Question

Two copper wires A & B of length 30m and 10m have radius 2cm and 1cm respectively. compare the resistance of two wires.

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Answer 1

QUᎬᏚᎢᏆON ?

✏ Two copper wires A & B of length 30m and 10m have radius 2cm and 1cm respectively. compare the resistance of two wires.

ᎪNᏚᏔᎬᎡ :-

✒ ғɪʀsᴛ ᴡɪʀᴇ ʜᴀs ʟᴇss ʀᴇsɪsᴛᴀɴᴄᴇ ᴀs ᴄᴏᴍᴘᴀʀᴇᴅ ᴛᴏ sᴇᴄᴏɴᴅ .

ᏚOᏞUᎢᏆON :-

Resistance [R] = ρ $\dfrac{l}{a}$

Where, ρ = resistivity is same for both copper wires.

L = length

A = Area of cross-section = π r²

For wire A,

R = $\dfrac{p \times 30}{3.14 \times {0.2}^{2} }$

▶ $\dfrac{ 30p}{3.14 \times {0.0004} }$

For wire B,

Ŕ = $\dfrac{p \times 10}{3.14 \times {0.01}^{2} }$

▶ $\dfrac{ 10p}{3.14 \times {0.0001} }$

Now,

➡ $\dfrac{R}{Ŕ}$ = $\dfrac{ 30p}{3.14 \times {0.0004}}$ × $\dfrac{3.14 × 0.0001} { 10p}$

➡ $\dfrac{3 × 0.0001}{0.0004}$

➡ $\dfrac{3 }{4}$

➡ $\dfrac{R}{Ŕ}$ = $\dfrac{3 }{4}$

➡ Ŕ = $\dfrac{4 }{3}$R

Hence, First wire has less resistance.

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Answer 2

Given :

➳ Length of wire A = 30m

➳ Length of wire B = 10m

➳ Radius of wire A = 2cm

➳ Radius of wire B = 1cm

To Find :

➨ Relation b/w resistance of wire-A and wire-B.

Concept :

➠ Resistance of a conductor is directly proportional to the length of conductor and inversely proportional to the area of cross section of conductor.

➠ Resistivity is a materialistic property of a conductor. It doesn't depend on the dimension of the conductor.

➠ Here, resistivity of both wires is same because they are made up from same material.

Mathematically,

◈ $\underline{\boxed{\bf{\pink{R=\rho\times\dfrac{l}{A}}}}}$

where,

◕ R denotes resistance

◕ $\rho$ denotes resistivity

◕ l denotes length

◕ A denotes are of cross section

Calculation :

$\dashrightarrow\sf\:\dfrac{R_A}{R_B}=\dfrac{L_A\times A_A}{A_A\times L_B}\\ \\ \dashrightarrow\sf\:\dfrac{R_A}{R_B}=\dfrac{L_A\times (\pi\times {R_B}^2)}{(\pi\times {R_A}^2)\times L_B}\\ \\ \dashrightarrow\sf\:\dfrac{R_A}{R_B}=\dfrac{L_A\times {R_B}^2}{{R_B}^2\times L_B}\\ \\ \dashrightarrow\sf\:\dfrac{R_A}{R_B}=\dfrac{30\times (1)^2}{(2)^2\times 10}\\ \\ \dashrightarrow\sf\:\dfrac{R_A}{R_B}=\dfrac{30}{40}\\ \\ \dashrightarrow\underline{\boxed{\bf{\purple{R_A:R_B=3:4}}}}\:\gray{\bigstar}$