l2 electric field, dipoles
TRANSCRIPT
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: Coulombโs 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 (โq).
The net electric force that Charges #2 and #3 exert on Charge #1 is in
A. the +x-direction. B. the โx-direction.
C. the +y-direction. D. the โy-direction.
E. none of the above
#1
#2
#3
+q
+q
โq
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 (โq).
The net electric force that Charges #2 and #3 exert on Charge #1 is in
A. the +x direction. B. the โx direction.
C. the +y direction. D. the โy direction.
E. none of the above
#1
#2
#3
+q
+q
โq x
y
๐น =1
4๐๐0๐1๐2
๐2 โ ๐1 2
Iclicker Question
๏ฟฝโ๏ฟฝ1โ2 ๏ฟฝโ๏ฟฝ2โ1
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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 ๐ถ๐, the angle between the thread supporting #2 and the vertical is ๐ถ๐. Choose the statement with which you agree: A. ฮฑ1 > ฮฑ2
B. ฮฑ1 < ฮฑ2
C. ฮฑ1 = ฮฑ2
D. You need to know the mass to answer the question.
#1 #2 ๐ผ1 ๐ผ2
๐น =1
4๐๐0๐1๐2
๐2 โ ๐1 2
Iclicker Question
๏ฟฝโ๏ฟฝ1โ2 ๏ฟฝโ๏ฟฝ2โ1
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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 ๐ผ1, the angle between the thread supporting #2 and the vertical is ๐ผ2. Choose the statement with which you agree: A. ฮฑ1 > ฮฑ2
B. ฮฑ1 < ฮฑ2
C. ฮฑ1 = ฮฑ2
D. You need to know the mass to answer the question.
#1 #2 ๐ผ1 ๐ผ2
X-components: Newtonโs 3d Law ๐น1โ2 = ๐น2โ1
Y-components: ๐1๐ = ๐2๐ = ๐๐
Concept of the Electromagnetic Field
Reasons to introduce electric (and magnetic) fields:
Coulombโs Law suggests an instantaneous interaction: no matter how far apart the charges are, they instantly โknowโ if the location of another charge has been changed (โaction-at-a-distanceโ). However, no information can travel faster than light. General cure for the โaction-at-a-distanceโ problem: charges generate โfieldsโ, 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 Coulombโs Law. However, we will need to modify Coulombโs 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 Sunโs 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 โaction-at-a-distanceโ between the charges, we can consider the interaction of a charge with the field created by all other charges at its location.
๐ธ ๐ = ๏ฟฝ๐ธ๐๐
๐
- the field at the location ๐ 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:
๏ฟฝโ๏ฟฝ๐๐๐ = ๏ฟฝ๏ฟฝโ๏ฟฝ๐๐
- the force on a charge at the location ๐ due to all other charges
๐น2โ1 =1
4๐๐0๐1๐2
๐2 โ ๐1 2 = ๐11
4๐๐0๐2
๐2 โ ๐1 2 = ๐1๐ธ2 ๐1
the ๐ธ field due
to ๐2 at the location of ๐1
Electric Field of a Point Charge
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๏ฟฝโ๏ฟฝ๐โ๐ ๐ =1
4๐๐0๐๐๐ 2 ๏ฟฝฬ๏ฟฝ
Consider two charges: +Q (at the origin) and +q (at ๐).
The force exerted by Q on q:
The electric field due to Q at the location ๐: ๐ธ๐ ๐ โก๏ฟฝโ๏ฟฝ๐โ๐ ๐
๐ =1
4๐๐0๐๐ 2 ๏ฟฝฬ๏ฟฝ
The E field due to Q at (โ ) ๐ is a vector that points along the vector of the force on a positive charge q placed at (โ ) ๐ (๐ธ is directed along ๏ฟฝฬ๏ฟฝ for +Q, along โ๏ฟฝฬ๏ฟฝ for โQ).
Units of the electric field: ๐ต๐ช
=โโโโโ
A few numbers:
โข At 1 m from a charge of 1C, the field would be 9ร109 N/C (ordinary materials break down in such a strong field).
โข The strength of the electric field near the Earthโs surface is 100-300 N/C (V/m).
๐ +๐
+๐ ๏ฟฝโ๏ฟฝ๐โ๐ ๐
๏ฟฝโ๏ฟฝ๐โ๐ ๐ = ๐๐ธ๐ ๐
(later weโll 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 (โsourceโ) to negative charges (โsinkโ), though there is no real displacement of matter in space.
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For a point charge, ๐ธ โ 1๐2
. Thus, the density of lines
โ 1๐2
. The area of a sphere centered at the charge โ ๐2. Thus, the total number of lines is fixed: they donโt โvanish into thin airโ, 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 โ 1/r2.
In 2D, the density of lines โ 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 donโt form sharp bends (there is only one tangent line to a field โcurveโ 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 (โ ) 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 girlโs head.
4. Girlโs 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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๐ = 2 ร ๐ธ๐๐2๐ ๐ ๐ ๐ = ๐ธ๐๐๐ ๐ ๐ ๐ = ๐ธ๐ธ๐ ๐ ๐ ๐
The net force on a dipole is zero; however, there is a non-zero torque:
๐ = ๏ฟฝโ๏ฟฝ ร ๐ธ
Potential energy of a dipole in an electric field: ๐ ๐ = โ๐ธ๐ธ๐๐๐ ๐
๐ธ โ the dipole moment
In the vector form: (๏ฟฝโ๏ฟฝ directed from โ 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 โ Hydrogen Bonding
As a result, itโs 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.
๐ธ โ 6 ร 10โ30 Cโ m
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๏ฟฝโ๏ฟฝ
๐โ =๐ธ๐
=6 ร 10โ30๐ถ๐0.6 ร 10โ10๐
= 1 ร 10โ19๐ถ โ 0.6๐
๐ = 0.96A ร ๐๐๐ 52.20 = 0.6A = 0.6 ร 10โ10๐
- 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, Gaussโ Law. ยงยง 22.1 โ 22.4.
Electric Dipoles