Chapter 9 Electrostatics (R)

Electric Charge

Elementary Charge

Charge is Quantized

Electric Force and Coulomb’s Law (Coulomb’s Law and Coulomb’s Constant)

The Principle of Superposition for Electric Forces (The Principle of Superposition)

Electric Field

Electric Force and Field

The Principle of Superposition for Electric Fields

Conductors, Insulators, and Polarization

Electrical Potential and Potential Energy

Electric Potential

Change in Electrical Potential Energy

Work done by Electric Field

The Principle of Superposition for Electric Potential

Electric Charge

An atom is composed of a central nucleus (which is itself composed of protons and neutrons) surrounded by a cloud of one or more electrons. The fact that an atom is held together as a single unit is due to the fact that protons and electrons have a special property: They carry electric charge, which gives rise to an attractive force between them.

Electric charge exists in two varieties, which are called positive and negative. By convention, we say that protons carry positive charge and electrons carry negative charge). (Neutrons are well-named: they’re neutral because they have no electric charge.) The charge of a proton is +e, where e is called the elementary charge, and the charge of an electron is -e. Notice that the proton and the electron carry exactly the same amount of charge: the only difference in their charge is that one is positive and the other is negative.

put the picture of the charges on page 256 here.

Elementary charge

e=1.6×10^-19 C

When an atom (or any other object) contains the same number of electrons as protons, its total charge is zero because the individual positive and negative charges add up and cancel. So, when the number of electrons (#e) equals the number protons (#p), the object is electrically neutral. We say that an object is charged when there’s an imbalance between the number of electrons and the number of protons. When an object has one or more electrons (#e>#p), the object is negatively charged, and when an object has an deficit of electrons (#e<#p), the object is positively charged. If a neutral atom has electrons removed or added, we say that it has been ionized, and the resulting electrically charged atom is called an ion. A positively charged ion is called a cation, and a negatively charged ion is called an anion (An object can also become charged by gaining or losing protons, but these are usually locked up tight within the nuclei of the atoms. In virtually all cases, objects become charged by the transfer of electrons)

Because an object can become charged only by losing or gaining electrons or protons, which can be “sliced” into smaller pieces with fractional amounts of charge, the charge on an object can only be a whole number of ±e’s; that is, charge is quantized. So for any object, its charge is always equal to n(±e), where n is a whole number. To remind us that charge is quantized, electric charge is usually denoted by the letter q (or Q).

Charge is quantized

q=n(±e)

where n=0,1,2…

It is interacting to note that this quantization of charge applies to all fundamental particles either found in nature or created in the laboratory (e.g., muons, pions, etc.)

In chemistry, it’s common to talk about the charge of an atom in terms of whole numbers like +1 or -2, etc. For example, we say that the charge of the fluoride ion, F^-, -1, and the charge of the calcium ion, Ca²⁺, is +2. This is just a convenient way of saying that the charge of the fluoride ion is -1 elementary unit (in other words, -1e), and the charge of the calcium ion is +2 elementary units, +2e. When we want to find the electric force between ions, we will express their charges in the proper unit (coulombs), and say, for example, that the charge of the fluoride ion is -1e= -1.6×10^-19 C and the charge of the calcium ion is +2e= +3.2×10^-19C.

Charge (Q) is expressed in the unit of Coulombs (C).

Finally, total electric charge is always conserved; that is, the total amount of charge before any process must always be equal to the total amount of charge afterward

put example 9-1 here

Electric Force and Coulomb’s Law

If two charged particles are a distance r apart (there is a picture on page 258), then the electric force between them F_E, is directed along the line joining them. The magnitude of this force is proportional to the charge (q1 and q2) and inversely proportional to r², as given by

Coulomb’s Law

F_E= k|q1q2|/r²

The proportionality constant is k, and in general, its value depends on the material between the particles. However, in the usual case where the particles are separated by empty space (or by air, for all practical purposes), the proportionality constant is denoted by k₀ and called Coulomb’s constant. This is a fundamental constant of nature (equal in magnitude, by definition, to 10^-7 times the speed of light squared), and its value is k₀= 9×10⁹ Nm²/C².

Coulomb’s Constant

k₀= 9 × 10⁹ Nm²/C²

This is this value of k you should use unless you’re specifically given another value (which would happen only if the charges were embedded in some insulated material that weakens the electric force).

The absolute value sign in the formula given the magnitude of the force, whether repulsive or attractive. If direction (e.g., + or -) needs to be assigned, it should be done based on the fact that like charges (two positives or two negatives) repel each other, and opposite charges (one positive and one negative) attract, Note that the two electric forces in each of the following diagrams form an action-reaction pair.

The Principle of Superpositon

Coulomb’s law tells us how to calculate the force that one charge exerts on another one. What if two (or more) charges affect a third one. For example, what is the electric force on q₃, in the following figure?

please attach the figure to this flashcard

This is the principle of Superposition:

The net electric force on a charge (q) due to a collection of other charges (Q’s) is equal to the sum of the individual forces that each of the Q’s alone exerts on q.

Electric Fields

There are several advantages to regarding electrical interactions in a slightly different way from the simple “charge Q exerts a force on charge q” mode of thinking. In this more sophisticated interpretation, the very existence of a charge (or a more general distribution of charge) alters the space around it, creating what we call an electric field in its vicinity. If a second charge happens to the be there or to roam by, it will feel the effect of the field created by the original charge. That is, we think of an electric force on a second charge q as exerted by the field, rather than directly by the original charge (s). Qualitatively, we can represent electric interactions as follows:

There is a picture at the top of page 265.

The charges creating the electric field is/are called the source charge(s); they’re the source of the electric field. You may like to think of a source charge as a spider and its electric field as the spider’s web. After a spider creates a web, when a small insect roams by, it is the web that ensnare’s the unfortunate bug, not the spider directly.

There is another picture on page 265.

This figure on the right above illustrates one way to picture an electric field, but a few words of explanation are needed. First: an electric field is a vector field, which means that at each point in space surrounding the source charge, we associate a specific vector. The length of this vector will tell us the magnitude or strength, or the field at that point, and the direction of the vector will tell us the direction of the resulting electric force that a positive test charge would feel if it were placed at that point. That’s the convention: although the charge that finds itself in an electric field can of course be positive or negative, for purposes of llustrating the field, we always think of a positive test charge. Because of this convention, electric field vectors always point away from positive source charges and towards negative ones. Also, the closer we are to the source charge, the stronger the resulting electric force a test charge would feel (because Coulomb’s law is an inverse-square law). So, we expect the electric field vectors to be long at points close to the source charge and shorter points farther away. The following figures illustrate the electric field due to a positive source charge and the electric field due to a negative source charge.

There is a picture on page 266.

We can use Coulomb’s law to find a formula for the strength of the electric field (stopped on page 266).

Electric Field

E_{by Q}= k|Q|/r²

In the formula for the force by Q on q, the variable r represents the distance from Q to q. However, if q is not there, what does r mean now? Answer: it’s simply the distance from Q to the point in space where we want to know the electric field vector.

There is a picture on page 267.

Then there is an example.

Electric Force and Field

F_{on q}= qE

Note that the absolute value symbol is useful when solving for the magnitude of force and electric field. The vector equation, F_E=qE, contains directional information and therefore does not need absolute values. Notice also from this formula that E=F/ |q|, so the units of E are N/Cm which you saw in Example 7-9. The equation E=F/q also gives us the definition of the electric field: It’s the force per unit charge.

Finally, before we get to some more examples, realize that we’ve had to important (boxed) formulas in this section on the electric field: E=k|Q|/r² and F=qE. In the first formula, Q is the first charge that makes the field, while the second formula, q is the charge that feels the field.

There is stuff at the bottom on page 269.

The Principle of Superposition for Electric Fields

The pictures we’ve drawn so far have been of electric fields crated by a single source charge. However, we can also have two or more charges whose electric fields overlap, creating one combined field. For example, let’s consider the electric dipole, which, by definition, is a pair of equal but opposite charges:

put the picture of the electric dipole on page 273.

We’d first find the electric field vector, E₊, at P due to the +Q charge along (ignoring the presence of the -Q charge) and then we’d find the electric field vector, E_-, at P due to the -Q charge alone (ignoring the presence of the +Q charges). The net electric field vector at P will then be the vector sum, E₊ + E_-.

put the picture of the electric dipole on page 273.

We can do this for as many points as we like to and obtain a diagram of the electric field as a collection of vectors. The diagram in terms of electric field lines would look like this:

there is a picture at the bottom of page 273.

Conductors, Insulators, and Polarization

Most everyday materials can be classified into one of two major categories: conductors or insulators (also known as dielectrics). A material is a conductor if it contains charges that are free to roam throughout the material. Metals are the classic and most important conductors. In a metal, one or more valence electrons per atom are not strongly bound to any particular atom and are thus free to roam. If a metal is placed an electric field, these free charges (called conduction electrons) will move in response to the field. Another example of a conductor would be a solution that contains lots of dissolved ions (such as saltwater).

Here’s an interesting property of conductors: Imagine that we place a whole bunch of electrons on a piece of metal. It’s now negatively charged. Since electrons repel each other, they’ll want to get as far away from each other as possible. As a result, all of this excess charge moves (rapidly) to the surface. Any net charge on a conductor resides on its surface. Since there’s no excess charge within the body of the conductor, there cannot be an electrostatic field inside a conductor. You can block out external electric fields simply by surrounding yourself with metal; the free charges in the metal will move to the surface to shield the interior and keep E=0 inside.

By contrast, an insulator (dielectric) is a material that doesn’t have free charges. Electrons are tightly bound to their atoms and thus are not free to roam throughout the material. Common insulators include rubber, glass, wood, paper, and plastic.

Now let’s study this situation: Start with a neutral metal sphere and bring a charge (a positive charge) Q nearby without touching the original metal sphere. What will happen? The positive charge will attract free electrons in the metal, leaving the far side of the sphere positively charged. Since the negative charge is closer to Q than the positive charge, there’ll be a net attraction between Q and the sphere. So, even will create a force of electrical attraction between them.

picture on the bottom of page 275

Now what if there sphere was made of glass (and insulator)? Although there aren’t free electrons that can move to the near side of the sphere, the atoms that make up the sphere will become polarized. That is, their electrons will feel a tug toward Q, causing the atoms to develop a partial negative charge pointing toward Q (and a partial positive charge pointing away from Q). The effect isn’t as dramatic as the mass movement of free electrons in the case of a metal sphere, but the polarization is still enough to cause an electrical attraction between the sphere and Q. For example, if you comb your hair, the comb will pick up extra electrons, making it negatively charged. If you place this electric field source near little bits of paper, the paper will become polarized and will then be attracted to the comb.

Electric Potential and Potential Energy

So far, we have viewed the electric field due to a source charge (or a more general charge distribution, such as a pair of charges of a plate) as a collection of vectors. This point of view allowed us to answer questions about other vector quantities, like force and acceleration. The basic equations for finding these quantities were F=qE and a=F/m=qE/m

What if we wanted to answer questions about scalar quantities, like energy, work, or speed? It turns out that the easiest way to answer these questions is to view the electric field in a different way in terms of a scalar field. First, it is useful to review a few facts about gravitational potential energy. As you’ll recall, if an object of mass m is dropped from rest from a height h and hits the ground, W_{by grav}= +mgh while ΔPE_grav=-mgh. Similarlly, if the object is lifted from the ground to a height hi, W_{by grav}= -mgh, W_{against grav}= +mgh and ΔPE_grav= +mgh.

there is a picture on page 276.

If an object moves “with nature” (i.e., in the direction… stopped on page 276).

Electric Potential

ϕ=kQ/r

Electric potential, often written as the Greek letter phi (φ) is the electric potential energy stored per unit charge at any given point in space

where k is the Coulomb constant, Q is the charge, and r is the distance.

there is a picture on page 277

Notice the differences between this formula and the one for the electric field. First, the potential is kQ divided by r, while the electric field kQ divided by r². Second, the electric field has a specific direction at each point (because it’s a vector quantity); the potential, on the other hand, is not a vector, so it has no direction. For this reason, no absolute value symbol is needed. The sign of the potential is important in determining the behavior of nearby charges if they are placed in the field. While the electric field has the same magnitude at every point at distance r from Q, the field has a different direction at every point on the circle (or, more generally, the sphere) of radius r centered on Q. Therefore, we’re forced to say that the electric field isn’t the same at every point a distance r from Q because the directions are all different. The potential, however, is easier because it has no direction. The potential is the same at every point that’s a distance r from Q.

There is a picture on page 278

The dashed circles shown in the figure on the right above are called equipotentials (“equal potentials’), because the potential is the same at every point on them. For example, the potential is equal to 8 units everywhere on the inner dashed circled, stopped on page 278.

Change in Electric Potential Energy

That is, the change in potential energy of a charge q that moves between two points whose potential difference is Δϕ is just given by the product, qΔϕ; it also can be expressed as qV, where V is defined as the change in potential and is known as the voltage. For example, let’s say a charge q=+0.03 C moves from a point on the inner circle to a point on the outer circle in the figure accompanying the preceding example:

stopped on page 280.

ΔPE= qΔϕ=qV

Core Components

• Δ PE: Change in electric potential energy, measured in Joules (J).

• q: Electric charge of the particle, measured in Coulombs (C).

• Δ φ or V: Electric potential difference (voltage), measured in Volts (V).

Key Rules

• Sign Matters: Charge (q) can be positive or negative.

• Voltage Sign: Voltage (V) is final potential minus initial potential (\(V_f - V_i\)).

• Energy Gain: Positive Δ PE means the particle gained potential energy.

• Energy Loss: Negative Δ PE means the particle lost potential energy.

• Kinetic Tradeoff: Lost potential energy converts into kinetic energy (Δ KE = -Δ PE).

Work Done by Electric Field

W_{by electric field}= -ΔPE

Now what about kinetic energy? Well, if there’s no friction (which will be the case for charges moving around in empty space) or other forces doing work as a charge moves, then mechanical energy is conserved: that is KE + PE will remain constant. And if KE + PE is constant, then Δ (KE + PE) will be zero. That is ΔKE will equal to -ΔPE. Since we know how to calculte the ΔPE, we can calculate the ΔKE by just changing the sign of ΔPE:

ΔKE=-ΔPE

So as long as you remember the fundamental formula for potential energy changes in an electric field, ΔPE= qΔϕ, you can answer questions about work or kinetic energy in an electric field by just using the formulas above.

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The Principle of Superposition for Electric Potential

The formula ϕ= kQ/r tells us how to find the potential due to a single point charge, Q. To find the potential in an electric field that’s created by more than one charge, we use the principles of superposition. In fact, applying this principle is even easier here than for electric forces and fields because potential is a scalar. When we add up individual potentials, we’re simply adding numbers: we’re not adding vectors.

Let’s illustrate with an example. In the figure below…stopped on page 284.