The Ring

Ring2
 

A homogeneous wire is used to make a ring with a jumper on the diameter AB. What is the percentage change in the resistance between points A and B, if the jumper is cut?

 
 
 
 
 
 

Ring3

Solution.

We denote the resistance of the upper and lower part of the ring as R1 and the resistance of the jumper R2.

We start with a first case when we have the ring with a jumper.
The total resistance of the parallel circuit is determined using the following equation:

1/Rtotal1 = 1/R1 + 1/R1 + 1/R2

Then,

Rtotal1 = (R1*R2/2)/(R1/2+R2)

In the case when the jumper is cut, the total resistance of the parallel circuit is

1/Rtotal = 1/R1 + 1/R1

or

Rtotal2 = R1/2

The circumference of the circle is given by the formula C = 2πr, where r is the radius of the circle.

Now, the resistance can be written by the resistance per unit length
R = λ*L, where λ is the resistance per unit length (Ω/m) and L is the length of the wire (m).

Then,

R1 = λ*πr (only half of the circle)

R2 = λ*2r (diameter)

Using this we will get

Rtotal1 = 2λπr/(π+4)

and

Rtotal2 = λπr/2

Finally, what percent of Rtotal1 is Rtotal2?

Rtotal2 is 178.5% of Rtotal1

The resistance increased by 78.5%

 

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