Solution:
We are given a piece of wire with resistance R, and it is cut into 5 equal parts. The parts are then connected in parallel. The equivalent resistance of this combination is R', and we are tasked with finding the ratio R/R'.
Step-by-step Solution:
1. Wire Resistance (R)
The wire has an initial resistance denoted by R.
2. Cutting the Wire into 5 Equal Parts:
When the wire is cut into 5 equal parts, the resistance of each piece becomes R/5 because resistance is proportional to length. Since the wire is cut into five equal parts, each part is one-fifth of the original length, which means the resistance of each part is reduced by a factor of 5.
3. Connecting the Pieces in Parallel:
When resistors are connected in parallel, the reciprocal of the equivalent resistance (R') is the sum of the reciprocals of the individual resistances. The formula for parallel resistors is:
1/R' = 1/R₁ + 1/R₂ + ... + 1/Rₙ
For 5 resistors, each with resistance R/5, the equation becomes:
1/R' = 1/(R/5) + 1/(R/5) + 1/(R/5) + 1/(R/5) + 1/(R/5)
Which simplifies to:
1/R' = 5 × 5/R = 25/R
4. Solving for R':
Now, we take the reciprocal of both sides to solve for R':
R' = R/25
5. Finding the Ratio R/R':
The ratio of the original resistance R to the equivalent resistance R' is:
R/R' = R/(R/25) = 25
Final Answer:
The ratio R/R' is 25.
