Question

What is electrical resistivity? Derive its SI unit? In a series electrical circuit comprising a resistor made up of a metallic wire, the ammeter reads $100mA$. If the length of the wire is doubled, how will the current in the circuit change? Justify your answer.

Answer 1

Hint:Resistance of a material is the opposition or the hindrance that the material offers to the flow of current through it. The formula for the resistance in terms of the resistivity of a substance or a material is to be applied in order to find the SI unit of resistivity. The current is found out using the ohm’s law relation and the resistivity formula.

Complete step by step answer:
The above problem revolves around the concept of resistivity of a material. In order to know the concept of resistivity the concept of resistance needs to be known. The resistance of a conductor is the property by virtue of which it opposes the flow of charges through it. The electrical resistivity was introduced from the concept of the factors on which the resistance depends on.

Since, the resistance depends on the length of the conductor and area of the conductor, that is, it is directly proportional to the length and inversely proportional to the area and thus a new proportionality constant which was the quantity known as resistivity was established.

Thus, the electrical resistivity of a material was defined as the resistance of a conductor of that material, having unit length and unit area of cross-section or in other words it is said to be the resistance offered by the unit cube of the material of a conductor.

The SI unit of resistivity is derived in the following way:
As per the definition the resistance of a material in terms of resistivity is given as:
$R=\rho {\displaystyle \frac{l}{A}}$
By rearranging the terms we get the resistivity to be:
$\rho ={\displaystyle \frac{R\times A}{l}}$
By applying the units of R, A and l which are ohm, $m^2$and $m$ respectively we get:
 $\rho ={\displaystyle \frac{ohm(\mathrm{\Omega })\times m^2}{m}}$
On simplifying further we get:
$\rho =ohm-meter=\mathrm{\Omega }m$
Hence, the unit of resistivity is given as ohm-meter.

The Ohm’s law establishes the relation between the potential difference and the current that is flowing through a conductor. It states that the current flowing through the conductor is directly proportional to the voltage applied across it ends when the physical conditions like temperature and pressure remain constant. The equation for ohm’s law is given as follows:
$V=IR$
By rearranging the terms we get:
$\Rightarrow I={\displaystyle \frac{V}{R}}$ ---------($1$)
The question mentions that a resistor is connected in series to an ammeter and the current through the circuit or the current that the ammeter reads is given. We are asked to find the new value of resistance when the length of the wire it is made up of is doubled. Hence:

Given, $$I=100\times {10}^{-3}A$$. This value of current is the one that the ammeter reads when the original value of resistance was considered, that is, before doubling the resistance. Hence when we apply the ohm’s law equation from equation ($1$) then we get:
$\Rightarrow 100\times {10}^{-3}={\displaystyle \frac{V}{R}}$ -------($2$)
When the resistance is doubled then there will be a change in the current flowing through the circuit but the voltage remains the same as it is the fixed source voltage connected to the circuit. Hence, the ohm’s law equation becomes:
$I^{\prime }={\displaystyle \frac{V}{R^{\prime }}}$

Since, the resistance is doubled the change in resistance is:
$R^{\prime }=2R$
Thus the above equation becomes:
$I^{\prime }={\displaystyle \frac{V}{2R}}$
$\Rightarrow I^{\prime }={\displaystyle \frac{1}{2}}\times {\displaystyle \frac{V}{R}}$
The value for the ratio of voltage by current from equation () is substituted in the above equation to get:
$\Rightarrow I^{\prime }={\displaystyle \frac{1}{2}}\times 100\times {10}^{-3}$
On solving further simplification we get:
$\therefore I^{\prime }=50\times {10}^{-3}=0.05A$

Thus 0.05\,A$ is the change in current when the resistance is doubled.

Note: The potential difference even though connected in series in this case will not change and this is a common mistake which is made while solving problems of this type. The resistivity of material is also known as the specific resistance of the material not only depends on the nature of the material and also depends upon the physical conditions like temperature and pressure.