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# Hard to Resist

6 replies to this topic

### #1 mmiguel

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Posted 03 September 2012 - 10:16 AM

A funky looking piece of electrical material has a length of 2 cm on the x-axis, which is the axis of current flow.
Assume a coordinate system where the object is sitting between x = -1 cm and x = 1 cm.
The cross-sectional shape of the object is circular, with radius varying as a function of x like so:
r(x) = 0.04 + 0.4*x^2+x^4
r(x) is in cm

A differential volume of this material has a resistance of R ohms in the the direction of current flow, where R varies with distance from the x-axis. If the distance of the differential element from the x-axis is p, then R(p) = 2*sqrt(p)
R(p) is in ohms

What is the total resistance of the whole object for current flowing along the x-axis?

Edited by mmiguel, 03 September 2012 - 10:20 AM.

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### #2 mmiguel

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Posted 03 September 2012 - 11:01 PM

A funky looking piece of electrical material has a length of 2 cm on the x-axis, which is the axis of current flow.
Assume a coordinate system where the object is sitting between x = -1 cm and x = 1 cm.
The cross-sectional shape of the object is circular, with radius varying as a function of x like so:
r(x) = 0.04 + 0.4*x^2+x^4
r(x) is in cm

A differential volume of this material has a resistance of R ohms in the the direction of current flow, where R varies with distance from the x-axis. If the distance of the differential element from the x-axis is p, then R(p) = 2*sqrt(p)
R(p) is in ohms

What is the total resistance of the whole object for current flowing along the x-axis?

Let me know if you want any hints
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### #3 bonanova

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Posted 03 September 2012 - 11:36 PM

Spoiler for approach

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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

### #4 Rob_Gandy

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Posted 04 September 2012 - 10:21 PM

A funky looking piece of electrical material has a length of 2 cm on the x-axis, which is the axis of current flow.
Assume a coordinate system where the object is sitting between x = -1 cm and x = 1 cm.
The cross-sectional shape of the object is circular, with radius varying as a function of x like so:
r(x) = 0.04 + 0.4*x^2+x^4
r(x) is in cm

A differential volume of this material has a resistance of R ohms in the the direction of current flow, where R varies with distance from the x-axis. If the distance of the differential element from the x-axis is p, then R(p) = 2*sqrt(p)
R(p) is in ohms

What is the total resistance of the whole object for current flowing along the x-axis?

Spoiler for approach

Spoiler for So...

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### #5 mmiguel

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Posted 05 September 2012 - 04:47 AM

Spoiler for So...

p is radius from x axis

R(p) is resistance in ohms as a function of radius from x-axis
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### #6 bonanova

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Posted 06 September 2012 - 06:56 AM

Wikipedia has a helpful elementary discussion of resistance [unit = Ohm] of an object as it relates to the resistivity [unit = Ohm-cm] of a material. The easy relationship is that the resistance in Ohms of a 1-cm cube of material is equal numerically to its resistivity in Ohm-cm. Resistance = resistivity x length / area in compatible units.

It thus makes sense to speak of a material having a resistivity that is [or, as in the case of the OP, is not] constant. Thus, in the formula R(p) = 2*sqrt(p), R refers to the resistivity of the material. After you've specified the material's shape, i.e. created an object, it's meaningful to speak of resistance.

Again, the distinction is that resistivity is a property of a material, independent of its shape; resistance is a property of an object. Resistance depends on the object material's resistivity and on the object's shape.

Spoiler for So ...

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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

### #7 mmiguel

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Posted 06 September 2012 - 07:41 AM

Wikipedia has a helpful elementary discussion of resistance [unit = Ohm] of an object as it relates to the resistivity [unit = Ohm-cm] of a material. The easy relationship is that the resistance in Ohms of a 1-cm cube of material is equal numerically to its resistivity in Ohm-cm. Resistance = resistivity x length / area in compatible units.

It thus makes sense to speak of a material having a resistivity that is [or, as in the case of the OP, is not] constant. Thus, in the formula R(p) = 2*sqrt(p), R refers to the resistivity of the material. After you've specified the material's shape, i.e. created an object, it's meaningful to speak of resistance.

Again, the distinction is that resistivity is a property of a material, independent of its shape; resistance is a property of an object. Resistance depends on the object material's resistivity and on the object's shape.

Spoiler for So ...

I was attempting to recreate a problem from an EE final I had 1.5 years ago based on memory, but I realized after I submitted that I had some minor problems in the statement. One which causes the actual calculation to be slightly more difficult than I anticipated (harder integral), and now.... units.

I would argue that the differential volume elements have shape and are objects, but this would not be helpful, because integration involves more than just summing stuff together - it also involves a change in units due to multiplication of differential elements of whatever units belong to the variable of integration. Thus if you intend to use something as part of an integrand, such that after integration it becomes what you originally had, what's in the integrand is actually a density of the original quantity vs. being portions of the original quantity.

Spoiler for

In essence, bonanova is correct.

Edited by mmiguel, 06 September 2012 - 07:48 AM.

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