Understanding the Effects of Wire Length and Diameter on Resistance: A Comprehensive Guide

When dealing with electrical circuits, understanding the impact of wire length and diameter on resistance is crucial. This article provides a detailed explanation and calculation for a scenario where the wire is both half the length and half the diameter of the original. We'll use the resistivity formula to determine the resistance in each case.

Resistivity and Resistance Formula

The resistance of a cylindrical conductor can be calculated using the following formula:

R  (frac{rho L}{A})

R: Resistance (rho): Resistivity of the material L: Length of the wire A: Cross-sectional area of the wire

Original Wire Resistance Calculation

Let's start with the original wire:

Length: (L) Diameter: (d)

The cross-sectional area (A_1) of the original wire is given by:

A_1  (pi left(frac{d}{2}right)^2  frac{pi d^2}{4})

Thus, the resistance (R_1) of the original wire is:

R_1  (frac{rho L}{A_1}  frac{rho L}{frac{pi d^2}{4}}  frac{4 rho L}{pi d^2})

Modified Wire with Halved Length and Diameter

Consider a new wire that is half the length and half the diameter of the original wire:

New length: (frac{L}{2}) New diameter: (frac{d}{2})

The new cross-sectional area (A_2) is calculated as follows:

A_2  (pi left(frac{d/2}{2}right)^2  pi left(frac{d}{4}right)^2  frac{pi d^2}{16})

Using the resistivity formula, the new resistance (R_2) is:

R_2  (frac{rho frac{L}{2}}{A_2}  frac{rho frac{L}{2}}{frac{pi d^2}{16}}  frac{rho L}{2} cdot frac{16}{pi d^2}  frac{8 rho L}{pi d^2})

Comparing the Two Resistances

Now we compare the new resistance (R_2) to the original resistance (R_1):

(R_1 frac{4 rho L}{pi d^2}) (R_2 frac{8 rho L}{pi d^2})

We see that:

R_2  2 R_1

Hence, the resistance of the wire that is half the length and half the diameter will be twice the resistance of the original wire.

Conclusion

This comprehensive guide not only explains the underlying principles but also provides a clear step-by-step calculation. By following these steps, you can easily determine the resistance of a wire under different conditions.

Key Takeaways

The resistance of a wire is directly proportional to its length and inversely proportional to its cross-sectional area. Halving the length of a wire halves its resistance, while halving the diameter of a wire quadruples its resistance (area). Combining these effects results in a wire with a resistance that is twice that of the original.

Bonus Exercise

To further reinforce your understanding, try the following bonus exercise:

Choose a specific resistivity (rho) and length (L), and diameter (d). Calculate the resistance (R_1) of the original wire. Calculate the resistance (R_2) of the wire with halved length and diameter. Verify that (R_2) is indeed twice (R_1).

This hands-on approach will help you grasp the concepts more effectively.