Finding the resistance of the halved rod

In summary, a 1.50 m cylinder with a radius of 1.10 cm has a resistivity that varies with distance according to the formula ρ(x) = a + bx^2. At the left end, the resistivity is 2.25x10^-8 Ωm and at the right end it is 8.50x10^-8 Ωm. The resistance of the entire rod is 1.71x10^-4. The electric field at the midpoint carrying a 1.75 A current is 1.76x10^-4 V/m. The resistance of the left half of the rod is 5.4677x10^-5 and the resistance of the right half
  • #1
Northbysouth
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2

Homework Statement


A 1.50 m cylinder of radius 1.10 cm is made of a complicated mixture of materials. Its resistivity depends on the distance x from the left end and obeys the formula ρ(x) = a + bx^2, where a and b are constants. At the left end, the resistivity is 2.25x10^-8 Ω m, while at the right end it is 8.50x10^-8 Ωm.

A) What is the resistance of this rod?

B) What is the electric field at its midpoint if it carries a 1.75 A current?

C) If we cut the rod into two 75.0 m halves, what is the resistance of each half?

Homework Equations






The Attempt at a Solution



I have solved parts A and B by doing the following:

A) R = ρL/A

First I found the constants a and b by solving for them, which I found to be:

a = 2.25x10^-8 Ωm
b = 2.78x10^-8 Ωm

A = πr^2
A = 3.8013x10-4 m2

Taking the integral of the Resistance equation gave me:

1/A*(ax + bx3/3)

I then took the integral from 0 to 1.50m, thereby giving me:

(1/3.801299x10-4)*3.38x10-8*1.50 + 2.78x10-8*1.53/3
= 1.71x10-4

This I know to be correct.

B)

∫E.dl = V
dV/dx = d/dx(IR)

1.75/A*(a + bx2)

Plugging in 0.75 for x gave me

E = 1.76x10^-4 V/m

This is also correct

C) This is where I'm stuck. I had thought that if I used the integral in part A to find the integral from 0 to 0.75 then that would be my resistance in the left hand side. Then I thought that I could subtract the left hand value from the value I found in part A to give me the right hand value, but this hasn't worked. My workings are as follow:

R = ρL/A

A = 3.8x10^-4 m2

Rleft hand1/A*(2.25x10^-8)(0.75) + (2.78x10^-8)(0.75^3/3)
= 4.43966x10^-5

Then Rright hand = 1.71x10^-4 - 4.43966x10^-5 = 1.266x10^-4

Unfortunately, this did not work and I' not sure what else to try.
 
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  • #2
I'm not sure how you've ended up with the value you did for the left had side resistance; The numbers in the formula look okay, but the calculated result does not. Calculator finger problems?
 
  • #3
Yes, I've just gotten the right answer. The correct calculation is as follows:

I know that the Resistance of the entire (1.50m) rod is 1.71x10^-4

Thus I used the integral form part A to calculate the value of R from 0 to 0.75, which gave me 5.4677x10^-5. This is the Resistance of the left hand piece of rod. I then subtracted this value from the resistance of the entire rod, thereby giving me a resistance of 1.16323x10^-4 for the right hand rod.

Thanks for your help. Sorry to waste your time.
 

1. How do you determine the resistance of a halved rod?

To find the resistance of a halved rod, you will need to measure its length, cross-sectional area, and resistivity. Then, use the formula R = (ρ * L)/A, where R is resistance, ρ is resistivity, L is the length, and A is the cross-sectional area.

2. What is the purpose of finding the resistance of a halved rod?

Finding the resistance of a halved rod can help determine its conductivity and how well it can conduct electricity. It can also be used to calculate the power dissipation and voltage drop in a circuit that includes the halved rod.

3. How does the resistance of a halved rod compare to that of a whole rod?

The resistance of a halved rod will be double that of a whole rod with the same dimensions and material. This is because the cross-sectional area of a halved rod is half of the whole rod, leading to a higher resistance.

4. Can you use the same formula to find the resistance of any halved object?

Yes, the formula R = (ρ * L)/A can be used to find the resistance of any halved object, as long as you have the necessary measurements of length, cross-sectional area, and resistivity.

5. How does the resistivity of the material affect the resistance of a halved rod?

The resistivity of a material is a constant that determines how well it can conduct electricity. A material with a higher resistivity will have a higher resistance, while a material with a lower resistivity will have a lower resistance, regardless of the dimensions of the halved rod.

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