Current Electricity

Electric Potential Difference
Electric Potential
Moving a positive test charge against the direction of an electric field is like moving a mass upward within Earth's gravitational field. Both movements would be like going against nature and would require work by an external force. This work would in turn increase the potential energy of the object. On the other hand, the movement of a positive test charge in the direction of an electric field would be like a mass falling downward within Earth's gravitational field. Both movements would be like going with nature and would occur without the need of work by an external force. This.motipn would result in the loss of potential energy. Potential energy is the stored energy of position of an object and it is related to the location of the object within a field. We will introduce the concept of electric potential and relate this concept to the potential energy of a positive test charge at various locations within an electric field.
Electric potential energy is dependent upon at least two types of quantities:
(1) Electric charge - a property of the object experiencing the electrical field, and
(2) Distance from source - the location within the electric field
While electric potential energy has a dependency upon the charge of the object experiencing the electric fields, electric potential is purely location dependent. Electric potential is the potential is purely location dependent. Electric potential is the potential energy per charge.
Electric Potential = PE / Q
The concept of electric potential is used to express the affect of an electric field of a source in terms of the location within the electric field. A test charge with twice the quantity of charge would possess twice the potential energy at a given location; yet its electric potential at that location would be the same as any other test charge. A positive test charge would be at a high electric potential when held close to a positive source charge and at a lower electric potential when held further away. In this sense, electric p9tential becomes simply a property of the location within an electric field
Electric Potential Difference
Consider the task of moving a positive test charge within a uniform electric field from location A to location B as shown in the diagram at the right. In moving the charge against the electric field from location A to location B, work will have to be done on the charge by an external force. The work done on the charge changes its potential energy to a higher value; and the amount of work which is done is equal to the change in the potential energy. As a result of this change in potential energy, there is also a difference in electric potential between locations A and B. This difference in electric potential is represented by the symbol V and is formally referred to as the electric potential difference.
By definition, the electric potential difference is the difference in electric potential (V) between the final and the initial location when work is done upon a charge to change its potential energy. In equation form, the electric potential difference is
V = VB - VA = Work / Charge = PE / Charge
The standard metric unit on electric potential difference is the volt, abbreviated V and named in honor of Alessandra Volta. One Volt is equivalent to one Joule per Coulomb .. Because electric potential difference is expressed in units of volts, it is sometimes referred to as the voltage.
Electric Current
As a physical quantity, current is the rate at which charge flows past a point on a circuit. As depicted in the diagram below, the current in a circuit can be determined if the quantity of charge Q passing through a cross section of a wire in a time t can. be measured. The current is simply the ratio of the quantity of charge and time.
Current is a rate quantity. In every case of a rate quantity, the mathematical equation involves some quantity over time. Thus, current as a rate quantity would be expressed mathematically as
Current = I = Q / t
Note that the equation above uses the symbol I to represent the quantity current.
The standard metric unit for current is the ampere. Ampere is often shortened to Amp and is abbreviated by the unit symbol A. A current of 1 ampere means that there is 1 coulomb of charge passing through a cross section of a wire every 1 second.
1 ampere = 1 coulomb / 1 second
Conventional Current Direction
The particles which carry charge through wires in a circuit are mobile electrons. The electric field direction within a circuit is by definition the direction which positive test charges are pushed. Thus, these negatively charged electrons move in the direction opposite the electric field. But while electrons are the charge carriers in metal wires, the charge carriers in other circuits can be positive charges, negative charges or both. In fact, the charge carriers in semiconductors, street lamps and fluorescent lamps are simultaneously both positive and negative charges traveling in opposite directions.
The direction of an electric current is by convention the direction in which a positive charge would move. Thus, the current in the external circuit is directed away from the positive terminal and toward the negative terminal of the battery. Electrons would actually move through the wires in the opposite direction. Knowing that the actual charge carriers in wires are negatively charged electrons may make this convention seem a bit odd and outdated. Nonetheless, it is the convention which is used world wide and one that a student of physics can easily become accustomed to.
The Nature of Charge Flow
Why does the light in a room or in a flsshlight light immediately after the switched is turned on? Wouldn't there be a noticeable time delay before a charge carrier moves from the switch to the light bulb filament? The answer is NO! and the explanation of why reveals a significant amount about the nature of charge flow in a circuit.
charge carriers in the wires of electric circuits are electrons. These electrons are simply supplied by the atoms of copper (or whatever material the wire is made of) within the metal wire. Once the switch is turned to on, the circuit is closed and there is an electric potential difference is established across the two ends of the externai circuit. The electric field signal travels at nearly the speed of light to all mobile electrons within the circuit, ordering them to begin marching. As the signal is received, the electrons begin moving along a zigzag path in their usual direction. Thus, the flipping of the switch causes an immediate response throughout every part of the circuit, setting charge carriers everywhere in motion in the same net direction. While the actual motion of charge carriers occurs with a slow speed, the signal which informs them to start moving travels at a fraction of the speed of light.
The electrons which light the bulb in a flashlight do not have to first travel from the switch through 10 cm of wire to the filament. Rather, the electrons which light the bulb immediately after the switch is turned to on are the electrons which are present in the filament itself. As the switch is flipped, all mobile electrons everywhere begin marching; and it is the mobile electrons present in the filament whose motion are immediately responsible for the lighting of its bulb. As those electrons leave the filament, new electrons enter and become the ones which are responsible for lighting the bulb. The electrons are moving together much like the water in the pipes of a home move. When a faucet is turned on, it is the water in the faucet which emerges from the spigot. One does not have to wait a noticeable time for water from the entry point to your home to travel through the pipes to the spigot. The pipes are already filled with water and water everywhere within the water circuit is set in motion at the same time.
Power
Electric circuits are designed to serve a useful function. The mere movement of charge from terminal to terminal is of little use if the electrical energy possessed by the charge is not transformed into another useful form. To equip a circuit with a battery and a wire leading from positive to negative terminal without an electrical device (light bulb, beeper, motor, etc.) would lead to a high rate of charge flow. Such a circuit is referred to as a short circuit. With charge flowing rapidly between terminals, the rate at which energy would be consumed would be high. Such a circuit would heat the wires to a high temperature and drain the battery of its energy rather quickly. When a circuit is equipped with a light bulb, beeper, or motor, the electrical energy supplied to the charge by the battery is transformed into other forms in the electrical device.
A light bulb, beeper and motor are generally referred to as a load. In a light bulb, electrical energy is transformed into useful light energy (and some non-useful thermal energy). In a beeper, electrical energy is transformed into sound energy. And in a motor, electrical energy is transformed into mechanical energy.
Power is the rate at which electrical energy is supplied to a circuit or consumed by a load. The electrical energy is supplied to the load by an energy source such as an electrochemical cell
Like current, power is a rate quantity. Its mathematical formula is expressed on a per time basis.
Power = Work Done on Charge / Time = Energy Consumed by Load / Time
whether the focus is the energy gained by the charge at the energy source or the energy lost by the charge at the load, electrical power refers to the rate at which the charge changes its energy. In an electrochemical cell (or other energy source), the change is a positive change (i.e., a gain in energy) and at the load, the change is a negative change (i.e., a loss in energy). Thus, power is often referred to as the rate of energy change and its equation is expressed as the energy change per time. Like mechanical power, the unit of electrical power is the watt, abbreviated W. (Quite obviously, it is important that the symbol W as the unit of power not be confused with the symbol W for the quantity of work done upon a charge by the energy source.) A watt of power is equivalent to the delivery of 1joule of energy every second. In other words:
1 watt = 1 joule / second
The kilowatt-hour
A kilowatt is a unit of power and an hour is a unit of time. So a kilowatt • hour is a unit of Power • time. If Power = Energy / time, then Power • time = Energy. So n unit of power • time is a unit of energy. The kilowatt • hour is a unit of energy. When an electrical utility company charges a household for the electricity which they used, they are charging them for electrical energy.
It is a common misconception that the utility company provides electricity in the form of charge carriers or electrons. The fact is that the mobile electrons which are in the wires of our homes would be there whether there was a utility company or not. The electrons come with the atoms that make up the wires of our household circuits. The utility company simply provides the energy which causes the motion of the charge carriers within the household circuits. And when they charge us for a few hundred kilowatt-hours of electricity, they are providing us with an energy bill.
Calculating Power
The relationship between power, current and electric potential difference can be derived by combining the mathematical definitions of power, electric potential difference and current. Power is the rate at which energy is added to or removed from a circuit by a battery or a load. Current is the rate at which charge moves past a point on a circuit. And the electric potential difference across the two ends of a circuit is the potential energy difference per charge between those two points. In equation form:
P = V * Q / t
In the equation above, there is a Q in the numerator and a t in the denominator. This is simply the current; and as such, the equation can be rewritten as
P = V * I
The electrical power is simply the product of the electric potential difference and the current.
Electrical Resistance
Journey of a Typical Electron
An electron's journey through a. circuit can be described as a zigzag path which results from countless collisions with the atoms of the conducting wire. Each collision results in the alteration of the path, thus leading to a zigzag type motion. While the electric potential difference across the two ends of a circuit encourages the flow of charge, it is the collisions of charge carriers with atoms of the wire that discourages the flow of charge. Different types of atoms offer a different degree of hindrance to the flow of the charge carriers which pass through it.
The journey of an electron through an external circuit involves a long and slow zigzag path which is characterized by several successive losses in electric potential. Each loss of potential is referred to as a voltage drop. Accompanying this voltage drop is a voltage boost occurring within the internal circuit - for instance, within the electrochemical cell.
Resistance
An electron traveling through the wires and loads of the external circuit encounters resistance. Resistance is the hindrance to the flow of charge
Variables Affecting Electrical Resistance
The total amount of resistance to charge flow within a wire of an electric circuit is affected by some clearly identifiable variables
1. the total length of the wires will affect the amount of resistance.
The longer the wire, the more resistance that there will be. There is a direct relationship between the amount of resistance encountered by charge and the length of wire it must traverse. After all, if resistance occurs as the result of collisions between charge carriers and the atoms of the wire, then there is likely to be more collisions in a longer wire. More collisions means more resistance.
2. The cross-sectional area of the wires will affect the amount of resistance.
Wider wires have a greater cross-sectional area. Water will flow through a wider pipe at a higher rate than it will flow through a narrow pipe. This can be attributed to the lower amount of resistance which is present in the wider pepe. This can be attributed to the lower amount of resistance which is present in the wider pipe. In the same manner, the wider the wire, the less resistance that there will be to the flow of electric charge. When all other variables are the same, charge will flow at higher rates through wider wires with greater cross-sectional areas then through thinner wires.
3. the material that a wire is made of.
Not all materials are created equal in terms of their conductive ability. Some materials are better conductors then other and offer less resistance to the flow of charge. Silver is one of the best conductors but is never used in wires of household circuits due to its cost. Copper and aluminum are among the least expensive materials with suitable conducting ability of a material is often indicated by its resistivity. The resistivity of a material is dependent upon the material's electronic structure and its temperature. For most (but not all) materials, resistivity increases with increasing temperature. The table below lists resistivity values for various materials at temperatures of 20 degrees Celsius.
Material
Resistivity
(ohm•meter)
Silver
1.59 x 10-8
Copper
1.7 x 10-8
Gold
2.4 x 10-8
Aluminum
2.8 x 10-8
Tungsten
5.6 x 10-8
Iron
10 x 10-8
Platinum
11 x 10-8
Lead
22 x 10-8
Nichrome
150 x 10-8
Carbon
3.5 x 105
Polystyrene
107 - 1011
Polyethylene
108 - 109
Glass
1010 - 1014
Hard Rubber
1013
As seen in the table, there is a broad range of resistivity values for various materials. Those materials with lower resistivities offer less resistance to the flow of charge; they are better conductors. The materials shown in the last five rows of the above table have such high resistivity that they would not even be considered to be conductors.
Ohm's Law
The predominant equation which pervades the study of electric circuits is the equation
V = I * R
In words, the electric potential difference between two points on a circuit (V) is equivalent to the product of the current between those two points (I) and the total resistance of all electrical devices present between those two points (R). Often referred to as the Ohm's law equation, this equation is a powerful predictor of the relationship between potential difference, current and resistance.
Ohm's Law as a Predictor of Current
The Ohm's law equation can be rearranged and expressed as
I = V / R
The current in a circuit is directly proportional to the electric potential difference impressedacross its ends and inversely proportional to the total resistance offered by the external circuit. The greater the battery voltage (i.e., electric potential difference), the greater the current. And the greater the resistance, the less the current. Charge flows at the greatest rates when the battery voltage is increasedand the resistance is decreased. In fact, a twofold increase in the battery voltage would lead to a twofold increase in the current (if an other factors are kept equal). And an increasein the resistance of the load by a factor of two would cause the current to decrease by a factor of two to one-half its original value
Because the current in a circuit is affected by the resistance, resistors are often used in the circuits of electrical appliances to affect the amount of current which is present in its various components. By increasing or decreasing the amount of resistance in a particular branch of the circuit, a manufacturer can increase or decrease the amount of current in that branch. Kitchen appliances such as electric mixers and light dimmer switches operate by altering the current at the load by increasing or decreasing the resistance of the circuit. Pushing the various buttons on an electric mixer can change the mode from mixing to beating by reducing the resistance and allowing more current to be present in the mixer. Similarly, turning a dial on a dimmer switch can increase the resistance of its built-in resistor and thus reduce the current.
Circuit Connections
Circuit Symbols and Circuit Diagrams
Electric circuits, whether simple or complex, can be described in a variety of ways. An electric circuit is commonly described with mere words. Saying something like "A light bulb is connected to a D-cell" is a sufficient amount of words to describe a simple circuit.
A final means of describing an electric circuit is by use of conventional circuit symbols to provide a schematic diagram of the circuit and its components. Some circuit symbols used in schematic diagrams are shown below.
A single cell or other power source is represented by a long and a short parallel line. A collection of cells or battery is represented by a collection of long and short parallel lines. In both cases, the long line is representative of the positive terminal of the energy source and the short line represents the negative terminal. A straight line is used to represent a connecting wire between any two components of the circuit. An electrical device which offers resistance to the flow of charge is generically referred to as a resistor and is represented by a zigzag line. An open switch is generally represented by providing a break in a straight line by lifting a portion of the line upward at a diagonal.
Circuit Connections
Two Types of Connections
When there are two or more electrical devices present in a circuit with an energy source, there are a couple of basic means by which to connect them. They can be connected in series or connected in parallel. Suppose that there are three light bulbs connected together in the same circuit. If -connected in series, then they are connected in such a way that an individual charge would pass through each one of the light bulbs in consecutive fashion. When in.series, cha!:.gepasses through every light bulb. If connected in parallel, a single charge passing through the external circuit would only pass through one of the light bulbs. The light bulbs are placed within a separate branch line, and a charge traversing the external circuit will pass through only one of the branches during its path back to the low potential terminal. The means by which the resistors are connected will have a major affect upon the overall resistance of the circuit, the total' current in the circuit, and the current in each resistor..
Parallel Circuits
When all the devices are connected using parallel connections, the circuit is referred to as a parallel circuit. In a parallel circuit, each device is placed in its own separate branch. The presence of branch lines means that there are multiple pathways by which charge can traverse the external circuit. Each charge passing through the loop of the external circuit will pass through a single resistor present in a single branch. When arriving at the branching location or node, a charge makes a choice as to which branch to travel through on its journey back to the low potential terminal.
Current
In a parallel circuit, charge divides up into separate branches such that there can be more current in one branch than there is in another. Nonetheless, when taken as a whole, the total amount of current in aUthe branches when added together is the. same as the amount of current at locations outside the branches. The rule that current is everywhere the same still works, only with a twist. The current outside the branches is the same as the sum of the current in the individual branches. It is still the same amount of current, only split up into more than one pathway.
In equation form, this principle can be written as
I total = I1 + I2 + I3 + ...
Where I total is the total amount of current outside the branches (and in the battery) and I1, I2, and I3 represent the current in the individual branches of the circuit.
This is illustrated in the examples. In the examples a new circuit symbol is introduced - the letter A enclosed within a circle. this is the symbol for an ammeter - a device used to measure the current at a specific point. An ammeter is capable of measuring the current while offering negligible resistance to the flow of charge.
Diagram A displays two resistors in parallel with nodes at point A and point B. Charge flows into point A at a rate of 6 amps and divides into two pathways - one through resistor 1 and the other through resistor 2. The current in the branch with resistor 1 is 2 amps and the current in the branch with resistor 2 is 4 amps. After these two branches meet again at point B to form a single line, the current again becomes 6 amps. Thus we see the principle that the current outside the branches is equal to the sum of the current in the individual branches holds true.
I total = I1 + I2
6 amps = 2 amps + 4 amps
Diagram B above may be slightly more complicated with its three resistors placed in parallel. Four nodes are identified on the diagram and labeled A, B, C and D. Charge flows into point A at a rate of 12 amps and divides into two pathways one one passing through resistor 1 and the other heading towards point B (and resistors 2 and 3). The 12 amps of current is divided into a 2 amp pathway (through resistor 1) and a 10 amp pathway (heading toward point B). At point B, there is further division of the flow into two pathways - one through resistor 2 and the other through resistor 3. The 10 amps of current approaching point B is divided into a 6 amp pathway (through resistor 2) and a 4 amp pathway (through resistor 3). Thus, it is seen that the current values in the three branches are 2 amps, 6 amps and 4 amps and that the sum of the current values in the ind~vidual branches is equal to the current outside the branches.
I total = I1 + I2 + I3
12 amps = 2 amps + 6 amps + 4 amps
A flow analysis at points C and D can also be conducted and it is observed that the surn of the flow rates heading into these points is equal to the flow rate which is found immediately beyond these points.
Equivalent Resistance
The actual amount of current always varies inversely with the amount of overall resistance. There is a clear relationship between the resistance of the individual resistors and the overall resistance of the collection of resistors.
This is the concept of equivalent resistance. The equivalent resistance of a circuit is the amount of resistance which a single resistor would need in order to equal the overall effect of the collection of resistors which are present in the circuit. For parallel circuits, the mathematical formul'a for computing the equivalent resistance (Req) is
1 / Req = 1 / Rl + 1 / R2 + 1 / R3 + ...where R1 R2, and R3 are the resistance values of the individual resistors which are connected in parallel. For instance, consider the application of the equation to the one case below.
Combination Circuits When all the devices in a circuit are connected by series connections, then the circuit is referred to as a series circuit. When all the devices in a circuit are connected by parallel connections, then the circuit is referred to as a parallel circuit. A third type of circuit involves the dual use of series and parallel connections in a circuit; such circuits are referred to as compound circuits or combination circuits. The circuit depicted at the right is an example of the use of both series and parallel connections within the same circuit. In this case, light bulbs A and Bare connected by parallel connections and light bulbs C and D are connected by series connections. This is an example of a combination circuit.
When analyzing combination circuits, it is critically important to have a solid understanding of the concepts which pertain to both series circuits and parallel circuits. Since both types of connections are used in combination circuits, the concepts associated with both types of circuits apply to the respective parts of the circuit. The main concepts associated with series and parallel circuits are organized in the table below.
Series Circuits
Parallel Circuits
The current is the same in every resistor; this current is equal to that in the bettery.
The voltage drop is the same across each parallel branch.
The sum of the voltage drops across the individual resistors is equal to the voltage rating of the battery.
The sum of the current in each individual branch is equal to the current outside the branches.
The overal resistance of the collection of resistors is equal to the sum of the individual resistance values, R
tot = R1 + R2 + R3 + ...
The equivalent or overall resistance of the collection of resistors is given by the equation
I/R eq = 1/R1 + 1/R2 + 1/R3 ...
Each of the above concepts has a mathematical expression. Combining the athematical expressions of the above concepts with the Ohm's law equation (V = I • R) allows one to conduct a complete analysis of a combination circuit.