A wire which is very long is made of copper is connected to a battery. Because it is very long, its resistance is not neglected and the power dissipated by the wire alone is 8.8 watts. The same wire is melted down and shaped into another wire which is a factor of 4 shorter than the first wire. When this wire is connected to the same battery, what is the power dissipated by this wire in Watts?
Ohm's law
Ohm’s law is a prominent concept in physics and electronics. It gives the relation between the current and the voltage. It is used to analyze and construct electrical circuits. Ohm's law states that the voltage across a conductor is directly proportional to the current flowing through it.
Path of Least Resistance
In a series of alternate pathways, the direction of least resistance is the actual or metaphorical route that offers the least resistance to forwarding motion by a given individual or body.
Let us first consider, what happens to the resistance of the copper wire after it had been melted and its length shortened by a factor of 4. Since the volume (cross sectional area X length) of the copper is conserved, its cross sectional area has to increase by a factor of 4.
We know Resistance R = , where denotes the resistivity of the material, which is its microscopic property and does not depend on its macroscopic dimension. Now the new wire has its length 4 times shortened and the cross sectional area increased by a factor of 4. Thus new resistance will be reduced by a factor of 16.
We assume that the battery is an ideal one and its voltage V across is terminal is constant.
Now power dissipation in the wire is given by . Now if the resistance is decreased by a factor of 16, the power dissipation will increase 16 times.
Now the power dissipation will be 8.8 Watts X 16 = 140.8 Watts.
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