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).

13. 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