# The Ring

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?

###### 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%