Question

Calculate the diameter of a wire of resistance 50 Ω, resistivity 3.14 × 10−8 Ωm and length 1m.
PLS FAST PLS FAAST

Answer 1

Solution :

⏭ Given:

✏ Resistance of wire = 50Ω

✏ Length of wire = 1m

✏ Resistivity of wire = 3.14 × 10$^{-8}$ Ωm

⏭ To Find:

✏ Diameter of wire

⏭ Formula:

✏ Relation between resistivity, resistance, length and area of cross section is given by...

$\bigstar \: \boxed{ \tt{ \pink{R = \frac{ \rho \times l}{A}}}} \: \bigstar$

⏭ Calculation:

$\implies \sf \: R = \dfrac{ \rho \times l}{ \pi \times {r}^{2} } \\ \\ \implies \sf \: 50 = \frac{ \cancel{3.14} \times {10}^{ - 8} \times 1}{ \cancel{3.14} \times {r}^{2} } \\ \\ \implies \sf \: {r}^{2} = \frac{ {10}^{ - 8} }{50} \\ \\ \implies \sf \: {r}^{2} = 0.02 \times {10}^{ - 8} \\ \\ \implies \sf \: \red{radius = 0.141 \times {10}^{ - 4} \: m} \\ \\ \mapsto \sf \: we \: know \: that \\ \\ \circ \sf \: diameter = 2 \times radius \\ \\ \circ \sf \: diameter = 2 \times 0.141 \times {10}^{ - 4} \\ \\ \circ \sf \: diameter = 0.28 \times {10}^{ - 4} \\ \\ \circ \: \boxed{ \tt{ \orange{diameter = 0.028 \: mm}}}$

Answer 2

Calculate the diameter of a wire of resistance 50 Ω, resistivity 3.14 × 10−8 Ωm and length 1m.

⏭ Formula:

$\leadsto \bigstar \: \boxed{ \tt{ \orange{R = \frac{ \rho \times l}{A}}}}$

$\begin{lgathered}\leadsto \sf \: R = \dfrac{ \rho \times l}{ \pi \times {r}^{2} } \\ \\ \mapsto \sf \: 50 = \frac{ \cancel{3.14} \times {10}^{ - 8} \times 1}{ \cancel{3.14} \times {r}^{2} } \\ \\ \implies \sf \: {r}^{2} = \frac{ {10}^{ - 8} }{50} \\ \\ \leadsto \sf \: {r}^{2} = 0.02 \times {10}^{ - 8} \\ \\ \mapsto \sf \: \purple{radius = 0.141 \times {10}^{ - 4} \: m} \\ \\ \leadsto \sf \: We \: know - \\ \\ \circ \sf \: Diameter = 2 \times Radius \\ \\ \circ \sf \: Diameter = 2 \times 0.141 \times {10}^{ - 4} \\ \\ \circ \sf \: diameter = 0.28 \times {10}^{ - 4} \\ \\ \circ \: \boxed{ \tt{ \blue{\leadsto{\boxed{D = 0.028 \: mm}}}}}\end{lgathered}$