Nine identical wires each of diameter d and length L are connected in parallel. The combination has the same resistance as a single similar wire of length $L$ but whose diameter is ____.

Open in App

Solution

Find the diameter:
In parallel,
$\frac{1}{R_{n e t}} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}} + . . . . + \frac{1}{R_{9}}$
All wires are identical so resistance will be equal.
So $\frac{1}{R_{n e t}} = \frac{1}{R} + \frac{1}{R} + \frac{1}{R} + . . . . + \frac{1}{R}$
$\Rightarrow \frac{1}{R_{n e t}} = \frac{9}{R}$
$R = \frac{L}{A}$
$A = π r^{2}$
$\Rightarrow A = \frac{{πd}^{2}}{4} (r = \frac{d}{2})$
So $R_{n e t} = \frac{L \times 4}{r d^{2} \times 9}$
Now $R_{n e t} = \frac{L}{A}$
Let the diameter of the new wire is $D$
$A^{'} = \frac{π D^{2}}{4}$
So $R_{n e t} = \frac{L \times 4}{r D^{2}}$
According to the question,
$\frac{L \times 4}{r D^{2}} = \frac{L \times 4}{r d^{2} \times 9}$
$\Rightarrow D^{2} = 9 d^{2}$
$\Rightarrow D = 3 d$

Nine identical wires each of diameter d and length L are connected in parallel. The combination has the same resistance as a single similar wire of length $L$ but whose diameter is $3 d$ .

Suggest Corrections

Similar questions

Q. Two metallic wires A and B are connected in parallel. Wire A has length l and radius r, wire B has a length 2l and radius 2r. Compute the ratio of the total resistance of parallel combination and the resistance of wire A.

Q. 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

d .

Two metallic wires A and B of the same material are connected in parallel. Wire A has length $l$ and radius $r$ ,
wire B has a length $2 l$ and radius $2 r$ .Compute the ratio of the total resistance of parallel combination and the
resistance of wire A.

Q. A conducting wire has length

^{'} L_{1}^{'}

and diameter

^{'} d_{1}^{'}

. After stretching the same wire length becomes

^{'} L_{2}^{'}

and diameter

^{'} d_{2}^{'}

. The ratio of resistances before and after stretching is

Q. Q13 Two metallic wires A and B are connected in parallel. Wire A has length l and radius r, wire B has a length 21 and radius 2r. Compute the ratio of the total resistance
of parallel combination and the resistance of wire A.

Also if the wires are connected in series compute the ratio of the series combination and resistance of wire A ?

Join BYJU'S Learning Program

Nine identical wires each of diameter d and length L are connected in parallel. The combination has the same resistance as a single similar wire of length L but whose diameter is ____.

Nine identical wires each of diameter d and length L are connected in parallel. The combination has the same resistance as a single similar wire of length $L$ but whose diameter is ____.