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A wire of resistance R has its length and cross section both doubled. Its resistance will become
4R
2R
R
R/4
R
The resistance of a wire is given by R=╧БALтАЛ. When the length L is doubled (LтА▓=2L) and the cross-sectional area A is also doubled (AтА▓=2A), the new resistance RтА▓ becomes RтА▓=╧Б2A2LтАЛ=╧БALтАЛ=R.
The resistance of a wire is given by R=╧БALтАЛ. When the length L is doubled (LтА▓=2L) and the cross-sectional area A is also doubled (AтА▓=2A), the new resistance RтА▓ becomes RтА▓=╧Б2A2LтАЛ=╧БALтАЛ=R.
R=╧БALтАЛ тАФ Ohm's law for resistance based on geometry
RтА▓=╧БnAnLтАЛ=R тАФ Proof of invariance under proportional scaling
The electrical resistance of a conductor is directly proportional to its length and inversely proportional to its cross-sectional area. By increasing both dimensions by the same factor (a scale factor of 2), the ratios cancel out, leaving the total resistance invariant.
Resistance is a property of the conductor's geometry and material resistivity (╧Б).
Scaling factor 'n' applied to both L and A results in no change to resistance.
If only length were doubled, resistance would increase by a factor of 4 (assuming volume conservation).
If only area were doubled, resistance would decrease by a factor of 4.
Uniform scaling preserves electrical properties.
Easier to analyze using geometric scaling factors.
Cable sizing in power distribution.
Design of heating elements and conductors.
Resistivity (╧Б) depends only on the material type and temperature, not dimensions.
Option A (4R) would occur if only length were doubled while maintaining volume (which would force the area to decrease).
C is correct тАФ Since the length and cross-sectional area are both scaled by the same factor, the ratio remains unchanged, resulting in the same resistance R.
Always verify if the problem implies volume conservation. If volume is constant and you double length, area MUST decrease, leading to 4R. If physical dimensions are explicitly stated as doubled (as in this case), use the direct formula ratios.