Idea Transcript
Circuits (MTE 120) (Spring 2010)
Akrem El-ghazal http://pami.uwaterloo.ca/~akrem/ University of Waterloo, Electrical and Computer Engineering Dep.
Electric Charge Objective: 1. To review and learn the properties of electric charge 2. To learn how to compute the force between two point charges (Coulomb‘s law). 3. To learn how to compute the force on a point charge due many point charges. Electric Charge (Q): Electric charge is one of the basic quantities in nature. An object is made up of fundamental building blocks named atoms; each is composed of three main particles: electron, proton, and neutron. A proton has a positive (+) charge, and an electron has a negative (-) charge. The neutron has zero net electric charge (0). +
Electron (-) Proton (+) Neutron (0)
+ + -
A positive charge is a deficiency of electrons and a negative charge is an excess of electrons. It is also important to note that only electrons are transferred from object to object, an example of this transfer is friction. The object with equal amount of electrons and protons is electrically neutral. Quantization of Charge The electric charge is said to be quantized: the charge of an object is an integer multiple of the elementary charge, usually denoted e, which is the electric charge carried by a single proton, or equivalently, the negative of the electric charge carried by a single electron. Any electric charge, Q, occurs as integer multiples of the elementary charge e: . N is the number of elementary charges, and e is the elementary charge.
Circuits (MTE 120) (Spring 2010)
The Unit of Electric Charge A big amount of charge is used as a standard unit for charge. This unit is called the Coulomb and is given the symbol C. One coulomb is equal to 6.24 × 1018 elementary charges (e).
Note: • The smallest amount of free charge is the charge of the e=1.6×10-19 C. •
In electrostatic the following sub-multiples are often used:
Mili-Coulomb Micro-Coulomb Nano-Coulomb Pico-Coulomb
=10 -3 C =10 -6 C =10 -9 C =10 -12 C
1 mC 1 C 1 nC 1 pC
Properties of Electric Charge •
•
Electric charge can neither be created nor destroyed, but it can be transferred from one object to another. The quantity of electric charge, the amount of positive charge minus the amount of negative charge in the universe, is always conserved. Two electrically charged objects exert a force on one another, called as electrostatic force: like charges repel and unlike charges attract each other.
2
Circuits (MTE 120) (Spring 2010)
Coulomb’s Law The magnitude of the electrostatic force of repulsion or attraction between two charges q1 and q2 is given by:
is the electrostatic constant:
,- is the permittivity constant:
| | | |
1 8.99 x 10) *. / 4" #$
#$ 8.85 x 10/
*.
r is the distance between the two charges q1 and q2
Remember: The force is a vector quantity and its direction is along the line between the two charges. Note: From the figure below: The force (F12) acting on q1 due to q2 is equal in magnitude and opposite in direction to the force (F21 )acting on q2 due to q1.
3
Circuits (MTE 120) (Spring 2010)
Example #1: What are the magnitude and the direction of the electrostatic force F12 acting on q1 due to q2.
Solution: The magnitude of the force:
00000
| | | | 2 x 10/ x 1 x 10/2 ) 8.99 x 10 x 5x 10/3 7.192 x 10/5 N
The direction of the force: Since both particles are positively charged, particle 1 is repelled by particle 2.Thus the direction of the electrostatic force F12 is in the negative direction of the x axis. Thus, the force F12 has the following magnitude and direction of the x axis:
7.192 x 10/5 N ,
180$
We can also write the F12 in unit-vector notation as
00000 7.192 x 10/5 N 8̂
4
Circuits (MTE 120) (Spring 2010)
Exercise #1 (homework) In the above example, what is the net force 0000000000 ,:;< acting on q2 due to q1? The Principle of Superposition The electrostatic force obeys the principle of superposition. If we have n charged particles, the net charge on any one of them, let us say particle 1, is given by the vector sum of the forces:
0000000000 00000 00000 00000 000000 ,:;< 3 5 . . . :
Example #2 What is the net force 0000000000 ,:;< acting on q1 due to q2 and q3?
Solution Step #1: Identify the direction of each force acting on q1 Because q1 and q2 have charge of same signs, q1 is repelled along the line between q1 and q2. Thus, force F12 is directed towards the negative x axis (1800) as shown in the figure below. Because q1 and q3 have charge of opposite signs, q1 is attracted toward q3 along the line between q1 and q3. Thus, force F13 is directed towards q3, at angle θ as shown in the figure below. It is a straight forward to calculate the angle θ :
3 θ tan/ A C 36.87$ 4 5
Circuits (MTE 120) (Spring 2010)
Step # 2: Calculate the magnitude (|F12| and |F13|) of each force acting on q1: 00000
00000
3
| | | | 1 x 10/ x 2 x 10/ ) 8.99 x 10 x 4x 10/3 | | |3 | 3
1.1238 x 10/) N
It is a straight forward to calculate the distance (r13) between q1 and q3: 3 E3 4 5 F
) 00000
3 8.99 x 10 x
1 x 10/ x 3 x 10/ 5x 10/3
1.079 x 10/) N
Step # 3: Calculate the net force by adding the forces as vectors: 0000000000 00000 00000 ,:;< 3
Since the forces 00000 GH 00000 3 are not directed along the same axis we CANNOT sum them up by combining their magnitude. Instead, we MUST add them as vectors: 00000 00000 Each force ( GH 3 has x and y components that can be calculated as follows:
3,I | 3 | cosM 1.079 x 10/) cos36.87$ 8.63 x 10/$ N
3,N | 3 | sinM 1.079 x 10/) N sin36.87$ 6.47 x 10/$ N ,I | | cos180 1.1238 x 10/) N 6
Circuits (MTE 120) (Spring 2010)
,N | 3 | sin180 0 N Now, sum up the components axis by axis: The sum of the x components: ,:;