Question

A wire must be replaced in a circuit by a new wire of the same material but nine times longer. If the new wire's resistance is the same as the original wire's, what is the diameter of the new wire?

O the same as the original diameter
O one half the original diameter
O one third the original diameter
O three times the original diameter
O nine times the original diameter

Answer 1

three times the original diameter

Step-by-step explanation:

From the wire's resistance formula, we can calculate the relation between the diameter of the wire and its length:

`R=\rho(l)/(\pi (d^2)/(4))\\d=\sqrt{\rho (4 l)/(\pi R)}\\`

Here, d is the wire's diameter,
`\rho` is the electrical resistivity of the material and R is the resistance of the wire. We have
`l'=9l`

`d'=\sqrt{\rho (4 l')/(\pi R)}\\d'=\sqrt{\rho (4 (9l))/(\pi R)}\\d'=3\sqrt{\rho (4 l)/(\pi R)}\\d'=3d`