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The resistance, R (in ohms) of a wire varies directly with the length, L (in cm) of the wire, and inversely with the cross-sectional area, A (in cm^2). A 500 cm piece of wire with a radius of 0.2cm has a resistance of 0.025 ohm. Find an equation that relates these variables in the form of\(R = \frac {\square\cdot \pi\cdot L}{\square} \)

BillHicks1207  Apr 5, 2017
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The resistance, R (in ohms) of a wire varies directly with the length, L (in cm) of the wire,
and inversely with the cross-sectional area, A (in cm^2).
A 500 cm piece of wire with a radius of 0.2cm has a resistance of 0.025 ohm.
Find an equation that relates these variables in the form of \(R = \frac {\square\cdot \pi\cdot L}{\square}\)

 

Let \(\rho\) = resistivity
Let L = length
Let A = cross sectional area
Let r = Radius

 

\(\begin{array}{|rcll|} \hline R &=& \dfrac{ \rho \cdot L } {A} \quad & \text{rewrite...}\\ \rho &=& \dfrac{ A \cdot R } {L} \quad & | \quad A = \pi r^2 \\ \rho &=& \dfrac{ \pi r^2 \cdot R } {L} \\ \rho &=& \dfrac{ r^2 \cdot R } {L} \cdot \pi \quad & | \quad L = 500\ cm \quad R= 0.025\ \Omega \quad r = 0.2\ cm \\ \rho &=& \dfrac{ 0.2^2 \cdot 0.025 } {500} \cdot \pi \\ \rho &=& \dfrac{ 0.04 \cdot 0.025 } {500} \cdot \pi \\ \rho &=& \dfrac{ 0.001 } {500} \cdot \pi \\\\ \mathbf{\rho} & \mathbf{=} & \mathbf{0.000002 \cdot \pi } \\ \hline \end{array}\)

 

Find an equation that relates these variables in the form of \(R = \frac {\square\cdot \pi\cdot L}{\square}\)

\(\begin{array}{|rcll|} \hline R &=& \dfrac{ \rho \cdot L } {A} \quad & \quad \rho = 0.000002 \cdot \pi \\\\ \mathbf{R} & \mathbf{=} & \mathbf{ \dfrac{ {\color{red}0.000002} \cdot \pi \cdot L } {{\color{red}A}} } \\ \hline \end{array}\)

 

 

laugh

heureka  Apr 6, 2017

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