An electric current is passed through a circuit containing two wires ofsame material, connected in parallel. If the lengths and radii of the wiresare in the ratio of 3:2 and 2:3, then the ratio of the current passingthrough the wire will
Question
An electric current is passed through a circuit containing two wires ofsame material, connected in parallel. If the lengths and radii of the wiresare in the ratio of 3:2 and 2:3, then the ratio of the current passingthrough the wire will
Solution
The current passing through a wire is given by Ohm's law, which states that the current (I) is equal to the voltage (V) divided by the resistance (R).
The resistance of a wire is given by the formula R = ρL/A, where ρ is the resistivity of the material, L is the length of the wire, and A is the cross-sectional area of the wire.
In this case, the wires are made of the same material, so the resistivity is the same for both. The lengths and radii of the wires are given in the ratio 3:2 and 2:3 respectively.
The cross-sectional area of a wire is given by the formula A = πr², where r is the radius of the wire.
So, the resistance of the first wire (R1) is ρ(3L)/(π(2r)²) and the resistance of the second wire (R2) is ρ(2L)/(π(3r)²).
The ratio of the currents passing through the wires (I1/I2) is equal to the inverse ratio of the resistances (R2/R1), because the voltage is the same for both wires (they are connected in parallel).
So, I1/I2 = R2/R1 = [ρ(2L)/(π(3r)²)] / [ρ(3L)/(π(2r)²)] = (2/3) / (3/4) = 8/9.
Therefore, the ratio of the current passing through the wires is 8:9.
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