**Electric current**

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, v* _{d} *is the drift speed, and

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 v* _{d} *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, D*V *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 *r*_{0} is
the resistivity at some reference
temperature *T*_{0} .

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 D*V *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 D*V **=* *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 *R*_{1} must also pass
through *R*_{2} .

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.