Question

An electric current is passed through a circuit containing two wires of the same material, connected in parallel. If the length and radii of the wires are in the ratio of $\dfrac 43\ and\ \dfrac 23$, then the ratio of the currents passing through the wire will be:
A. 3
B. 1/3
C. 8/9
D. 2

Answer 1

Resistance of the material is its unique property by which it opposes the flow of electric charge. It depends upon the dimensions of the conductor or material through which charge is flowing and also on material. That means different materials have different resistance. In a parallel combination of resistors, the current distributes in such a way that the potential difference across both the wires remains the same.

Formula used: $R = \dfrac{\rho l}{A},\ V = iR$

Complete step-by-step solution:
Given that the lengths are in ratio $\dfrac43$and radius in ratio $\dfrac 23$.
Hence we can write $\dfrac{l_1}{l_2} = \dfrac 43$
Also, the ratio of area = $\dfrac {A_1}{A_2} =\dfrac{\pi r_1^2}{\pi r_2^2} = \left(\dfrac {r_1}{r_2}\right)^2 = \left(\dfrac23 \right)^2 = \dfrac 49$
Now, resistance $R = \dfrac{\rho l}{A}$
For two materials, we can write:
$\dfrac{R_1}{R_2} = \dfrac{\dfrac{\rho l_1}{A_1}}{\dfrac{\rho l_2}{A_2}} = \dfrac{A_2}{A_1}\dfrac{l_1}{l_2}$
Now, we know $\dfrac{l_1}{l_2} = \dfrac{4}{3}$
And$\dfrac{A_1}{A_2} = \dfrac49$
So, $\dfrac{A_2}{A_1} = \dfrac 94$
Putting these values in ratio of resistances, we get
$\dfrac{R_1}{R_2} = \dfrac{9}{4}\times \dfrac43 =3$
Now, as resistances are in parallel, we can write $V_1 = V_2$
Hence by ohm’s law, $V=iR$
So, $i_1 R_1 = i_2 R_2$
Or $\dfrac{i_1}{i_2} = \dfrac{R_2}{R_1} = \dfrac13$
Hence ratio of current is 1/3, option B. is correct.

Additional information: When two resistors are connected in series, all the charge flowing through one resistance is the same as flowing from the other as there is a single path to flow to charge. In other words, the current is the same through the resistors, we can say that they’re in parallel. Similarly, if the potential difference across the resistors is the same, we can say that the resistors are in parallel.

Note: Here, in the question, we have one extra term ($\rho$), called resistivity. Resistivity is the property of the material. It may be defined as resistance per unit volume. It is independent of the size, shape, and density of the material. Hence it’s a unique property of a material and is the same for a particular material. It is also called specific resistance.