A uniform wire of resistance R is shaped into a regular n sided polygon, where
n is even. Find the equivalent resistance between
(a) opposite corners of the polygen and
(b) adjacent corner of polygon
Answers
When, a uniform wire of resistance R is shaped into a regular n-sided polygon, the resistance of each side of polygon is
n
R
Let, R
1
and R
2
be the resistances between two parts of polygon between opposite corners. Polygon will get divided into
2
n
sides on both sides. Therefore,
R
1
=R
2
=(
2
n
)(
n
R
)
R
1
=R
2
=
2
R
The two parts are parallel to each other hence, equivalent resistance between two opposite corners is
R
′
=
R
1
+R
2
R
1
R
2
R
′
=
2
R
+
2
R
(
2
R
)(
2
R
)
R
′
=
4R
R
2
=
4
R
Also, the polygon is equivalent to the combination of two resistances.
The resistance of one side, R
1
=
n
R
The resistance of (n−1) side, R
2
=
n
(n−1)R
Since, the two parts are parallel, hence equivalent resistance between two adjacent corners is
R
′′
=
R
1
+R
2
R
1
R
2
R
′′
=
n
R
+
n
(n−1)R
(
n
R
)(
n
(n−1)R
)
R
′′
=
n
2
(n−1)R
2
×
R+nR−R
n
R
′′
=
n
2
(n−1)R