a piece of wire has a resistance of R , it is cut into three pieces of equal length,and the pieces are twisted together parallel to each other. what is the resistance of the resulting wire in terms of R
nine
R/3
As we know that
R is cutted into 3 equal parts and they are parallel to each other
But we have to remember that we have resultant in terms of R
Thus
Rn=1/R1+1/R2+1/R3
As R1=R2=R3=R
Thus 1/R+1/R+1/R=1/Rn
1/Rn=3/R
Rn=R/3
To find the resistance of the resulting wire after twisting the three pieces together, we need to understand how resistance works in parallel circuits and how the length of a wire affects its resistance.
1. Resistance in Parallel Circuits:
When resistors are connected in parallel, the total resistance (R_total) is given by the formula:
1/R_total = 1/R1 + 1/R2 + 1/R3 + ...
In our case, we have three pieces of wire, so we will use this formula to calculate the equivalent resistance.
2. Resistance of a Wire:
The resistance of a wire is proportional to its length. Let's assume the initial length of the wire is L and the resistance is R. If the wire is cut into three equal pieces, each piece will have a length of L/3.
Since the wire pieces are twisted together in parallel, the total length of their combined path remains the same, which is L. This means that each wire piece contributes a length of L/3 to the total length.
Now, we can calculate the resistance of each piece after it is cut:
R_piece = R * (L_piece / L)
Since all three pieces are of equal length, L_piece = L/3.
R_piece = R * (L/3) / L
R_piece = R/3
Therefore, each piece of the wire has a resistance of R/3.
3. Resistance of the Resulting Wire:
Now that we know each piece has a resistance of R/3, we can calculate the total resistance of the resulting wire by using the formula for resistors in parallel:
1/R_total = 1/(R/3) + 1/(R/3) + 1/(R/3)
1/R_total = 3/R + 3/R + 3/R
1/R_total = 9/R
R_total = R/9
So, the resistance of the resulting wire, when three pieces are twisted together in parallel, is R/9.