JEE Main · 2022 · 25 Jul 2022 · Shift 1 · Current Electricity

Eight copper wire of length

l

and diameter d are joined in parallel to form a single composite conductor of resistance R . If a single copper wire of length

2 l

have the same resistance (

R

) then its diameter will be

____

Answer: 4

Solution

The resistance of each wire of length

l

and diameter

d

is:

r = ρ \frac{l}{A} = ρ \frac{4 l}{π d ^{2}}

When 8 wires are connected in parallel, the equivalent resistance is:

R_{parallel} = \frac{r}{8} = \frac{ρl}{2 π d ^{2}}

For a single wire of length

2 l

and diameter

d_{1}

with the same resistance

R

R = ρ \frac{2 l \times 4}{π d _{1}^{2}} = \frac{8 ρl}{π d _{1}^{2}}

Equating the two expressions:

\frac{ρl}{2 π d ^{2}} = \frac{8 ρl}{π d _{1}^{2}}

\frac{1}{2 d ^{2}} = \frac{8}{d _{1}^{2}}

d_{1}^{2} = 16 d^{2}

d_{1} = 4 d

JEE Main · 2022 · 25 Jul 2022 · Shift 1 · Integer answer