Question

An electric current is passed through a circuit. The circuit has two wires of the same material that are connected in parallel. If the lengths and radii of the wires are in the ratio 4 : 3 and 2 : 3 respectively, then the ratio of the currents passing through the wire will be-

1. 3 : 1
2. 2 : 3
3. 1 : 3
4. 3 : 5

Answer 1

Correct Answer - Option 3 : 1 : 3

The correct answer is option 3) i.e. 1 : 3

CONCEPT:

• Resistance: The hindrance to the flow of current offered by a material is called electrical resistance.
  • The SI unit of electrical resistance is the ohm (Ω). The resistance of a conducting wire is given by the equation:

\(R = ρ\frac{l}{A}\)

Where R is the resistance, l is the length of the wire and A is the cross-sectional area of the material.

CALCULATION:

​Let I₁ and I₂ be the current through the resistors R₁ and R₂.

Since the two resistors are of the same material, their resistivity is the same. ⇒ ρ₁ = ρ₂

The ratio of resistances = \(\frac{R_1}{R_2} = \frac{ρ_1\frac{l_1}{A_1}}{ρ_2\frac{l_2}{A_2}}\)

\(\Rightarrow \frac{R_1}{R_2} = {\frac{l_1}{l_2}}\times {\frac{A_2}{A_1}}\)

\(\Rightarrow \frac{R_1}{R_2} = {\frac{l_1}{l_2}}\times {\frac{\pi r_2 ^2}{\pi r_1 ^2}} ={\frac{l_1}{l_2}}\times {(\frac{r_2 }{ r_1})^2} \)

\(\Rightarrow \frac{R_1}{R_2} = {\frac{4}{3}}\times {(\frac{3}{2})^2} \)

\(\Rightarrow \frac{R_1}{R_2} = \frac{3}{1}\)

For resistances connected in parallel, the voltage drop across their ends will be the same.

From Ohm's law, V = IR, and since V is constant, \(R \propto \frac{1}{I}\)

\(\Rightarrow \frac{R_1}{R_2} \propto \frac{I_2}{I_1} \)

\(\Rightarrow \frac{I_2}{I_1} = \frac{1}{3}\)