Question

A wire has linear resistance $\rho$(in Ohm/m). Find the resistance r between points A and B if the side of the larger square is ‘d’.

A) $\dfrac{\rho d}{\sqrt2}$
B) $\sqrt2\rho d$
C) $2rd$
D) None of these

Answer 1

Hint:The problem is from the electricity part of physics. We can apply the concept of parallel combination and series combination of resistance here. Use the equation for effective resistance in parallel and series combinations.


Formula Used:
Equivalent resistance for a series resistance circuit:
${R_E} = {R_1} + {R_2} + {R_3}$
Equivalent resistance for a parallel resistance circuit:
$\dfrac{1}{{{R_E}}} = \dfrac{1}{{{R_1}}} + \dfrac{1}{{{R_2}}} + \dfrac{1}{{{R_3}}}$
Where ${R_E}$= equivalent resistance and ${R_1},{R_2},{R_3}$ = component resistance.


Complete answer:
The equivalent resistance is a single resistance which can replace all the component resistances in a circuit in such a manner that the current in the circuit remains unchanged.

Figure 1
Let’s half side of the resistor is r. Where $r = \dfrac{{\rho d}}{2}$
We can redraw the circuit diagram and it will be.

Figure 2
Apply the equations for parallel and series combinations and the equivalence resistance is calculated as,
${R_E} = \dfrac{1}{2}\left[ {2r + \dfrac{{2r \times r\sqrt 2 }}{{2r + r\sqrt 2 }}} \right] = \dfrac{1}{2}\left[ {2r + \dfrac{{2r \times r\sqrt 2 }}{{(2 + \sqrt 2 )r}}} \right]$
${R_E} = \dfrac{1}{2}\left[ {\dfrac{{2r \times (2 + \sqrt 2 )r + 2{r^2}\sqrt 2 }}{{(2 + \sqrt 2 )r}}} \right]$
${R_E} = \dfrac{1}{2}\left[ {\dfrac{{4r + 2r\sqrt 2 + 2r\sqrt 2 }}{{2 + \sqrt 2 }}} \right]$
${R_E} = \dfrac{1}{2}\left[ {\dfrac{{4r\sqrt 2 + 4r}}{{2 + \sqrt 2 }}} \right]$
${R_E} = \dfrac{{4r}}{2}\dfrac{{\sqrt 2 + 1}}{{2 + \sqrt 2 }} \times \dfrac{{2 - \sqrt 2 }}{{2 - \sqrt 2 }}$
${R_E} = 2r\left( {\dfrac{{2\sqrt 2 - {{(\sqrt 2 )}^2} + 2 - \sqrt 2 }}{{{2^2} - {{(\sqrt 2 )}^2}}}} \right)$
${R_E} = 2r\dfrac{{\sqrt 2 }}{2} = r\sqrt 2 $
Substituting the value of r and the ${R_E}$will become,
\[{R_E} = \dfrac{{\rho d\sqrt 2 }}{2} = \dfrac{{\rho d}}{{\sqrt 2 }}\]

Hence, the correct option is Option (A).


Note: Resistance is a measure of the opposition to current flow in an electrical circuit. Resistance blocks the flow of current. The S.I unit of resistance is ohms. The current decreases as resistance increases. On the other hand, the current increases as the resistance decreases. While solving this one also has to pay close attention to the connection between resistances.