6.
Three wires of same material having lengths in ratio
of 2 : 3:4 and mass in ratio 3 : 4 : 5 are
connected in parallel combination across a V volt
supply, then the ratio of current flowing through
them is
(1) 75:64: 54
(2) 108 : 64:45
(3) 80 : 135: 192
(4) 54 : 64 : 75
Answers
Answer is 108:64:45 option 2
Below the proof.
The ratio of current flowing through them is (2) 108 : 64 : 45.
Explanation:
The current is given by the formula:
I = V/R
The ratio of current is:
I₁ : I₂ : I₃ = V/R₁ : V/R₂ : V/R₃
I₁ : I₂ : I₃ = 1/R₁ : 1/R₂ : 1/R₃ → (Equation 1)
The resistance is given by the formula:
R = ρL/A
Where,
L = Length and A = Area
The ratio of resistance is given as:
R₁ : R₂ : R₃ = ρL₁/A₁ : ρL₂/A₂ : ρL₃/A₃
R₁ : R₂ : R₃ = L₁/A₁ : L₂/A₂ : L₃/A₃
Area = Volume/Length
Volume ≈ Mass
Now,
A₁ : A₂ : A₃ = 3/2 : 4/3 : 5/4
On substituting the value of lengths and mass in ratio, we get,
R₁ : R₂ : R₃ = 2/(3/2) : 3/(4/3) : 4/(5/4)
R₁ : R₂ : R₃ = 4/3 : 9/4 : 16/5
On substituting the above values in equation (1), we get,
I₁ : I₂ : I₃ = 1/(4/3) : 1/(9/4) : 1/(16/5)
I₁ : I₂ : I₃ = 3/4 : 4/9 : 5/16
∴ I₁ : I₂ : I₃ = 108 : 64 : 45