Calculating resistance between two bolts connected to a metal plate

[foges]

Homework Statement

Given is a metal plate with two metal bolts attached to it, the attachement is assumed to be idealy conducting. A current I flows between the two bolts. Find a general formula for the Resistance between the two bolts (depending upon: a) the thickness of the plate (delta) b) the radius of the Bolt (r) c) the distance between the center-points of both bolts (l) d) and the specific conductivity of the metal that the plate is made of (kappa))

Here is what it looks like:

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|[COLOR="#black"]aaaaa[/COLOR]<>[COLOR="#black"]aaaaaaaaaaaaaaaaaaaaa[/COLOR]<>[COLOR="#black"]aaa[/COLOR]|
| aaaaa2r [COLOR="#black"]aaaaaaaaaaaaaaaaaaa[/COLOR]2r[COLOR="#black"]aaaaa|[/COLOR]
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|__________________________________|

Edit: ignore the white a's. i tried making them spaces, but the BB removed the spaces

Homework Equations

Here is the answer, but how do you get there:

R = ln( 2(l-r)/ (2r) )/( pi * kappa * delta)

The Attempt at a Solution

the ln() must come from integrating one over circumference: 1/( r * 2 * pi) from (2r) to (2(l-r)) but why?



Thanks

 

[KataKoniK]

for the question! It's always great to see someone interested in the mathematical reasoning behind a formula.

First of all, let's start by defining some variables to make things clearer:

- t = thickness of the plate (delta)
- r = radius of the bolt
- l = distance between the center points of both bolts
- k = specific conductivity of the metal plate (kappa)
- R = resistance between the bolts

Now, let's think about what factors could affect the resistance between the bolts. The first thing that comes to mind is the thickness of the plate. Intuitively, we can imagine that a thicker plate would provide more resistance than a thinner one. This is because the current has to travel through more material, which would slow it down and create more collisions with the atoms in the metal, thus increasing the resistance.

Next, let's consider the radius of the bolt. A larger radius would mean that the current has to travel through a larger cross-sectional area of the bolt, which would decrease the resistance. On the other hand, a smaller radius would mean a smaller cross-sectional area and thus a higher resistance.

Now, let's think about the distance between the bolts. As the distance increases, the current has to travel through a longer path, which would increase the resistance. Similarly, if the distance decreases, the resistance would decrease as well.

Finally, we have the specific conductivity of the metal plate. This is a measure of how easily the material allows electric current to flow through it. A higher conductivity would mean a lower resistance, and vice versa.

Now, let's try to put all of these factors together and see if we can come up with a formula for the resistance between the bolts.

We know that resistance is given by the following formula:

R = (resistivity * length) / (cross-sectional area)

In our case, the length would be the distance between the bolts (l), and the cross-sectional area would be the area of the bolt (pi * r^2). The resistivity is related to the specific conductivity by the following formula:

resistivity = 1 / (specific conductivity)

So, we can rewrite the formula for resistance as:

R = (1 / (specific conductivity)) * (l / (pi * r^2))

Now, let's substitute the variables we defined earlier:

R = (1 / k) * (l / (pi * r^2))

But we're not quite there yet

 

[piercas]

for sharing this problem! I would approach this problem by first identifying the physical principles at play. In this case, we are dealing with electrical resistance, which is a measure of how much a material resists the flow of electric current. It is dependent on the material's properties, such as conductivity, as well as the physical dimensions of the material.

To solve this problem, we can use Ohm's law, which states that the resistance (R) is equal to the voltage (V) divided by the current (I). In this case, we are given the current (I) and we need to find the resistance (R). So we can rearrange Ohm's law to solve for R:

R = V/I

Next, we need to find the voltage (V) between the two bolts. This can be done using the concept of potential difference, which is the difference in electrical potential energy between two points. In this case, the two bolts are at different distances from each other, so there will be a difference in potential energy between them.

To find this potential difference, we can use the formula V = IR, where I is the current and R is the resistance. We know the current (I) and we can use the formula given in the problem to find the resistance (R).

Now, we need to find the potential difference between the two bolts. This can be done by considering the electric field between the two bolts. The electric field is a measure of the force per unit charge, and it is dependent on the distance between the two bolts.

So, we can use the formula for electric field (E = kQ/r^2) to find the potential difference (V) between the two bolts. Here, k is the Coulomb's constant, Q is the charge on one of the bolts (assuming it is the same for both bolts), and r is the distance between the two bolts.

Now, we can substitute our expression for V into the formula V = IR to get an expression for R in terms of the given parameters:

R = (kQ/r^2) * (ln(2(l-r)/(2r)) / (pi * kappa * delta))

Here, we can see that the ln() term comes from the integration of the electric field over the circumference of the bolts. This is because the electric field is not constant over the entire circumference, so we need to integrate to get the total potential difference between the two