Question

Two metallic wires A and B are connected in parallel. Wire A has length I and radius r, wire B has
a length 21 and radius 2r. Compute the ratio of the total resistance of parallel combination and the
resistance of wire A.

Answer 1

Correct Question

Two metallic wires A and B are connected in parallel. Wire A has length I and radius r, wire B has a length 2l and radius 2r. Compute the ratio of the total resistance of parallel combination and the resistance of wire A.

Solution-

For wire A:

Length = l and radius = r

For wire B:

Length = 2l and radius = 2r

Both the wire are connected in parallel. We have to find the ratio of the total resistance of parallel combination and the resistance of wire A.

Now,

R = p l/A

Here; R is Resistance, p is rho (resistivity), l is length and A is Area of cross-section.

Also, A = πr². So,

R = p l/πr²

For wire A:

R1 = p l/πr² ....................(1st equation)

For wire B:

R2 = p 2l/π(2r)²

R2 = p 2l/4πr²

R2 = 2/4 p l/πr²

R2 = 1/2 p l/πr² ..............(2nd equation)

As, both R1 and R2 are connected in parallel. So,

1/Rp = 1/R1 + 1/R2

1/Rp = πr²/pl + 2πr²/pl

1/Rp = 3πr²/pl

Rp = 1/3 p l/πr²

As per given condition, we have to find the ratio of Rp and R1.

So,

Rp/R1 = [1/3 (p l/πr²)]/(p l/πr²)

Rp/R1 = 1/3

Therefore, the ratio of Rp and R1 is 1:3.