Physics, asked by nishtharaman5454, 7 months ago

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

Answered by vimarshraina007
2

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

Similar questions