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Two metallic wires A and B are connected in parallel. Wire A has a length $l$ and radius $r$ , wire B has a length of $2 l$ and radius of $2 r$ . Which of the following is the ratio of the total resistance of parallel combination and the resistance of wire A?

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3 : 1

1 : 3

2 : 1

1 : 2

answer is B.

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Detailed Solution

The ratio of the total resistance of parallel combination and the resistance of wire A is

1 : 3

.
Given:
Wire A: Length

= l

Radius

= r

Wire B;
Length

= 2 l

Radius

= 2 r

Resistance of A:

R_{A} = \frac{ρ \times l}{π \times r^{2}}

Resistance of B:

R_{B} = \frac{ρ \times 2 l}{π \times {(2 r^{})}^{2}}

The resistance

R_{p}

for the parallel combination must be written as:

\frac{1}{R_{p}} = \frac{1}{R_{A}} + \frac{1}{R_{B}}

= \frac{π r^{2}}{ρ \times l} + \frac{2 π r^{2}}{ρ \times l}

\frac{1}{R_{p}} = \frac{ρ \times l}{3 π r^{2}}

Hence the ratio of

R_{A}

R_{B}

will be

1 : 3

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