Charge and Electric Force

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Learning Objectives

After reading this page, watching the videos and reviewing the exercises, you will be able to:

• Understand the nature of charge.
• Know Coulomb's Law: the magnitude and direction of the Electric Force.
• Apply the concepts to some problems.

What leads us to think that there are such things as charges, anyway?

Think about what you have learned about “forces.” A force can be thought of as a push or a pull that may act on an object. Usually, the force that acts on some object “A” is caused by some other object which is in direct contact with object “A.” For instance, a hammer might strike a nail; then the hammer is exerting a force on the nail. According to Newton’s third law, the nail also exerts a force on the hammer (equal in magnitude, opposite in direction). This type of force is called a “contact” force.
There is another type of force, in which direct contact between objects is not required. For example the force of gravity or magnets interacting while not in direct contact. These forces are sometimes called “action-at-a-distance” forces.
Now consider some other phenomena you have seen. When you rub a comb through your hair, you may have noticed the comb pick up small bits of paper. You may have seen balloons appearing to “stick” to walls, without any glue. You have probably seen various types of plastic materials (such as thin plastic wrap) adhere to other objects, as if they were “attracted” to each other across space.

In experiments using certain simple materials such as rubber and glass rods, when they are rubbed with fur or silk, certain phenomena consistently recur:
(1) when two of these objects made from identical materials are prepared the same way (e.g., rubber rubbed with fur), the objects appear to exert small repulsive forces on each other
(2) when two of these objects made from different materials are held near each other (e.g., a rubber rod held near a glass rod), the objects appear to exert attractive forces on each other.
This is due to the “electrical” force. It can be attractive, or repulsive.

Two Types of Charge

The experiments described above, and many other similar ones, lead us to conclude that there are two different properties of matter that, in some sense, cause this force. When two objects with property “A” are near each other, they push each other apart (i.e., exert repulsive forces on each other). The same thing happens when two objects with property “B” are near each other. But when an object with property “A” is near an object with property “B,” they exert attractive forces on each other. These properties of matter have been called “charge,” and instead of “A” charge and “B” charge, the terms “positive” [symbol: +] and “negative” [symbol: –] charge are used. The symbols usually used for charge are q or Q, and it is measured in units called “coulombs” [symbol: C].

Two Types of Material

Very broadly, materials can be described as either conductors or insulators.
Conductors allow charge (usually free electrons) to move easily. Conductors are often metals. This is why wires are made of metal.
Insulators can be charged, but the charge is trapped in or on the material and cannot move easily. Plastics are often good insulators. This is why metal wires are covered in plastic to isolate them.
Semi-conductors fall somewhere in between.

Nature of the Electric Force: Coulomb's Law

From the experiments with “charged” objects (i.e., objects with the charge property), we are led to conclude that the magnitude of the electrical force depends strongly on the distance between the objects. The effects of the repulsive and attractive forces are much more noticeable when the charged objects are close together, than when they are far apart. Many careful experiments have led to the following relationship for the magnitude of the electrical force between two objects separated by a distance r, when one object has charge q1 and the other object has charge q2:
$|F_{electric}|=k{\frac {|q_{1}||q_{2}|}{r^{2}}}$
This relationship is called “Coulomb’s law.” Here, the letter k represents a proportionality constant, which has the value $9\times 10^{9}Nm^{2}/C^{2}$.
You may see k written in terms of another constant, the permittivity of free space $\varepsilon _{0}$. In fact $k={\frac {1}{4\pi \varepsilon _{0}}}$.

Note how the function depends on the variables. The bigger the charges, the bigger the electric force. However, the function is an inverse r squared function. Therefore the bigger the r the smaller the force, the smaller the r the much bigger the force.

The absolute value signs on the charge symbols are there because the magnitude of the force does not depend on whether the charges are positive or negative – it only depends on the amount of charge present on the objects. The direction of the force does depend on the charge sign.
Note that the electrical force does not depend on the mass of the objects, or any other property – only on the charge.

Note: When doing problems it is recommended to calculate the magnitude, and then think about the direction of the force separately

Direction of the Electric Force

The direction of the force depends on the relative types of the charges. If q1 and q2 both have the same type of charge – positive or negative – then the force between them is repulsive, and is directed along a straight line connecting the two charges (see figures below, left and center). If one charge is positive is the other is negative, then the force is attractive, but still directed along the line connecting the charges (see right figure below).

In the SI system of units, distance is measured in meters (m), charge is measured in coulombs (C), and force is measured in newtons (N). In these units, the constant k has the value $9\times 10^{9}Nm^{2}/C^{2}$. So, for instance, the magnitude of the electrical force between an object with a charge of 3 C and one with a charge of 6 C, separated by a distance of 4 m, is:

$F_{electric}=k{\frac {q_{1}q_{2}}{r^{2}}}=9\times 10^{9}Nm^{2}/C^{2}{\frac {3C6C}{(4m)^{2}}}=1.01\times 10^{10}N$

This is a huge force, but a charge of 3 C is far larger than would be found on any ordinary object.

Charge is quantized

Although typical amounts of charge are much smaller than that, it turns out that charge never appears in quantities below a certain minimum value. This value, symbolized by the letter e, is called the “elementary charge.” It is equal to $1.6\times 10^{-19}C$. This can be thought of as the “minimum package size” for charge. Charge always appears in quantities that are some integer multiple of e, e.g. 35e, 17984e,$3\times 10^{6}e$ , etc. That is to say charge is quantized, the charge always comes as a multiple of a base unit.
So, can you obtain a quantity of charge equal to 3.5 e? [Answer: No].
Atoms are composed of smaller, “sub-atomic” particles, such as the proton (p), neutron (n), and electron (e). It turns out that while the neutron has no charge (and so does not experience electrical forces), both the proton and the electron have an amount of charge whose magnitude is e. However, the proton has +e (a positive charge) and the electron has –e (a negative charge). The fact that the charges on these two particles are the same magnitude is curious because their masses are so different (the proton has a mass nearly 2000 times larger than that of the electron).

Charge is Conserved

An important principle about charge that that has been discovered through many experiments is called “conservation” of charge. This principle states that the total amount of charge in any closed system never changes. A “closed” system is one in which charges can neither enter nor leave. By “total amount of charge,” we mean the algebraic sum of all positive and negative charge quantities.

Example: If a sealed box initially contains positive charges equal to +6 e, and negative charges equal to –11 e, the total amount of charge in that box must always remain at –5 e. It is possible that some of the positively charged particles and some of the negatively charged particles may actually disappear – such phenomena do occur (and energy is then given off in some form) – but nonetheless, the charge on the particles that would remain must always sum up to –5 e in this case.

The Electric Force obeys the Superposition Principle

An important property of electrical forces is known as the “superposition” principle. This states that the net electrical force acting on any charged object is equal to the vector sum of all the individual forces on that object, where each individual force results from the interaction of the object and one other charged object. Each individual interaction is unaffected by any of the other interactions.
That is to say: each interaction is independent.
Algebraically, this can be expressed as:

${\vec {F}}_{net}=\Sigma {\vec {F}}={\vec {F}}_{1}+{\vec {F}}_{2}+{\vec {F}}_{3}...+{\vec {F}}_{n}$

Here, the net force on the object is found from the vector sum of the forces from n other charged objects. This equation implies that the net x component of the force equals the sum of the individual x components, and that the net y component equals the sum of the individual y components.

Flying insects pick up static charge (also spiders on LSD...)

Need to review Vector Algebra? See here

Short Example Questions

Example 1: Suppose each grid square below is one meter long, and suppose that all three charges are identical, that is: Q1 = Q2 = Q3 = $3\mu C$.
What is the net force acting on charge Q2?

Answer: Net force equals the vector sum of [force from Q3] and [force from Q1]. These forces are represented by left and right arrows, respectively, in the diagram below:

Since all three charges are identical, we can use the same symbol Q to represent all of them; that is: Q1 = Q2 = Q3 = Q.
You have to consider both the magnitude and direction of each force.
The arrow pointing to the right is the force due to charge Q1; its magnitude is
$|F_{1\ on\ 2}|={\frac {kQ1Q2}{(12)^{2}}}={\frac {kQ^{2}}{144}}$;
the arrow pointing to the left is the force due to the charge Q3; its magnitude is
$|F_{3\ on\ 2}|={\frac {kQ2Q3}{(6)^{2}}}={\frac {kQ^{2}}{36}}\ or\ 4{\frac {kQ^{2}}{144}}$;
so it is four times larger than the magnitude of the rightward-pointing force: because it is twice as close.

Note that we are only looking at the forces acting on Q2. Q1 experience forces from Q2 and Q3. Q3 experiences forces from Q1 and Q2. Newton's 3rd Law!
These forces are directed along the line connecting the charges, and so we can see that each force in this case only has an x component – the y components are zero. The rightward pointing force has a positive x component, and the leftward pointing force has a negative x component. Then the net x component is equal to $F_{x_{net}}={\frac {kQ^{2}}{144}}-4{\frac {kQ^{2}}{144}}=-3{\frac {kQ^{2}}{144}}=-1.68mN$ .
This would be represented by an arrow pointing to the left (negative x direction) with a length of three grid squares (three times the length of the rightward pointing force). That arrow would then represent the net electrical force acting on Q2, the middle charge.

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Example 2: Suppose an object with zero net charge (i.e., exactly equal quantities of positive and negative charges) is located in the neighborhood of another object with zero net charge. What will be the net electrical force experienced by either object?

Answer: There will be virtually no net electrical force on either object, because all of the repulsive and attractive forces will cancel each other out (i.e., their vector sum will be nearly zero). Show this by considering two objects, each containing two protons and two electrons, and drawing all of the force vectors on all of the charged particles. You should be able to see that the net force on each object is nearly zero, as long as they are not located too close together.

Long Answer Example Problem

Three charges $q1=1.0\mu C$, $q2=-2.0\mu C$ and $q3=3.0\mu C$ are held fixed on a horizontal plane at positions (x=-1,y=3)cm for q1, (0,0)cm for q2 and (2,1)cm for q3.
Find the net electric force that q2 and q3 exert on q1. [Ans: 12.8 N@205deg]
Video Solution Courtesy of John Abbott College