Electric Charge and Coulomb's Law

From MyMCAT

Jump to: navigation, search

Contents

Introduction

Probably everyone is familiar with the concept of electric charges. Static shocks, lightning, and batteries all have to do with charge. But how do you quantify charge? What does having something with charge mean and what can it do?

Charge

To define charge more formally, we must understand a few basic concepts of charge:

  • There are two kinds of charge, positive and negative. (Protons carry positive charge, while electrons carry negative charge.)
  • Like charges repel, unlike charges attract. (Electrons go towards protons.)
  • Positive overall charge implies there are more protons than electrons in a molecule, while negative charge implies the reverse.
  • Charge is quantized. All values of charge come in multiples of the elementary charge e. Electrons have a charge of -e, while protons have a charge of +e.

Charge is generally given the symbol q, but because the charge of an electron is a rather small number, it is more often measured measured in Coulombs (C), where e = 1.60 x 10-19Coulombs. As an example, a Ca2+ ion would have a charge of 2e, or 3.20 x 10-19 Coulombs.

The Law of Conservation of Charge

The Law of conservation of charge states that the net charge of an isolated system remains constant.

If a system starts out with an equal number of positive and negative charges, there's nothing we can do to create an excess of one kind of charge in that system unless we bring in charge from outside the system (or remove some charge from the system). Likewise, if something starts out with a certain net charge, say +100 e, it will always have +100 e unless it is allowed to interact with something external to it.

Conductors versus Insulators

Most things are electrically neutral; they have equal amounts of positive and negative charge. Metals are good conductors of electric charge, while plastics, wood, and rubber are not. They're called insulators. Charge does not flow well through insulators, which is why wires you plug into a wall socket are covered with a protective rubber coating. Charge flows along the wire, but not through the plastic coating to you.

Most materials are either conductors or insulators. The difference between them is that in conductors, the outermost electrons in the atoms are so loosely bound to their atoms that they're free to travel around. In insulators, on the other hand, the electrons are much more tightly bound to the atoms, and are not free to flow. Semi-conductors are a very useful intermediate class, not as conductive as metals but considerably more conductive than insulators. By adding impurities to semi-conductors the conductivity can be specifically controlled. (It is this process that make modern day electronics possible!)


Coulomb's Law

We have mentioned that charges attract and repeal each other, but so far have provided a method for determining by how much or how strong this attraction/repulsion is. Coulomb's law provides the empirical calculations necessary to determine this. To calculate the force exerted by one charge, q1 on another charge q2 we use the equation

F = \frac{kq_1q_2}{r^2}

where r is the distance between the charges and k is the electrostatic constant (related to the electric constant) and is equal to 8.99 x 109NC2m-2.

This formula says that the magnitude of the force is directly proportional to the magnitude of the charges of each object and inversely proportional to the square of the distance between them. When dealing with just two charges, the direction of the force is always either directly towards or against the other charge (depending on if they are attracting or repealing each other).

This force is a vector, so when more than one charge exerts a force on another charge, the net force on that charge is the vector sum of the individual forces. Remember, too, that charges of the same sign exert repulsive forces on one another, while charges of opposite sign attract. This can be determined by the sign of F. F is positive when either two positive values or two negative values for q and Q were used and thus it is a repulsive force. F is negative when two values of opposite sign are used, in which case, F is an attractive force.


For the following two questions, consider the following situation:
Two charges, a +5 µC charge and a -6 µC charge, are 2 meters apart.

1. What is the force felt by the -6 µC charge? (k is 9.0 x 109 N•m2 / C2)

0.065 N
-0.065 N
6.5 N
-6.5 N
→ \begin{align}F_{\text{q1 on q2}} &= \frac{kq_{1}q_{2}}{r^2} \\ &= \frac{\left( 9.0 \times 10^{9} \right) \left( 5 \times 10^{-6} \right) \left( -6 \times 10^{-6} \right) }{ \left( 2 \right) ^2} \\ &= - 0.065

2. How would the force change if the charges were moved to a final distance of half the original distance?

The force felt would not change
The force felt would double
The force felt would half
the force felt would quadruple
→ In looking at Coulomb's Law, the expression is over the radius squared, thus if the distance (or radius) is 1/2 the original then the radius squared term is reduced to 1/4 the original, and so the whole force term quadruples.

3. A positive charge of q = 3.0 uC is pulled on by two negative charges. One is 0.050m to the left of q with a charge of -2.0uC while the other is -4.0 uC and 0.030m to the right of q. What is the total force exerted on the positive charge q? (k is 9.0 x 109 N•m2 / C2)

54.0 N to the left
54.0 N to the right
140.4 N to the left
140.4 N to the right
→ Coulomb's Law tells us what the Force felt between two charges is. Furthermore, this force is a vector and as such we can add these vectors together if there are multiple charges involved. In this case, we need to look at ALL the forces being felt by the positive charge q. Therefore we will need to determine the force of each of the negative charges on q and then sum these vectors together. We will use "q" to denote the positive charge, and q1 and q2 to denote the other two negative charges.
→ F_{\text{q1 on q}} = \frac{kq_{1}q}{r_{1}^2}
→ F_{\text{q2 on q}} = \frac{kq_{2}q}{r_{2}^2}
→ F_{net} = F_{\text{q1 on q}} + F_{q2 on q}
→ If we avoid the math until the end, we can simply the expressions to reduce the number of calculations necessary as follows,
→ \begin{align}F_{\text{q1 on q}} &= kq \left( \frac{q_{1}}{r_{1}^{2}} \right) \\ &= kq \left( \frac{-2 \times 10^{-6}}{0.05^{2}} \right) \\ &= kq \left( \frac{-2 \times 10^{-6}}{2.5 \times 10^{-3}} \right) \\ &= kq \left( -0.8 \times 10^{-3} \right) \end{align}
→ \begin{align}F_{\text{q2 on q}} &= kq \left( \frac{q_{2}}{r_{2}^{2}} \right) \\ &= kq \left( \frac{-4 \times 10^{-6}}{0.03^{2}} \right) \\ &= kq \left( \frac{-4 \times 10^{-6}}{9 \times 10^{-4}} \right) \\ &= kq \left( -4.4 \times 10^{-3} \right) \end{align}
→ Now that we have expressions for the two forces in play, we must look at their magnitudes and decide how the two are interaction. Both are attractive forces, but they are acting in different directions. Thus, when we sum the forces together we should take this into consideration and the final net force will be a subtraction (because they partially cancel each other). Assuming right is the positive direction, we can subtract the magnitude of q2's force from the magnitude of q1's force.
→ \begin{align}F_{net} &= F_{\text{q2 on q}} - F_{q1 on q} \\ &= kq \left( \left( 4.4 \times 10^{-3} \right) - \left( 0.8 \times 10^{-3} \right) \right) \end{align}
→ Plugging in the final values for k and q, yields -140.4N. The directions we used were that left was positive while right was negative (as seen in the Fnet equation where the right force was subtracted from the left force). We could also notice that the charge to the right is both larger and closer, and therefore we would expect the force to be felt in that direction more than the other.

Your score is 0 / 0
Retrieved from "http://www.mymcat.com/wiki/Electric_Charge_and_Coulomb%27s_Law"