A square and an equilateral triangle have the same perimeter a and the same resistance per unit length. Find the ratio of equivalent resistances between adjacent vertices, $R_square/R_triangle.$
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A square and an equilateral triangle have the same perimeter a and the same resistance per unit length. Find the ratio of equivalent resistances between adjacent vertices, $R_square/R_triangle.$

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G1 Physics · Accepted Answer

For the square, side=a/4. Between adjacent corners the two paths have resistances λa/4 and 3λa/4 in parallel, giving R s q u a r e = 3 λ a / 16. R_square=3λa/16. R s ​ q u a r e = 3 λa /16. For the triangle, side=a/3; paths λa/3 and 2λa/3 in parallel give R t r i a n g l e = 2 λ a / 9. R_triangle=2λa/9. R t ​ r ian g l e = 2 λa /9. Thus R s q u a r e / R t r i a n g l e = ( 3 / 16 ) ( 9 / 2 ) = 27 / 32. R_square/R_triangle=(3/16)(9/2)=27/32. R s ​ q u a r e / R t ​ r ian g l e = ( 3/16 ) ( 9/2…

A square and an equilateral triangle have the same perimeter a and the same resistance per unit length. Find the ratio of equivalent resistances between adjacent vertices, R s q u a r e / R t r i a n g l e . R_square/R_triangle. R s ​ q u a r e / R t ​ r ian g l e .

A square and an equilateral triangle have the same perimeter a and the same resistance per unit length. Find the ratio of equivalent resistances between adjacent vertices, R s q u a r e / R t r i a n g l e . R_square/R_triangle. R s q u a r e / R t r ian g l e .