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A wire has a resistance of R. If its length is increased by a factor of 8 and its cross- sectional area is decreased by a factor of 4, what is its new resistance (R')?
R' = 8 R
R' = 16 R
R' = 32 R
R' = 2 R
R' = 32 R
The resistance of a uniform wire is directly proportional to its length (l) and inversely proportional to its cross-sectional area (A). When the length is increased by a factor of 8 (lтА▓=8l) and the cross-sectional area is decreased by a factor of 4 (AтА▓=A/4), the new resistance becomes RтА▓=╧БA/48lтАЛ=32(╧БAlтАЛ)=32R.
The resistance of a uniform wire is directly proportional to its length (l) and inversely proportional to its cross-sectional area (A). When the length is increased by a factor of 8 (lтА▓=8l) and the cross-sectional area is decreased by a factor of 4 (AтА▓=A/4), the new resistance becomes RтА▓=╧БA/48lтАЛ=32(╧БAlтАЛ)=32R.
Imagine water flowing through a garden hose. If you make the hose 8 times longer, the water faces much more frictional resistance. If you simultaneously pinch the hose to make its cross-sectional area 4 times smaller, the flow restriction multiplies further, making it extremely difficult for water to pass through.
Remember 'Length is High, Area is Low, Resistance will Grow' тАФ length goes to the numerator, area goes to the denominator.
R=╧БAlтАЛ тАФ Resistance formula in terms of resistivity, length, and cross-sectional area
RтА▓=╧БA/48lтАЛ=32R тАФ Scaled resistance calculation
According to Ohm's law and material properties, resistance is defined by the formula R=╧БAlтАЛ, where ╧Б is the electrical resistivity of the material. When a wire is stretched or modified, its volume remains constant only if mass and density are constant, but here the problem explicitly states the length and cross-sectional area changes independently. Substituting the modified dimensions into the resistance formula reveals that the scaling factors multiply directly, yielding a 32-fold increase in resistance.
Resistance is directly proportional to length (RтИЭl).
Resistance is inversely proportional to cross-sectional area (RтИЭA1тАЛ).
When dimensions change independently, the new resistance is found by substituting the scaling factors into the formula.
Multiplying a length factor of 8 and an inverse area factor of 4 yields a total factor of 8├Ч4=32.
Allows rapid calculation of electrical properties without full re-measurement
Fundamental principle used in resistor design and transmission line analysis
Assumes uniform cross-section and homogeneous material properties
Does not account for temperature-dependent resistivity variations unless specified
Electrical power transmission design
Strain gauges and load cells sensing mechanical deformation
Resistivity (╧Б) is an intrinsic property of the material and does not change when dimensions are altered.
Option A (RтА▓=8R) only considers the length factor and ignores the area reduction.
Option B (RтА▓=16R) is an arithmetic error in combining the factors.
Option D (RтА▓=2R) incorrectly divides length by area or uses wrong arithmetic.
C is correct тАФ The new resistance is 32 times the original resistance because resistance is directly proportional to length and inversely proportional to cross-sectional area (RтА▓=╧БA/48lтАЛ=32R).
In competitive exams, always be careful whether a wire is 'stretched' (where volume V=l├ЧA remains constant) or 'reshaped/replaced' with given independent dimensions. Here, length and area were changed independently.