The resistance of a wire of uniform diameter | vedclass

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(B) The resistance $R$ of a wire is given by the formula $R = \rho \frac{L}{A}$,where $\rho$ is the resistivity,$L$ is the length,and $A$ is the cross

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$R$

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(B) The resistance $R$ of a wire is given by the formula $R = ho \frac{L}{A}$,where $ho$ is the resistivity,$L$ is the length,and $A$ is the cross-sectional area.Since the wire has a uniform diameter $d$,the cross-sectional area $A = \pi (d/2)^2 = \frac{\pi d^2}{4}$.Thus,$R \propto \frac{L}{d^2}$.For the first wire: $R_1 = R$,$L_1 = L$,$d_1 = d$.For the second wire: $L_2 = 4L$,$d_2 = 2d$.Taking the ratio: $\frac{R_2}{R_1} = \frac{L_2}{L_1} 	imes \left( \frac{d_1}{d_2} ight)^2$.Substituting the values: $\frac{R_2}{R} = \frac{4L}{L} 	imes \left( \frac{d}{2d} ight)^2 = 4 	imes \left( \frac{1}{2} ight)^2 = 4 	imes \frac{1}{4} = 1$.Therefore,$R_2 = R$.

The resistance of a wire of uniform diameter $d$ and length $L$ is $R$. The resistance of another wire of the same material but diameter $2d$ and length $4L$ will be

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