Question 3: State Ohm’s law. Discuss its scope and validity (a) Discuss resistivity and conductivity of a material (b) How does resistance change with temperature?

George Simon Ohm experimentally established a fundamental relation between the voltage and the current in a metallic conductor, called Ohm’s Law. It states,   “The current I in a metal wire is directly proportional to the voltage V applied across its ends, provided the physical state such as the temperature of the conductor is kept constant.”

Mathematically,

V = IR

Here R is a constant known as the resistance of the conducting material. Resistance depends upon the nature (such as which element the conductor is), dimensions (such as the cross-sectional area is small or large) and physical state (such as temperature) of the conductor.

Scope and validity of the law

For conductor that obeys Ohm’s Law are called ohmic conductor. The graph of current I as a function of voltage V is a straight line passing through the origin in ohmic conductors. Reversing the voltage, the current also reverses. The slope of the line is given by

However, it is noted that Ohm’s law is not valid for all conducting material. Certain conductors are found not obeying Ohm’s law. They are called non-ohmic material. For non-ohmic conductors, the I-V graph is not a straight line or it may not pass through the origin. They may also conduct poorly or not at all when the voltage is reversed. Some of the examples of non-ohmic conductors are described below.

(1) Filament Bulb:

When filament bulb is turned on, temperature of the filament increases, thereby increasing the resistance which causes a current decrease. The graph bends as V increases indicating a given change of V causes a corresponding smaller change in the current at large values of V.

(2) Thermistor

Thermistors are made of semi-conductors and the I – V graph bends upward, showing more current with correspondingly low voltage increase. The covalent bonds in semi-conductors break due to increase of thermal energy with the passage of current, producing a current pulse.

(3) Semiconductor diode

The I-V graph of semi conductor diodes shows the current passes when the potential difference is applied in one direction and is almost zero in the opposite direction. A semiconductor has a small resistance when the PD is applied in one way and a large resistance when it is reversed. This one way property makes it useful as a rectifier for changing AC to DC. The graph does not pass through the origin.

### Resistivity or specific resistance of a material

“Resistivity is the resistance of a conductor having unit cross section and unit length.” It is denoted by ρ

Explanation

Resistivity of a material depends on the nature, dimensions and temperature of the material. At constant temperature, R is found to vary directly with its length L and inversely with its cross sectional area A.

ρ  is the resistivity or specific resistance of the material. Its unit is (Ω .m2/m) = Ω -m.

We note that

1. Resistivity of metals and alloys are small. Therefore, metals and alloys are good conductor of current.
2. Resistivity of insulators is very large. Therefore, they almost do not conduct any current.
3. Resistivity of semi-conductors is in between the conductors and insulators.
4. Resistivity of different substances varies over a wide range from 10-4 Ω m to 104 Ω m.

### Conductivity of a substance

Conductivity, like resistivity, describes the electrical properties of the material. It is reciprocal of resistivity and is denoted by σ.

σ = 1/ρ = L/RA

Its unit is moh/m or Siemens /m. Greater conductivity, less resistivity, and a good conductor the material is.

### Effect of temperature on resistance

It has been found that in case of metals, the resistance increases with increase in the temperature over a wide range below and above 00C. The increase is slight and nearly linear. Let’s suppose,

Resistance of the metal at reference temperature, say, 00C = R0

Resistance of the metal at some other temperature like t0C = Rt

Therefore, change in resistance = Rt-R0 = ΔR

Change in temperature of the metal = t-0 = t0C = t

Experiments have shown that,

This implies that ΔR ∝ R0Δt
OR ΔR = αR0Δt
⟹ Rt – R0 = αR0Δt
Rt = R0 + αR0Δt
Rt = R0 (1+αΔt)

Here α is the constant of proportionality and is called temperature coefficient of resistance. Its value ( α = ΔR/R0Δt) can be calculated from the above equation. α can be defined as the change in resistance per unit resistance per degree rise in temperature, based upon the reference temperature at 00 C. Its unit is K-1.

#### Temperature and Resistivity (also temperature coefficient of resistivity):

The resistivity or specific resistance ρ of most solids increases linearly with increasing temperature.
Suppose
ρ0 is resistivity at t1 = 00 C
ρt is resistivity at t2 = t0C
Change in resistivity = Δρ = ρt – ρ0
It is found experimentally that, Δρ ∝ ρ0 and Δt and hence,
Δρ = αρ0Δt ⇒ ρt – ρ0 = αρ0Δt ⇒ ρt = ρ0(1 + αΔt)
Also from the above equation Here α is the proportionality constant called temperature coefficient of resistivity. It can be defined as “the fractional change in resistivity per degree rise in temperature of the material”. Its unit is K-1. It helps us to distinguish different material from one another.

The temperature coefficient of resistivity (α) is positive for metals.