A potential difference V is applied to a wire of cross-sectional area A, length L, and conductivity . We want to change the applied potential difference and draw out the power so the power dissipated is increased by a factor of 30 and the current is increased by a factor of 4. We have to find the new values of (a) the length and (b) the cross-sectional area.

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Let the original resistance of the wire be R. When a potential difference V is applied to the ends of the wire the current in it will be

and the power dissipated will be given by

Wire is stretched and potential difference is applied so that the current through the wire becomes , such that

Let the changed resistance of the wire that ensures that when the current is the power dissipated becomes

This gives

As the wire is stretched its volume remains unchanged.

Therefore,

Resistance of a wire of cross-sectional area A, length L, and conductivity of the material is given by

Therefore, we have the relation

Using the result

we have

and