Introduction And Ohm’s Law

CURRENT ELECTRICITY

Flow of electric charge constitutes electric current. For a given conductor, if '$\delta$Q' charge flows through a cross-section of area A in time '$\delta$t', then the average electric current through the conductor is given as

I = $\frac{\delta Q}{\delta t}$ and its instantaneous value is $\frac{dQ}{dt}$.

MECHANISM OF CURRENT FLOW IN METALLIC CONDUCTOR

When an external potential difference is applied across a metallic conductor then an electric field is set up within the conductor.

Applied electric field $\to$ Force on electrons $\to$ drift of electrons

Due to the externally applied electric field electrons drift with an average velocity called drift velocity. This causes an electric current

Total charge crossing a cross-section in one second is equal to

I = neAv_d. Here Av_d is the volume of a cylinder of cross-section A length v_d and ne is charge density of charge carriers (e.g. electrons).

The current density is defined by J = I/A

Example1 : A steady current passes through a cylindrical conductor. Is there an electric field inside the conductor ?

Solution : Yes; No doubt under steady state conditions in electrostatics when a conductor is charged, electric field inside it is zero as metal is an equipotential surface. However when a potential difference is applied across a conductor and a steady current flows though it, the condition no longer remains static and there exists an electric field inside the conductor.

OHM'S LAW

It states that "the potential difference across a conductor is directly proportional to the current flowing through it at a given temperature".

$\Rightarrow$ At constant temperature $\frac{V}{I}=\text{constant(R)}$

the constant 'R' is called resistance of the conductor.

Resistivity ($\rho$) and conductivity ($\sigma$):

The resistance R of a given conductor is directly proptinal to length ($\ell$) and inversitional proptional cross-sectional area (A) such that R = $\rho$ $\frac{\ell }{A}$, where $\rho$ = resistivity of the material of the given conductor. Its S.I. unit is $\Omega$ m. Reciprocal of resistivity is called the electrical conductivity ($\sigma$) of the material, thus

$\sigma$ = $\frac{1}{\rho }$ = $\frac{\ell }{RA}$whereas reciprocal of resistance is called conductance of the given conductor. S.I. unit of conductivity $\sigma$ is ($\Omega$ - m)^-1 and is usually written as mho/m.

Temperature Dependence of Resistivity:

The conductivity of a metal decreases as its temperature is increased. Thus resistivity$\rho$increases with the rise in temperature. If $\rho$_T and r₀ represent the resistivities at temperatures T and T₀ respectively, then for small temperature variations,

_$\rho$T= $\rho$₀ [1 + $\alpha$(T - T₀)]

Where $\alpha$ is called the temperature coefficient of resistivity. The resistivity varies over a very wide range. For metals (good conductor)$\rho$$\approx$ 10^-8 $\Omega$-m and for insulators$\rho$$\approx$ 10¹⁷ $\Omega$-m

Semiconductors (silicon, germanium ) have intermediate value much smaller than insulator but much larger than metals. Temperature coefficient of resistivity is negative for semiconductors and positive for the metals. For superconductors resistivity is zero.

Thermistor:

A thermistor is a semiconductor electronic device in which the resistance decreases as its temperature increases. This is used as a thermometer.

The temperature coefficient of resistivity is negative for semiconductors, hence thermistors are usually prepared from oxides of various metals such as nickel, iron, cobalt and copper etc. A thermistor is used to detect small changes in temperature of the order of even 10^{-3 0}C.

Colour code for carbon Resistors:

The four bands indicate digit -1, digit-2, multiplier and tolerance respectively and the values of different colours are given in the following table.

Resistance code (in $\Omega$)

Sometimes the carbon resistor indicates only three bands and the tolerance is missing from the colour code. This means tolerance has to be taken as 20%.

Example2: Find the resistance of a carbon resistor if the colour code from left to right indicates brown, yellow, green and gold.

Solution: Use diagram

$\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$

1 4 x 10⁵ $\pm$5%

R = (14 x 10⁵ $\pm$ 5%) $\Omega$

= (1.4 x 10⁶ + 0.07 x 10⁶ ) $\Omega$

= (1.4 $\pm$ 0.07)M$\Omega$