Chapter 20: Electrostatic Fields and Forces

Equations Introduced and used in this Topic: 

[latex]\overrightarrow{F}_{el} =\dfrac{kq_{1}q_{2}}{d^{2}}[/latex]             [latex]\overrightarrow{F}_{el}=q\overrightarrow{E}[/latex]             [latex]\overrightarrow{E}=\dfrac{kq}{d^{2}}[/latex]             [latex]V =\dfrac{kq}{d}[/latex]

q  =  (+ 1.602 x 10^-19 C)(number of electrons/protons)

Bohr Radius  (a₀, r_Bohr) =  5.29177 x 10^-11 m    Excited State Radius  r = n² a_o 

Where

F  is generally the Net Force acting on the charged object

F_el  is the Electrostatic Force, measured in Newtons (N)

E   is the Electric Field, measured in Newtons per Coulomb (N/C) or Volts per Metre (V/m)

k   is the Coulombs Constant, measured in Newton Metres ^squared per Coulomb ^squared (Nm²/C²)

q, q₁ & q₂   are the Charges you are studying, measured in Coulombs (C)

q  if negative indicates that it has an excess of electrons (or shortage of protons)

q   if positive indicates that it has an excess of protons (or shortage of electrons)

d   is the Distance away from the Charge Center, measured in Metres (m)

V   is the Potential Difference, measured in Volts (V) or Joules/Coulombs (J/C)

a   is the Acceleration of the object measured in Metres per Second squared (m/s²)

The static charge on an object can be calculated by the number of

excess electrons or protons time + 1.602 x 10^-19 C

Elementary Particles 4

Important Constants (round to needed accuracy)

Coulombs Constant                            k =  8.9875517873681764 x 109 Nm²/C²

Electron Volt                                        1 eV = 1.60217653(14) x 10^-19 J

Bohr Radius                                         (a₀, r_Bohr) =  5.29177 x 10^-11 m

20.1 Electrostatic Forces⁵

Equations Introduced and Used in this Section:

Historical records indicate awareness of numerous electrical phenomena for over 3000 years but the explanations for them were quite different from how we understand them today.  For instance, Thales of Miletus (624-546 BC) thought we were seeing the soul of a piece of amber that, when rubbed on fur, would attract small objects. Likewise, lightning, St. Elmo’s Fire, and the electric ray were well-known electrical phenomena with equally mystical natures.

The first recorded study of electrical phenomena, which is different from the study of mathematics and astronomy, was published by William Gilbert (1544-1603) three years before his death from what is believed to have been the bubonic plague.  The name for electricity came from Gilbert’s use of the Latin word “electrica” (ἤλεκτρον) meaning “like amber”. While electrica had been used since the 13th century, Gilbert defined the name respecting its attractive or repulsive properties.

From Gilbert’s work, three rules of electrostatics were developed:

• There are only two kinds of electric charge

• Two objects charged alike (having the same kind of charge) repel each other

• Two objects charged oppositely attract each other.

Explaining the nature of charged objects was more challenging.  Gilbert speculated that the “amber effect” was the result of the effluvium (small particles that flowed from one charged object to another that did not have any mass or volume). Over the next century, multiple small advances were made by numerous philosopher/scientists with one, Charles François de Cisternay du Fay (1698-1739), proposing that there were two types of electric charge: a positive charged fluid and a negative charged fluid. However, it was Benjamin Franklin (1706-1790) who postulated that electric charge was but a single fluid, not that an object having an excess of this fluid was positively charged and an object having a deficit of this fluid was negatively charged.

We now know that the Franklin one-fluid model is fundamentally correct in that it is the electron that can be understood as the source of electric charge. Because of this construct, everything in conventional electricity is defined in terms of the movement of this excess charge. However it was not until the late 1890s that evidence accumulated to support what electric charge actually was. By this time, the battery, lightbulb, telephone and a host of other electrical inventions had been created, and all were understood using Franklin’s concept of the movement of excess fluid. The legacy of Franklin’s understanding can still be found in electrical terminology, including: positive and negative charge, charging and discharging, conductors and condensers, the lightning rod, and the flow of electric charge called electric current.  Franklin⁶ and William Watson (1715-1787) both proposed the conservation of electric charge. This concept was later interpreted as an isolated system wherein the net charge remains constant and, if matter is created or destroyed, charge is created and destroyed in equal amounts.

Quantifying the force that exists between similar or oppositely charged objects was proposed by Joseph Priestly (1733-1804) to follow an inverse square law as had been proposed by Isaac Newton for gravity. Priestly posited this relationship in his 700 page book The History and Present State of Electricity (1767). Priestly’s inverse square law became formalized by Charles-Augustin de Coulomb (1736-1806), who demonstrated evidence of its authenticity using a device called the torsion balance.  Coulomb later was able to demonstrate that an inverse square law relationship also existed between the poles of bar magnets.

In algebraic form, Coulomb’s Law ^7, 8  is stated as:

[latex]\overrightarrow{F}_{el} =\dfrac{kq_{1}q_{2}}{d^{2}}[/latex]

Where:

k =  8.9875517873681764 x 10⁹ Nm²/C²    (Coulomb’s Constant)

q  =  (+ 1.602 x 10^-19 C)   (number of electrons/protons)

d  =  the distance between the charge centers (measured in metres)

F_el = find          [latex]\overrightarrow{F}_{el} =\dfrac{kq_{1}q_{2}}{d^{2}}[/latex]

k = 8.99 x 10⁹ Nm²/C²     [latex]\overrightarrow{F}_{el} =\dfrac{(8.99\times10^{9}Nm^{2}/C^{2})(4.2\times10^{-6} C)(4.5\times10^{-6} C)}{(5.0\times10^{-3}m)^{2}}[/latex]

q₁ =  4.2 x 10^-6 C

q₂ = 4.5 x 10^-6 C       [latex]\overrightarrow{F}_{el}[/latex]= 6800 N directed away from each other (both are positive charges)

d = 5.0 x 10^-3 m

YOU NEED TO ADD ARROW AND TEXT TO THIS PICTURE

The Lithium Atom ⁹

20.2 Electric Fields (Point Charges)¹⁰

Equations Introduced or Used in this Topic:

[latex]\overrightarrow{E}=\dfrac{kq}{d^{2}}[/latex]             [latex]\overrightarrow{F}_{el}= q\overrightarrow{E}[/latex]              [latex]\overrightarrow{F} = m\overrightarrow{a}[/latex]             [latex]\overrightarrow{F}_{el}=\dfrac{kq_{1}q_{2}}{d^{2}}[/latex]

Like the concept of gravitational fields covered in Chapter 17.2, electrostatics share a similar concept of electric field (also magnetic fields).  Historically, the first reference to such a concept was by Gilbert when he was describing the sphere of influence of a charged object. The equation quantifying the electric field can be derived in similar fashion to the derivation of the gravitational field.  This can be shown as follows:

[latex]\overrightarrow{F}_{el}=q\overrightarrow{E}[/latex]       is equated to      [latex]\overrightarrow{F}_{el}=\dfrac{kq_{1}q_{2}}{d^{2}}[/latex]

The direction of the electric field as defined by a negative or positive sign depends on the net charge of the object, whether or not the overall charge of the object is negative or positive, and how it would react on a small positive test charge placed in the electric field¹¹. This means that the electric field can be written as either a positive or a negative quantity.  In practice, many will ignore the positive and negative signs and just draw the field as attractive if the net charge is negative or repulsive if the net charge is positive.

[latex]\overrightarrow{E}=\underline{+}\dfrac{kq_{1}}{d^{2}}[/latex]

[latex]\overrightarrow{E} ∝ 1/d^{2}[/latex]

The units of electrical field strength are N/C.   This equation relates to the electric field at any distance away from a charged object and is a inverse square law relationship.

[latex]\overrightarrow{E}=\text{ 0 N/C}[/latex]

When we look at the electric field strength as we move inside the charged object’s surface, the electric field strength drops to zero¹². Anything inside the charged object is shielded from the electric field; this is why a radio signal on a vehicle fades out when driving through a tunnel or inside the metal frame of a bridge.

Electric fields can have multiple variations of positive and negatively charged objects.  Examples showing field lines of various combinations of positive and negative charges are shown below.¹³

These field lines show the direction of the force that a small positive test charge would experience if placed near these charged objects. Also note that as the field lines get closer together, the field strength gets stronger and as the field lines move farther apart from each other, the field weakens.

20.3  The Hydrogen Atom

       Particle                                 Mass                                            Charge

Electron (e- or ß-)       9.10938356(11) x 10^-31 kg       -1.6021766208 x 10-19 C (-e)

Proton (p, p+ or N+)     1.672621898(21) x 10^-27 kg     +1.6021766208 x 10-19 C (+e)

Bohr Radius  (a₀, r_Bohr) =  5.29177 x 10^-11 m    Excited State Radius  r = n² a_o

[latex]F_{el} =\dfrac{kq_{1}q_{2}}{d^{2}}F[/latex]         [latex]Fc =\dfrac{mv^{2}}{r}[/latex]           [latex]E_{k} = ½ mv^{2}[/latex]              [latex]p = mv[/latex]

The hydrogen atom is important to an understanding of both the structure of atoms and the quantum nature of matter.  One of the first clues to the nature of the hydrogen atom came from the emission spectrum (the specific colors) that hydrogen gas emits when heated. In 1885 Jacob Balmer (1825-1898) discovered that the visible wavelengths associated with the specific emitted colors fit a simple formula. By manipulating the formula, Balmer found other series of wavelengths, both visible and invisible.

Ernest Rutherford (1871-1937) solved the remaining part of the mystery of the nature of atomic structure. Rutherford worked at three universities (including McGill in Montreal) and later at the Cavendish Laboratory. He discovered that atoms were mostly space, with an inner hard core surrounded by orbiting electrons, much like planets orbiting a sun at the center. This concept, the Rutherford Planetary Model of the Atom, can be added to his concepts of alpha, beta and gamma rays, protons and half-life radioactivity.

Combining the results of Balmer and Rutherford, Niels Bohr (1885-1962) used the energy of the emitted light spectra to come up with a model of the atom that certain average orbitals around the nucleus or core were allowed.  Bohr’s work predicted the size of the hydrogen atom and initiated an understanding of the nature of the movement of the electron around the atom. Rutherford’s Planetary Model and Balmer’s emission spectra were now grounded in a common conceptual understanding.

In less than two decades, Werner Heisenberg (1901-1976), Paul Dirac (1902-1984) and Wolfgang Pauli (1900-1958) together laid out the foundations of quantum mechanics by focusing on the frequencies and intensities of the hydrogen spectra transitions, and the elliptical effect of the electron orbits on the hydrogen spectra, with the mathematical genius to put it all together.

The following questions relate to the Hydrogen Atom

20.4 Electric Potential of Point Charges ¹⁵

The images below show the magnitude of the Electrical Potential of both a positive and negative charge as one moves away from the charge.

Electric potential for charged objects is similar to the gravitational potential, in that instead of moving without friction through some displacement in a gravitational field, an object instead moves through a displacement inside an electrical field.  There are two possibilities for this electric field: either the field is constant like the gravity one finds when moving through small changes in height on Earth, or the field varies similarly to the movement of a spaceship either towards or away from the Earth.

This results in two different equations for electric potential: the electric potential moving away from a charged object, or the electric potential in a constant electric field. This section looks at only the electric potential moving away from a charged object, that as such experiences a variable electric field.

Electric potential for a charged object can be used to find the energy change if the amount of charge moving through the potential difference is known.  It is important to remember the distinction between electric potential and electric potential energy. They are different.

The equation for electric potential is one that is most useful in a number of physics situations that look at the energy required to remove an electron from an atom, or to move from one allowed orbital to another for an electron in orbit around an atom.

While the following examples relate to the hydrogen atom, applications of this equation extend far beyond this.

REFERENCES

5. In Research News – Super-Strong Electric Forces May Have Helped Tiny Clumps of Dust Seed the Planets: https://www.smithsonianmag.com/smart-news/super-strong-electric-forces-may-have-helped-tiny-clumps-dust-seed-planets-180973756/