In this lecture we closely follow the textbook of D.Halliday, R.Resnick and J.Walker which is one of the best in the field.
Whenever there is a net flow of charge through some region, a current is said to exist. The current is the rate at which charge flows through this surface.
The electric current I in a conductor is defined as
where dQ is the charge that passes through a cross-section of the conductor in a time dt. The SI unit of current is the ampere (A), where 1 A = 1 C/s.
It is conventional to assign to the current the same direction as the flow of positive charge.
If the ends of a conducting wire are connected to form a loop, all points on the loop are at the same electric potential, and hence the electric field is zero within and at the surface of the conductor. Because the electric field is zero, there is no net transport of charge through the wire, and therefore there is no current.
The current in the conductor is zero even if the conductor has an excess of charge on it. However, if the ends of the conducting wire are connected to a battery, all points on the loop are not at the same potential. The battery sets up a potential difference between the ends of the loop, creating an electric field within the wire. The electric field exerts forces on the conduction electrons in the wire, causing them to move around the loop and thus creating a current. It is common to refer to a moving charge (positive or negative) as a mobile charge carrier. For example, the mobile charge carriers in a metal are electrons.
The average current in a conductor is related to the motion of the charge carriers through the relationship
where n is the density of charge carriers, q is the charge on each carrier, vd is the drift speed, and A is the cross-sectional area of the conductor.
In a classical model of electrical conduction in metals, the electrons are treated as molecules of a gas. In the absence of an electric field, the average velocity of the electrons is zero. When an electric field is applied, the electrons move (on the average) with a drift velocity vd that is opposite the electric field. We can think of the atom–electron collisions in a conductor as an effective internal friction (or drag force) similar to that experienced by the molecules of a liquid flowing through a pipe stuffed with steel wool. The energy transferred from the electrons to the metal atoms during collision causes an increase in the vibrational energy of the atoms and a corresponding increase in the temperature of the conductor.
The magnitude of the current density J in a conductor is the current per unit area:
The current density in a conductor is proportional to the electric field according to the expression
The proportionality constant s is called the conductivity of the material of which the conductor is made. The inverse of s is known as resistivity r (r=1/s). The above equation is known as Ohm’s law, and a material is said to obey this law if the ratio of its current density J to its applied electric field E is a constant that is independent of the applied field.
The resistance R of a conductor is defined either in terms of the length of the conductor or in terms of the potential difference across it:
where l is the length of the conductor , s is the conductivity of the material ofwhich it is made, A is its cross-sectional area, DV is the potential difference across it, and I is the current it carries.
One can express the resistance of a uniform block of material as
The SI unit of resistance is volts per ampere, which is defined to be 1 ohm (Ω); that is, 1 Ω = 1 V/A. If the resistance is independent of the applied potential difference, the conductor obeys Ohm’s law.
Most electric circuits use devices called resistors to control the current level in the various parts of the circuit.
Every ohmic material has a characteristic resistivity that depends on the properties of the material and on temperature.
The resistivity of a conductor varies approximately linearly with temperature according to the expression
where a is the temperature coefficient of resistivity and r0 is the resistivity at some reference temperature T0 .
There is a class of metals and compounds whose resistance decreases to zero when they are below a certain temperature Tc , known as the critical temperature. These materials are known as superconductors.
If a potential difference DV is maintained across a resistor, the power, or rate at which energy is supplied to the resistor, is
Because the potential difference across a resistor is given by DV = IR, we can express the power delivered to a resistor in the form
The electrical energy supplied to a resistor appears in the form of internal energy in the resistor.
A constant current can be maintained in a closed circuit through the use of a source of emf (electromotive force), which is a device (such as a battery or generator) that produces an electric field and thus may cause charges to move around a circuit. The resistor represents a load on the battery because the battery must supply energy to operate the device.
The current depends on both the load resistance R external to the battery and the internal resistance r.
If R is much greater than r, as it is in many real-world circuits, we can neglect r.
If two or more resistors can connected together one after the other, they are said to be in series. In a series connection, all the charges moving through one resistor must also pass through the second resistor. For a series combination of resistors, the currents in the two resistors are the same because any charge that passes through R1 must also pass through R2 .
The equivalent resistance of three or more resistors connected in series is
This relationship indicates that the equivalent resistance of a series connection of resistors is always greater than any individual resistance.
Consider two resistors connected in parallel, that is, corresponding ends together. When resistors are connected in parallel, the potential differences across them are the same.
The equivalent resistance of resistors in parallel is given by
We can see from this expression that the equivalent resistance of two or more resistors connected in parallel is always less than the least resistance in the group.
Household circuits are always wired such that the appliances are connected in parallel. Each device operates independently of the others so that if one is switched off, the others remain on. In addition, the devices operate on the same voltage.
A device that measures current is called an ammeter. The current to be measured must pass directly through the ammeter, so the ammeter must be connected in se ries with other elements in the circuit. When using an ammeter to measure direct currents, you must be sure to connect it so that current enters the instrument at the positive terminal and exits at the negative terminal.
A device that measures potential difference is called a voltmeter. The potential difference between any two points in a circuit can be measured by attaching the terminals of the voltmeter between these points without breaking the circuit. Again, it is necessary to observe the polarity of the instrument. The positive terminal of the voltmeter must be connected to the end of the resistor that is at the higher potential, and the negative terminal to the end of the resistor at the lower potential.