Electric charges and Coulomb s law

Transcription

1 Electric charges and Coulomb s law 1. Introduction The field of physics that deals with stationary electric charges (i.e. at rest) is called Electrostatics. Electrostatics describes the interaction between stationary electric charges by Coulomb s law. 2. Electric charge There are two types of electric charges: positive charge, like the charge of proton, of positron, etc... and negative charge, like the charge of an electron. Charges of the same nature (positive and positve, or negative and negative) repel each other, while charges of different nature (positive and negative) attract each other. The unit of electric charge is Coulomb (C) Charge quantization In the middle of the eighteenth century, electric charge was thought to be as a continuous fluid. Then it was proved experimentally by Robert Millikan, that the charge is discrete, i.e. any charge has to be a multiple of some fundamental elementary charge which is thought to be the charge of the electron. So according to this concept, electric charge can be given or taken in a whole number (integer) of discrete elementary quantities (called quanta), and not continuously (like pouring a fluid) Charge distribution Electric charge can be concentrated in a point, or can be distributed along a line, over a surface, or over a volume, with a certain density. When the distribution is uniform, the charge density is constant. When the distribution is non-uniform, the charge density is a function of the coordinates. In some cases, the charge density can even be a function of time. We define: λ to be the linear charge density (C/m) σ to be the surface charge density (C/m 2 ) ρ to be the volume charge density (C/m 3 ) So, we can obtain the expression of the total charge q by integration: 1

2 Over a line of length ab Or over a surface area S Or over a volume V ˆ b q = λ dl (1) a ˆ q = σ ds (2) ˆ q = s v ρ dv (3) 3. Electric conductors and insulator Electric conductors are materials that permit electric charge to move within them. Electric insulators are those materials that do not permit electric charge to move within them. Note that most metals are good conductors, while most non- metals are insulators. 4. Charging an object 4.1. Charging by friction When two bodies are rubbed against each other, electrons are usually transfered from one body to another. So the body that lost electrons, becomes positive (since it lost negative charge), and the body that gained electrons becomes negative (since it gained negative chages). Charging by friction is easily observed when rubbing insolators with each other, as in insolators, charges are accumulated and can not easily move within Charging by contact When two conducting objects are brought in contact, any existing charge on either of them is redistributed over the two objects. So, a transfer of charges occurs between them Charging by induction The presence of an electrically charged body near a conductor causes a redistribution of charges on the conductors known as induced charge. Figure 1 illustrate this idea. 2

3 Figure Coulomb s law Coulomb s law is a quantitative description of the interaction between electric charges, i.e. the force between two or more electric charges Formation of Coulomb s law Consider two charges q 1 and q 2. The force F between them is observed to have the following properties: 1. The magnitude of F is proportional to the magnitudes of the two quantities of charges q 1 and q The magnitude of F is inversely proportional to the square of the distance r between two charges q 1 and q The interaction between the two charges q 1 and q 2 is the pair of action-reaction force F, which is along the straight line joining q1 and q 2. The force exerted by q 2 on q 1 is: F 21 = ± F21 u r21 where u r21 = r 1 r 1 is the unit vector along the line joining q 1 and q 2, and is directed towards q 1 (See Figure 2) Figure 2. 3

4 And the force exerted by q 1 on q 2 is: F 12 = ± F12 u r12 where u r12 = r 1 r 1 is the unit vector along the line joining q 1 and q 2, and is directed towards q 1 (See Figure 3) Figure If q 1 and q 2 have the same sign (same nature, e.g. positive and positive or negative and negative), then F is repulsive. If q1 and q 2 are opposite in sign (different natures, e.g. positive and negative), then F is attractive. 5. The magnitude of the force F depends on the nature of the surrounding medium Conclusion From properties (1) and (2), we can deduce the magnitude of F to be: F = k q 1 q 2 (4) where k is a constant. From properties (3) and (4), we can deduce the direction of F: and F 12 = k q1 q 2 F 21 = k q1 q 2 r12 (5) r21 (6) Note that if q 1 and q 2 are of the same sign, then the force F (either F12 or F21 ) is directed to outward of the segment joining the charges q 1 and q 2, since we can write their expressions as: And F 12 = +k q 1 q 2 F 21 = +k q 1 q 2 r12 (7) r21 (8) 4

5 Otherwise, if q 1 and q 2 have opposite signs, then the force F (either F12 or F21 ) is directed to inward of the segment joining the charges q 1 and q 2, since we can write their expressions as: And F 12 = k q 1 q 2 F 21 = k q 1 q 2 r12 (9) r21 (10) From the property (5), we can define different constants k for different media. In particular we can define k = 1 4πɛ (11) where ɛ is the permittivity of the medium. For free space (vacuum) ɛ = F/m (Faraday per meter). 6. Principle of Superposition Coulomb s law describes the interaction between two stationary electric charges. But in practice, interaction occures between more than two chages. Suppose that we have some electric charges q 1, q 2, q 3, etc... What force do they exert on another charge Q? The solution of this problem is given by the Principle of Superposition, which states that: The interaction between any two electric charges is completely unaffected by the presence of other charges. This means that we can find F1, which is the force exerted by q 1 on Q, as if there are no other charges, then find F2 in the same manner, then F3, etc... Then the total force exerted by all the changes q 1, q 2, q 3, etc... on Q, is the vector sum of F1, F2, F3, etc..., that is: N F total = Fi, where Fi = 1 4πɛ qi Q i=1 ri 2 u i (12) Here u i is the unit vector corresponding to the interaction between the two charges q i and Q, that is: u i = r Q r i r Q r i Physics Zone by Farid Minawi 5