Introduction
In this lesson, our attention will be on moving charges and the effects they produce on their surroundings, We will define electrical current, electrical resistance and capacitance, Ohm's law and Kirchoff's law, resistor series, and parallel circuits.
Electrical Current
A new equilibrium situation arises in electricity. In thermal phenomena, as particles move around and collide with each other, we say they reach thermal equilibrium when the temperature becomes the same across all parts of the system. In the case of electrical phenomena, the charges are the ones determining a situation of equilibrium. Charges move between two points when a potential difference (electrical potential energy per charge) exists between the two ends of a conductor. If the ends are at the same potential, there is no flow of charge through the conductor. The flow of charge rapidly stops if we do not insure that the potential difference is maintained between the ends of the conductor by supplying energy to the conductor through a battery or an electrical generator such as a Van der Graaff generator. The maximum potential difference given out by a battery is called electromotive force (emf), is symbolized by E, and is measured in volts by an instrument called a voltmeter. A device that can measure more than one electrical property (such as resistance, voltage, and current) is called a multimeter.
A battery is usually symbolized by a diagram such as that shown in Figure 14.1, where the two electrodes are charged with positive and negative charges, and the current, as the convention we agreed upon tells us, travels from the positive electrode out in the circuit toward the negative electrode.
The charges moving through the conductor are, in a majority of cases, the free electrons in the conductor; for this reason, they are also called conduction electrons. Positive charges moving through the conductor are usually positive ions because the protons are strongly bonded to the nucleus.
The flow of charge can be continuous and at a constant rate, and the current is called direct current, or dc. Or it can flow in alternating directions, and then the current is called alternative current, or ac.
Electrical Current
The electrical current is the amount of charge that passes through the cross-sectional area of the conductor per unit time.
where Δq is the total charge that passes through the area and Δt is the time considered. Electrical current is measured in coulombs per second, or amperes, A, named after Andrè-Marie Ampère (1775–1836).
History Note
Historically,itis considered that the flow of electrons is opposite to the electrical current, because in the beginning, it was believed that positive charges–and not electrons–determine the electrical current through a conductor.
Example
In a conducting wire, electrons move as shown in Figure 14.2. Determine the electric current if 108 electrons move through the cross-sectional area A in 1 ms.
Solution
First, convert all quantities to SI and then set your formulas to determine I.
The electron charge is:
qe = –1.6 · 10–19 C
And because we have a number of 108 electrons, the total charge is:
Δq = 108 · qe = 108 · (–1.6 · 10–19C) = –1.6 · 10–11C
Δt = 1 ms = 10–3s
I = ?
By the definition of the electrical current:
Because the current is created by electrons moving through the conductor, and considering the previous note, the current is taken to be opposite to the velocity v.
Electrical Resistance and Ohm's Law
Mechanical motion or heat transfer are sometimes associated with loss of energy (as, for example, with friction). The same situation is encountered in electricity. The same battery hooked up to different wires will establish currents of different values. Therefore, we can introduce the concept of resistance: electrical resistance.
Electrical resistances are measured in volts per ampere, and the unit is called an ohm (Ω) after Georg Ohm (1789–1854).
The electrical resistance varies with some factors, but for most materials, the ratio stays the same for a large range of currents and voltages. In these cases, the resistance is a constant, and this observation is known as Ohm's law:
= constant
Electrical resistance R varies with physical properties of the wires (length and cross-sectional area) and with the nature of the material (coefficient known as resistivity, p). The value of the resistivity helps classify materials into conductors, semiconductors, and insulators. Electrical conductors have small resistivities and conduct electrical current well (examples include: copper and silver), whereas insulators hinder the flow of charges (examples include: wood and Teflon). Semiconductors have a varying resistivity, and in certain conditions, they can behave as conductors and in other conditions as insulators. These are intrinsic properties of the materials and are highly dependent on the type of bonding between subatomic particles.
With this new insight, we can also define the electrical resistance in the following way. Electrical resistance R is proportional to the electrical resistivity, p, of the material and the length and it is inversely proportional to the cross-sectional area A.
Electrical Resistance
Electrical resistance is the ratio of the voltage v applied across the ends of a material to the current I established through the material.
Example
Determine the unit for the electrical resistivity.
Solution
We start with the definition and solve for resistivity by multiplying each side with A/L:
So, the emerging unit is
A few examples of the most encountered conductor materials and their resistivity are listed in Table 14.1.
Most diagrams containing a resistor are going to represent it in a manner similar to that shown in Figure 14.3.
Electrical Resistance
Electrical resistance R is proportional to the electrical resistivity, p, of the material and the length and is inversely proportional to the cross-sectional area A.
Electrical Capacitance
Wires are not the only electrical devices that perform a function when a potential difference is applied. Another important category of devices is capacitors.
Capacitors are systems composed of two conductors placed in proximity but not in contact. The space between the conductors is filled with insulating materials called dielectrics. The plates of the capacitor are charged with equal and opposite charges. An electrical field is established as shown in Figure 14.4.
The amount of charge that a capacitor can store on each plate is proportional to the voltage V applied and the capacitance C.
q = C· V
If we solve for the capacitance:
The capacitance is measured in SI, in coulombs per volt in a unit called a farad (F), named after Michael Faraday (1791–1867). Usual capacitances are in the range of microfaradds to picofarads.
Dielectrics placed in between the plates increase the capacitance and therefore the charge of the capacitor. They decrease the field established between the plates. A measure of this property is the dielectric constant K that represents the ratio of the field without and with dielectric. Therefore, this coefficient is unitless.
Similar to the dependence noted previously for electrical resistance, the electrical capacitance is dependent upon intrinsic properties of the dielectric and on the geometry of the plates. For a parallel plate capacitor, such as the one shown previously in Figure 14.4, the dependence is summarized in the expression:
The other constant is called permittivity of vacuum and is expressed by:
ε0 = 8.8 · 10–12C2/(N · m2)
Electrical Capacitance
The electrical capacitance is proportional to the dielectric constant · K and the area of the plates A and inversely proportional to the distance between the plates d.
Example
How does the increase in the distance between the two parallel plates of a capacitor affect capacitance?
Solution
If the distance between the plates increases two times, then we have an initial distance d0 and a final distance 2 · d0, and then we can compare the two capacitances:
Hence, the capacitance decreases by 2.
Table 14.2 lists some of most common insulators and shows the value of their dielectric constant.
Considering the definition of the capacitance and the few examples in the table, you can say that the capacitance will increase when a dielectric other than a vacuum is placed between the plates, and it increases elecby the value of the dielectric constant.
Kirchoff's Laws
Conservation of charge is expressed in the Kirchoff's laws. In order to introduce these laws, we must define an electrical circuit. An electrical circuit is a set of electrical consumers and electrical sources that form a closed path in which an electrical current can be established. A closed electrical circuit will have batteries or other electrical sources, resistors and capacitors, switches, and conductors. The point where two or more conductors meet is called a junction, and a closed path is called a loop.
The junction law expresses the fact that the total current entering a junction has to be equal to the current leaving the junction. In other words, the charge arriving at the junction should be the same as the amount of charge leaving the junction. In a closed circuit, the voltage difference at the ends of the circuit provides the energy for charge circulation through the circuit.
In a closed circuit, the total potential drop on the consumers is equal to the potential supplied from the sources. And again we have a rule very similar to the conservation of energy we worked with in mechanics and heat. The closed circuit is the analog of the previous mechanical and thermal isolated systems.
The two laws provide us with sufficient equations to be able to solve different characteristics of a circuit.
Example
In a simple circuit, we have two identical light bulbs and two identical batteries connected to each other as in Figure 14.5. Considering the electrical resistance is 100 Ω, find out the emf of the batteries and the currents through all the branches of the circuit. The current through the top resistor is 2 mA.
Solutions
First, determine the given quantities and then set up the equation to solve for the unknowns. The currents are:
R = 100 Ω
I1 = 2 mA
E = ?
I2 and I3 = ?
According to the convention of positive-charge motion determining the direction of the current, we will choose a convenient direction for the three currents in this circuit. When we analyze the results regarding the currents, we are able to see if our choice was valid for this circuit by looking at the sign. If the result for a current is negative, the current flows in the opposite direction to the one considered. As an example: If the result in the current below will be I1 = –2A, then the direction of the real current is toward the left, not the right as we considered.
Using Kirchoff's laws, we consider first the junctions A and B. They are equivalent in this case, as they connect the same branches of the circuit. Let's look at the currents entering junction A: I2 and I3. And exiting junction A: I1. Then the first law is:
- I2 + I3 = I1
And it is the same for the junction B.
Now we can see there are three loops in the circuit, and we will consider a direction for looping (which is completely arbitrary, and therefore, I have decided to use clockwise). First, one will be considering the upper half of the circuit:
The second loop is the lower part of the circuit:
And lastly, we can disregard the middle part and have a circuit composed only of the outside conductors and electrical components.
Out of the total number of loops N, only N – 1 are independent (as a condition, we need to be able to write independent equations), in this case 3 loops – 1 = 2 loops.
We will look at Figure 14.6 and 14.7.
For Figure 14.6, there is only a resistor (batteries have no resistance considered in this example, and they run at maximum energy, emf).
- I1 · R + I3 · 0 = E
For Figure 14.7, there is again one resistor but two batteries. Around the circuit, the two batteries are supplying opposite voltages to the loop.
- I2 · 0 + I3 · R = E – E
And so, we have three equations and three unknowns, and we can solve for each of the unknowns. From the third of these equations (Equation 3), we can solve for I3.
I2 · 0 + I3 · R = E – E = 0
I3 · R = 0
Because R is a nonzero resistor, the only possibility is for the current to be zero:
I3 = 0 A
Next, we use Equation 2, and solve for E:
I1 · R + I3 · 0 = E
I1 · R + 0 = E
I1 · R = E
2 mA · 100 Ω = E
E = 200 mV
And lastly, we can use the junction law to figure the current through the middle branch of the circuit.
I2 + I3 = I1
I2 + 0 = 2 mA
I2 = 2 mA
All currents are positive, so the direction considered in the diagram was correct. As you can see, no current runs through the lower branch of the battery.
Series and Parallel Resistor Circuits
The previous examples show, as practice does, that circuits usually have more than one component. The question is, can we find a way to simplify calculations using Kirchoff's laws? The answer is yes, and the elements can be coupled together into two types of connections: series and parallel connections. We will now investigate the series and parallel connections for electrical resistances.
A series connection has all resistors connected in a line and the same amount of current passes through each resistor as shown in Figure 14.11.
If the three resistors are of different resistances, the voltage drop on each of them is different (they oppose differently charge motion). According to Ohm's law, the voltage is:
V1 = R1 · I, V2 = R2 · I, V3 = R3 · I
The voltage drop on the entire line of resistors has to be the same as the sum of all the voltages in the previous equation:
V = V1 + V2 + V3
V = R1 · I + R2 · I + R3 · I
V = (R1 + R2 + R3) · I
If we imagine the entire circuit represented by just one resistor that has the same effect on the current as do the three, we can consider the previous equation to be written as:
V = Rs · I
If we put together the last equations, we have:
(R1 + R2 + R3) · I = Rs · I
R1 + R2 + R3 = Rs
The resistance of an equivalent circuit of resistors connected in series is equal to the sum of the resistances. More generally, the previous equation can be written as:
where i is the number of resistors connected in series.
A parallel connection has multiple resistors hooked to the same junction (see Figure 14.12), and the current splits, taking the path of least resistance, literally! The same equations of currents and voltages in junctions and loops can be set to determine the equivalent parallel connection resistance.
The current in the resistors is different and dependent on the value of the resistors, but the voltage drop between the ends of each resistance is the same, as the voltage on the parallel connection, V.
The resistance of an equivalent circuit of resistors connected in parallel is equal to the sum of the inverse of the resistances. More generally, the previous equation can be written as:
where i is the number of resistors connected in parallel.
Example
Two resistors, R1 = 100 mΩ and R2 = 50 mΩ are connected first in series and then in parallel to the same battery. When is the equivalent resistance larger: in the series or in the parallel connection?
Solution
To answer this problem, set up the equations for each of the series and parallel connections for the two resistors.
R1 + R2 = Rs
100 + 50 = Rs
Rs = 150 mΩ
Rp = 33 mΩ
Rs > Rp
Actually, the above result is true for all resistors, and the parallel connection will always yield a smaller value of the equivalent resistance than does a series connection made up of the same resistors as in the parallel one.
Practice problems of this concept can be found at: Electrical Current Practice Questions
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