Question

Two metallic wires $A$ and $B$ are connected in series, wire A has length $L$ and radius $r$ while $B$ has length $2L$ and the radius $2r$. If both the wires are of same material then the ratio of the total resistance of series combination to the resistance of the wire $A$.
A. $\dfrac{3}{4}$
B. $\dfrac{3}{2}$
C. $\dfrac{6}{2}$
D. $\dfrac{6}{5}$

Answer 1

Hint: Resistance is directly proportional to the length of the wire and inversely proportional to the cross-section area. The above line is valid only if the material properties are the same throughout the material or wire we are talking about.

Complete step by step answer:
Resistance is directly proportional to the length of the wire.
$R\,\propto \,L\,\,\,\,.....(i)$
Resistance is inversely proportional to the cross section area of the material or wire.
$R\,\propto \,\dfrac{1}{A}\,\,\,\,.....(ii)$
From $(i)$ and $(ii)$
$R\, = \dfrac{{\rho L}}{A}\,\,\,\,.....(iii)$ ,
Here, $\rho $ is the proportionality constant called resistivity. Its value remains constant for the particular material.

Now according to the question, wire A has length $L$ and radius $r$ . Therefore according to the equation $(iii)$, resistance of the wire A is
${R_1}\, = \,\dfrac{{\rho L}}{{\pi {r^2}}}\,\, = \dfrac{{\rho L}}{A}\,\,\,......(iv)$
And wire B has length $2L$ and radius $2r$ . Therefore according to the equation $(iii)$ , resistance of the wire B is
${R_2}\, = \,\dfrac{{\rho (2L)}}{{\pi {{(2r)}^2}}}\,\, = \,\dfrac{{2\rho L}}{{4\pi {r^2}}}\, = \dfrac{{\rho L}}{{2A}}\,\,\,......(v)$

With the series combination of wires A and B,
${R_{result}}\, = \,{R_1} + \,{R_2}$
$ \Rightarrow \,{R_{result}}\, = \,\dfrac{{\rho L}}{A}\, + \dfrac{{\rho L}}{{2A}}\,$
On further calculation we will get resistance as,
$ \Rightarrow \,{R_{result}}\, = \,\dfrac{{3\rho L}}{{2A}}\,$
Now we are supposed to find the ratio,
$\dfrac{{{R_{result}}}}{{{R_1}}}\,\, = \,\,\dfrac{{\,\,\dfrac{{3\rho L}}{{2A}}\,\,\,}}{{\dfrac{{\rho L}}{A}\,}} \\
\therefore \dfrac{{{R_{result}}}}{{{R_1}}}\,\, = \,\dfrac{3}{2}$
The required ratio is $\dfrac{3}{2}$ .

Hence, option B is correct.

Note: Always remember that resistivity is a constant of proportionality and has specific value for any given material. Resistivity of any material depends on the temperature and the nature of that material. And also resistance can also be defined by ohm’s law where we consider the equation of current, voltage and resistance.