Question

Two wires of equal length, one of copper and other of tungsten, are to carry the same current when connected across the same voltage. Then the ratio of Diameter of copper to the diameter of tungsten will be

Answer 1

Answer :

Given -

• Two wires
• One is of Copper and the other one is Tungsten
• They are connected across same voltage
• They carry same current

To Find -

• Ratio of diameter of copper to the diameter of tungsten ?

Solution -

Both of the wires have same voltage and current.

Using Ohm's Law :

For Copper Wire :- V = I/R(Cu)

➜ R(Cu) = I/V

For Tungsten Wire :- V = I/R(W)

➜ R(W) = I/V

As, both of them have same Voltage, hence :

⇒ V = I/R(Cu) = I/R(W)

⇒ R(Cu) = R(W) = I/V

⇒ R(Cu) = R(W)

Hence, The resistance of both the wires will be same.

We know that, R = rho*l/A

⇒ R = rho*l/πr²

⇒ R = [rho(Cu)]*[l/πr²(Cu)] = [rho(W)]*[l/πr²(W)]

⇒ rho(Cu)/πr²(Cu) = rho(W)/πr²(W)

⇒ rho(Cu)*πr²(W) = rho(W)*πr²(Cu)

⇒ πr²(W)/πr²(Cu) = rho(W)/rho(Cu)

⇒ r²(W)/r²(Cu) = rho(W)/rho(Cu)

⇒ r(W)/r(Cu) = √[rho(W)/rho(Cu)]

⇒ 2*r(W)/r(Cu) = 2*√[rho(W)/rho(Cu)]

⇒ d(W)/d(Cu) = 2√[rho(W)/rho(Cu)]

Reciprocating,

⇒ d(Cu)/d(W) = 1/2*√[rho(Cu)/rho(W)]

Hence, Ratio of diameter of copper (Cu) to the diameter of tungsten (W) is 1/2*√[rho(W)/rho(Cu)].