Question

Two wires A and B with circular cross-section are made of the same metal and have equal lengths, but the resistance of wire A is five times greater than that of wire B. What is the ratio of the radius of A to that of B?

Answer 1

The ratio of radius of A to that of B is
`1:√(5)`.

Step-by-step explanation:

The resistance of a wire is given by :

`R=\rho (l)/(A)`

l is length of wire

A is area of cross section

Resistance of wire A is :
`R_A=\rho (l_A)/(A_A)` .......(1)

Resistance of wire B,
`R_B=\rho (l_B)/(A_B)` .....(2)

As
`l_A=l_B` and
`R_A=5R_B` (given)

From equation (1) and (2) and putting given condition, we get :

`(R_A)/(R_B)=(A_B)/(A_A)\\\\(R_A)/(R_B)=(r_B^2)/(r_A^2)`

Since,
`R_A=5R_B`. So,

`(5R_B)/(R_B)=(r_B^2)/(r_A^2)\\\\(5)/(1)=(r_B^2)/(r_A^2)\\\\((r_A)/(r_B))^2=(1)/(5)\\\\(r_A)/(r_B)=(1)/(√(5) )`

So, the ratio of radius of A to that of B is
`1:√(5)`.