General Physics PHY 2140

1
General Physics (PHY 2140)
Lecture 4

• Electrostatics
• Electric flux and Gausss law
• Electrical energy
• potential difference and electric potential
• potential energy of charged conductors

http//www.physics.wayne.edu/apetrov/PHY2140/
Chapters 15-16
2
Lightning Review

• Last lecture
• Properties of the electric field, field lines
• Conductors in electrostatic equilibrium
• Electric field is zero everywhere within the
  conductor.
• Any excess charge field on an isolated conductor
  resides on its surface.
• The electric field just outside a charged
  conductor is perpendicular to the conductors
  surface.
• On an irregular shaped conductor, the charge
  tends to accumulate at locations where the radius
  of curvature of the surface is smallest.
• Review Problem Would life be different if the
  electron were positively charged and the proton
  were negatively charged? Does the choice of signs
  have any bearing on physical and chemical
  interactions?

3
15.10 Electric Flux and Gausss Law

• A convenient technique was introduced by Karl F.
  Gauss (1777-1855) to calculate electric fields.
• Requires symmetric charge distributions.
• Technique based on the notion of electrical flux.

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15.10 Electric Flux

• To introduce the notion of flux, consider a
  situation where the electric field is uniform in
  magnitude and direction.
• Consider also that the field lines cross a
  surface of area A which is perpendicular to the
  field.
• The number of field lines per unit of area is
  constant.
• The flux, F, is defined as the product of the
  field magnitude by the area crossed by the field
  lines.

AreaA
E
5
15.10 Electric Flux

• Units Nm2/C in SI units.
• Find the electric flux through the area A 2 m2,
  which is perpendicular to an electric field E22
  N/C

Answer F 44 Nm2/C.
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15.10 Electric Flux

• If the surface is not perpendicular to the field,
  the expression of the field becomes
• Where q is the angle between the field and a
  normal to the surface.

N
q
q
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15.10 Electric Flux

• Remark
• When an area is constructed such that a closed
  surface is formed, we shall adopt the convention
  that the flux lines passing into the interior of
  the volume are negative and those passing out of
  the interior of the volume are positive.

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Example

• Question
• Calculate the flux of a constant E field (along
  x) through a cube of side L.

y
1
2
E
x
z
9

• Question
• Calculate the flux of a constant E field (along
  x) through a cube of side L.
• Reasoning
• Dealing with a composite, closed surface.
• Sum of the fluxes through all surfaces.
• Flux of field going in is negative
• Flux of field going out is positive.
• E is parallel to all surfaces except surfaces
  labeled 1 and 2.
• So only those surface contribute to the flux.
• Solution

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15.10 Gausss Law

• The net flux passing through a closed surface
  surrounding a charge Q is proportional to the
  magnitude of Q
• In free space, the constant of proportionality is
  1/eo where eo is called the permittivity of of
  free space.

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15.10 Gausss Law

• The net flux passing through any closed surface
  is equal to the net charge inside the surface
  divided by eo.
• Can be used to compute electric fields. Example
  point charge

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16.0 Introduction

• The Coulomb force is a conservative force
• A potential energy function can be defined for
  any conservative force, including Coulomb force
• The notions of potential and potential energy are
  important for practical problem solving

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16.1 Potential difference and electric potential

• The electrostatic force is conservative
• As in mechanics, work is
• Work done on the positive charge by moving it
  from A to B

B
A
d
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Potential energy of electrostatic field

• The work done by a conservative force equals the
  negative of the change in potential energy, DPE
• This equation
• is valid only for the case of a uniform electric
  field
• allows to introduce the concept of electric
  potential

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Electric potential

• The potential difference between points A and B,
  VB-VA, is defined as the change in potential
  energy (final minus initial value) of a charge,
  q, moved from A to B, divided by the charge
• Electric potential is a scalar quantity
• Electric potential difference is a measure of
  electric energy per unit charge
• Potential is often referred to as voltage

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Electric potential - units

• Electric potential difference is the work done to
  move a charge from a point A to a point B divided
  by the magnitude of the charge. Thus the SI units
  of electric potential
• In other words, 1 J of work is required to move a
  1 C of charge between two points that are at
  potential difference of 1 V

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Electric potential - notes

• Units of electric field (N/C) can be expressed in
  terms of the units of potential (as volts per
  meter)
• Because the positive tends to move in the
  direction of the electric field, work must be
  done on the charge to move it in the direction,
  opposite the field. Thus,
• A positive charge gains electric potential energy
  when it is moved in a direction opposite the
  electric field
• A negative charge looses electrical potential
  energy when it moves in the direction opposite
  the electric field

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Analogy between electric and gravitational fields

• The same kinetic-potential energy theorem works
  here
• If a positive charge is released from A, it
  accelerates in the direction of electric field,
  i.e. gains kinetic energy
• If a negative charge is released from A, it
  accelerates in the direction opposite the
  electric field

A
A
d
d
q
m
B
B
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Example motion of an electron
What is the speed of an electron accelerated from
rest across a potential difference of 100V? What
is the speed of a proton accelerated under the
same conditions?

• Observations
• given potential energy difference, one can find
  the kinetic energy difference
• kinetic energy is related to speed

Given DV100 V me 9.1110-31 kg mp
1.6710-27 kg e 1.6010-19 C Find ve? vp?
Vab
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16.2 Electric potential and potential energy due
to point charges

• Electric circuits point of zero potential is
  defined by grounding some point in the circuit
• Electric potential due to a point charge at a
  point in space point of zero potential is taken
  at an infinite distance from the charge
• With this choice, a potential can be found as
• Note the potential depends only on charge of an
  object, q, and a distance from this object to a
  point in space, r.

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Superposition principle for potentials

• If more than one point charge is present, their
  electric potential can be found by applying
  superposition principle
• The total electric potential at some point P due
  to several point charges is the algebraic sum of
  the electric potentials due to the individual
  charges.
• Remember that potentials are scalar quantities!

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Potential energy of a system of point charges

• Consider a system of two particles
• If V1 is the electric potential due to charge q1
  at a point P, then work required to bring the
  charge q2 from infinity to P without acceleration
  is q2V1. If a distance between P and q1 is r,
  then by definition
• Potential energy is positive if charges are of
  the same sign and vice versa.

q2
q1
r
P
A
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Mini-quiz potential energy of an ion
Three ions, Na, Na, and Cl-, located such, that
they form corners of an equilateral triangle of
side 2 nm in water. What is the electric
potential energy of one of the Na ions?
Cl-
?
Na
Na
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16.3 Potentials and charged conductors

• Recall that work is opposite of the change in
  potential energy,
• No work is required to move a charge between two
  points that are at the same potential. That is,
  W0 if VBVA
• Recall
• all charge of the charged conductor is located on
  its surface
• electric field, E, is always perpendicular to its
  surface, i.e. no work is done if charges are
  moved along the surface
• Thus potential is constant everywhere on the
  surface of a charged conductor in equilibrium

but thats not all!
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• Because the electric field in zero inside the
  conductor, no work is required to move charges
  between any two points, i.e.
• If work is zero, any two points inside the
  conductor have the same potential, i.e. potential
  is constant everywhere inside a conductor
• Finally, since one of the points can be
  arbitrarily close to the surface of the
  conductor, the electric potential is constant
  everywhere inside a conductor and equal to its
  value at the surface!
• Note that the potential inside a conductor is not
  necessarily zero, even though the interior
  electric field is always zero!

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The electron volt

• A unit of energy commonly used in atomic, nuclear
  and particle physics is electron volt (eV)
• The electron volt is defined as the energy that
  electron (or proton) gains when accelerating
  through a potential difference of 1 V
• Relation to SI
• 1 eV 1.6010-19 CV 1.6010-19 J

Vab1 V
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Problem-solving strategy

• Remember that potential is a scalar quantity
• Superposition principle is an algebraic sum of
  potentials due to a system of charges
• Signs are important
• Just in mechanics, only changes in electric
  potential are significant, hence, the point you
  choose for zero electric potential is arbitrary.

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Example ionization energy of the electron in a
hydrogen atom
In the Bohr model of a hydrogen atom, the
electron, if it is in the ground state, orbits
the proton at a distance of r 5.2910-11 m.
Find the ionization energy of the atom, i.e. the
energy required to remove the electron from the
atom.
Note that the Bohr model, the idea of electrons
as tiny balls orbiting the nucleus, is not a very
good model of the atom. A better picture is one
in which the electron is spread out around the
nucleus in a cloud of varying density however,
the Bohr model does give the right answer for the
ionization energy
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In the Bohr model of a hydrogen atom, the
electron, if it is in the ground state, orbits
the proton at a distance of r 5.29 x 10-11 m.
Find the ionization energy, i.e. the energy
required to remove the electron from the atom.
The ionization energy equals to the total energy
of the electron-proton system,
Given r 5.292 x 10-11 m me 9.1110-31 kg
mp 1.6710-27 kg e 1.6010-19
C Find E?
with
The velocity of e can be found by analyzing the
force on the electron. This force is the Coulomb
force because the electron travels in a circular
orbit, the acceleration will be the centripetal
acceleration
or
or
Thus, total energy is
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16.4 Equipotential surfaces

• They are defined as a surface in space on which
  the potential is the same for every point
  (surfaces of constant voltage)
• The electric field at every point of an
  equipotential surface is perpendicular to the
  surface

• convenient to represent by drawing
  equipotential lines

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