Resistivity equation relating to equations for resistors

[lpettigrew]

Homework Statement

Hello, I have been trying to answer this question but I am really struggling. As you can see, I have attempted to formulate a solution but I do not think that it is correct and I am grasping at straws really.

The equation R = pL /A can be related to the equations for resistors in series and parallel. Consider a wire of resistivity p, length x and diameter d.
1. Write an equation for resistance R in terms of k, x and d, where k is a constant.

2. If n wires of length x are joined together, this is the same as joining n resistors in series. Show this mathematically by using the equation for resistors in series to express the resistance of a wire of length nx and comparing it to the equation for resistivity.

3. If n wires of diameter d are joined side-by-side, this is the same as joining resistors in parallel. By what factor does it effectively increase the cross-sectional area of the wire? Use the equation for resistors in parallel to demonstrate.

However, as can be inferred I am very confused and have just jotted down some rough ideas but I would really appreciate some further help and explanation!

Relevant Equations

R = pL/A

1. Would k be p/π? (since resistivity of a material is a constant property and π is a constant)
I understand that typically R = pL/A
Therefore, would the equation be R = k(x/(d/2^2))
Confessedly, I truly am baffled.2. R total = sum of individual resistors (in series)
R total = R1 + R2+R3...+Rn etc.
If all of the resistances are the same;
R total=nR (where n is the number of components with resistance R)

R=pL/A or R=px/π*(d/2^2)

When L=x, then n lengths of wire x have resistance of:
R total= n(px/A) - Total resistance
R total = R total = n(px/A) - (pL/A) = nR
Total resistance = n*the resistance of one wire

3. Honestly, I do not know where to begin, I have attempted rearranging and substituting the equation but I ended up confusing myself more so.
I do understand that in parallel the total resistance is equal to the sum of the reciprocal of the resistances of the components.
R total = 1/ R1+ 1/R2+1/R3...+1/Rn etc.

 

[DrClaude]

1. Would k be p/π? (since resistivity of a material is a constant property and π is a constant)
I understand that typically R = pL/A
Therefore, would the equation be R = k(x/(d/2^2))

Put all constants in k.

2. R total = sum of individual resistors (in series)
R total = R1 + R2+R3...+Rn etc.
If all of the resistances are the same;
R total=nR (where n is the number of components with resistance R)

R=pL/A or R=px/π*(d/2^2)

When L=x, then n lengths of wire x have resistance of:
R total= n(px/A) - Total resistance
R total = R total = n(px/A) - (pL/A) = nR
Total resistance = n*the resistance of one wire

I don't understand your subtraction in there. Also, once you have R interns of k, x, and d, you should continue with that notation.

There are different ways to approach that question. I suggest you start with one case, resistors in series or one long wire, and show that you can transform the equation into the other case.

3. Honestly, I do not know where to begin, I have attempted rearranging and substituting the equation but I ended up confusing myself more so.
I do understand that in parallel the total resistance is equal to the sum of the reciprocal of the resistances of the components.
R total = 1/ R1+ 1/R2+1/R3...+1/Rn etc.

Same as for question 2. Start with one formulation, resistors in parallel or wires stuck together, and show that you can rearrange to get the other case.

 

[kuruman]

Put all constants in k.

The problem says "Write an equation for resistance R in terms of k, x and d, where k is a constant." Constant d should not be part of k.

 

[DrClaude]

The problem says "Write an equation for resistance R in terms of k, x and d, where k is a constant." Constant d should not be part of k.

I see more than just k, x, and d on the RHS of @lpettigrew's formula

 

[kuruman]

I see more than just k, x, and d on the RHS of @lpettigrew's formula

Ah, yes. It's so obvious that becomes transparent.

 

[lpettigrew]

Put all constants in k.I don't understand your subtraction in there. Also, once you have R interns of k, x, and d, you should continue with that notation.

There are different ways to approach that question. I suggest you start with one case, resistors in series or one long wire, and show that you can transform the equation into the other case.Same as for question 2. Start with one formulation, resistors in parallel or wires stuck together, and show that you can rearrange to get the other case.

Hello, thank you for your reply. I am really confused could you explain what you mean about taking a different approach. I do not know why I included that subtraction upon reflection either.

How would I rearrange the equation for resitivity in terms of k, x and d?

Sorry to be so uncertain

 

[DrClaude]

Hello, thank you for your reply. I am really confused could you explain what you mean about taking a different approach. I do not know why I included that subtraction upon reflection either.

It is difficult for me to follow your demonstration. This is why I gave you a hint as to how it can be done. You can simply try and clean ups what you wrote in post #1.

How would I rearrange the equation for resitivity in terms of k, x and d?

I would move the numerical constant into k.

 

[Tom.G]

I do understand that in parallel the total resistance is equal to the sum of the reciprocal of the resistances of the components.
R total = 1/ R1+ 1/R2+1/R3...+1/Rn etc.

Uhmm... you might want to review that for future reference. When more resistors are added in parallel does R_TOTAL increase or decrease?

Cheers,
Tom

 

[lpettigrew]

hen more resistors are added in parallel does RTOTAL increase or decrease?

By adding more resistors in parallel the equivalent resistance decreases and the toal current increases.

 

[lpettigrew]

It is difficult for me to follow your demonstration. This is why I gave you a hint as to how it can be done. You can simply try and clean ups what you wrote in post #1.I would move the numerical constant into k.

Sorry I am really confused and do not know where to start from. Can we just start afresh approaching the question instead of modifying my previous attempt?

 

[DrClaude]

Sorry I am really confused and do not know where to start from. Can we just start afresh approaching the question instead of modifying my previous attempt?

Yes, go ahead.

 

[lpettigrew]

I do not know where to begin really. When you say that the numerical constants should be in k, are you referring to π?

 

[phinds]

By adding more resistors in parallel the equivalent resistance decreases and the toal current increases.

Current increases only provided that the rest of the circuit can provide more current. What can also happen is that voltage decreases. This is extraneous to the heart of your discussion but you should be careful about making categorical statements that are actually limited by circumstances.

 

[lpettigrew]

Current increases only provided that the rest of the circuit can provide more current. What can also happen is that voltage decreases. This is extraneous to the heart of your discussion but you should be careful about making categorical statements that are actually limited by circumstances.

Thank you for your informative statement. In actuality, I did not know that an increase in current was a circumstantial and was merely thinking along the lines of what I had read in my textbook.

 

[DrClaude]

I do not know where to begin really. When you say that the numerical constants should be in k, are you referring to π?

There is a numerical factor (some power of 2) that should be moved to k.

That gives you an equation for the resistance of one particular resistor of length x. What is the resistance of n such resistors? What is the resistance of one resistor of length nx? Compare the two.

 

[lpettigrew]

Are you referring to the numerical power (a power of 2) being d/2^2?

 

[Tom.G]

By adding more resistors in parallel the equivalent resistance decreases and the toal current increases.

Correct.

However your first post has this:

R total = 1/ R1+ 1/R2+1/R3...+1/Rn etc.

If there are two parallel resistors, say 1Ω and 2Ω, that formula says

R total = 1/1 + 1/2
R total = 1+½
R total = 1½Ω,
which is an increase in value.

That's why I suggested you might want to review the situation.

Cheers,
Tom

 

[DrClaude]

Are you referring to the numerical power (a power of 2) being d/2^2?

Correct.

 

[lpettigrew]

Correct.

So does k = (d/2)^2?
In which case the equation would be R=px/π*k

 

[vela]

Write an equation for resistance R in terms of k, x and d, where k is a constant.

You're being asked to express R in terms of only k, x, and d. No other variables, like $\rho$, should appear in the expression. When you separate out the factors that depend on $x$ and $d$, you should be able to see that what remains is a constant, which you identify as the constant $k$.

 

[lpettigrew]

You're being asked to express R in terms of only k, x, and d. No other variables, like $\rho$, should appear in the expression. When you separate out the factors that depend on $x$ and $d$, you should be able to see that what remains is a constant, which you identify as the constant $k$.

Thank you very much for your explanation that has helped massively.
So with the original equation being R=px/π(d/2)^2
So, the constant would be ρ/π?
R=k(x/(d/2)^2) (which I posted earlier
DrClaude told me to move all numerical constants into k, would this become:
R=√2k(x/d) ?

 

[DrClaude]

DrClaude told me to move all numerical constants into k, would this become:
R=√2k(x/d) ?

You still have a numerical constant there! Also, where does the square root come from?
Consider your original equation:

Therefore, would the equation be R = k(x/(d/2^2))

The only thing you need to do is to make the "divide by 2" part of k.

 

[lpettigrew]

You still have a numerical constant there! Also, where does the square root come from?
Consider your original equation:

The only thing you need to do is to make the "divide by 2" part of k.

So, R=k/2(x/d)?

I am very confused

 

[vela]

Everything that's not an $x$ or a $d$ should end up in $k$. That includes numerical constants.

By the way, it's not helping that you're either making really basic algebraic mistakes or typing them in wrong. Double-check your work before clicking on "post reply". It would also help if you showed your work step by step so we can identify possible mistakes.