Question

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A wire of resistance R is transformed into circle. Arc AB subtends 120 degree angle at center. Find resistance of Arc AB.

Answer 1

θ=120°
r=R
.........
Resistance per radian=R/2π.
Resistance of θ segment=R/2π.θ

$\frac{r}{2\pi} .120$

=
$\frac{r}{\pi} .60$
=

Answer 2

The resistance of the wire depends on the length of the wire.

R = ρl/A

where ρ → resistivity

l → length of wire

A → area of cross section of it

When the wire is made into a circle, its resistance (R) does not change as there is no change in length or area of cross section.

But the resistance of arc AB changes as the length of it changes.

Since the arc subtends 120° at the centre, the wire's length becomes ¹/₃ rd of total.

Hence R' = ρ(l/2)/A = ¹/₃ (ρl/A) = ¹/₃ R

Thus, the resistance becomes R/3.

Hope it helps!!