Calculus I homework question about electric current pulse velocity dependent on wire gauge.
I understand the calculus I think using L'Hospital's rule, but when the problem asks me "what is the significance of this limit" I have no clue.
Also the limits make no sense because they are saying that if a wire has no insulation then it has no current in the wire, and it is saying that if the wire has a conductor radius that is 6 times radius of the conductor and insulator radius combined then the signal can travel 64.5 times faster than the speed of light, which everyone knows it is impossible to travel faster than the speed of light.
Is this the whole reason why its a L'Hospital's rule problem since traveling faster than the speed of light is impossible so faster than the speed of light can be considered infinite velocity and if the time is infinite then we have infinite distance over infinite time so then we have indeterminate form "infinity over infinity"?
Also the problem has a negative c, which c is obviously the velocity of light. How can the velocity of light be negative?
Here is the problem word for word:
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A metal cable has radius r and is covered by insulation, so that the distance from the center of the cable to the exterior of the insulation is R. The velocity v of an electrical impulse in the cable is
v = - c ( ( r / R ) ^ 2 ) ln ( r / R )
where c is a positive constant. Find the following limits and interpret your answers.
(a) lim ( v ) as R --> r+
(b) lim ( v) as r --> 0+
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So if the velocity v is in meters per second and if c is a constant in meters per second, r is the radius of the conductor in meters, and R is the radius of both the conductor and the insulator in meters, then meters squared over meters squared cancels out and meters over meters cancels out leaving meters/second from c as the unit of velocity for v.
So to test this I plugged in numbers in my calculator to try to make sense of these limits.
So for test values to see if the transmission speed is reasonable with the limits I used the gauge of telephone wire (24 gauge).
Telephone wire is a copper wire so it is a non-ferrous metal. Ferrous metal that is 24 gauge is...no suprises...about 0.024 inches thick. Non-ferrous metal that is 24 gauge is about .0201 inches thick.
24 gauge copper wire in metric is about 0.559 millimeters.
So lets say that the insulation of the telephone wire is half of that radius thick, which we will call 0.3 millimeters.
So then our r would be 0.6 mm and our R would be 0.6 mm + 0.3 mm = 0.9 mm.
So plugging those values into the equation including the negative sign, we get
( ( - 3.0 E 8) m/s ) ( ( 0.0006 m / 0.0009 m ) ^ 2 ) ( ln ( 0.0006 m / 0.0009 m ) )
= 54, 062, 014.41 m / s
To find out how fast this is compared to the speed of light, we divide by 3.0 E 8
This is .18 times as fast as the speed of light.
but what if we were allowed to destroy matter and we found a way with quantum physics to make a wire with no insulation but still having insulation properties such that R is less than r instead of greater than r? Then if R was less than r we would have exceeded the speed of light.
( ( - 3.0 E 8) m/s ) ( ( 0.0006 m / 0.0001 m ) ^ 2 ) ( ln ( 0.0006 m / 0.0001 m ) )
= 1.935 E 10 m/s = 64.5 times faster than the speed of light.
Is that why it is currently impossible for a wire to conduct faster than the speed of light because it is currently impossible to destroy matter creating an R that is less than r ?
but what if the wire had no insulation, such that r = R ?
Then the equation would suggest that ln ( r / R ) = ln (1) = 0, so the velocity is 0 when r = R.
That doesn't make any sense, because the electricity company uses bare wires on its electric poles and they sure have current flowing.
Having no velocity of current impulses would mean that no current is flowing.
I understand the calculus I think using L'Hospital's rule, but when the problem asks me "what is the significance of this limit" I have no clue.
Also the limits make no sense because they are saying that if a wire has no insulation then it has no current in the wire, and it is saying that if the wire has a conductor radius that is 6 times radius of the conductor and insulator radius combined then the signal can travel 64.5 times faster than the speed of light, which everyone knows it is impossible to travel faster than the speed of light.
Is this the whole reason why its a L'Hospital's rule problem since traveling faster than the speed of light is impossible so faster than the speed of light can be considered infinite velocity and if the time is infinite then we have infinite distance over infinite time so then we have indeterminate form "infinity over infinity"?
Also the problem has a negative c, which c is obviously the velocity of light. How can the velocity of light be negative?
Here is the problem word for word:
-----------------------------------------------------------------------------------------------------------------------------------
A metal cable has radius r and is covered by insulation, so that the distance from the center of the cable to the exterior of the insulation is R. The velocity v of an electrical impulse in the cable is
v = - c ( ( r / R ) ^ 2 ) ln ( r / R )
where c is a positive constant. Find the following limits and interpret your answers.
(a) lim ( v ) as R --> r+
(b) lim ( v) as r --> 0+
----------------------------------------------------------------------------------------------------------------------------------
So if the velocity v is in meters per second and if c is a constant in meters per second, r is the radius of the conductor in meters, and R is the radius of both the conductor and the insulator in meters, then meters squared over meters squared cancels out and meters over meters cancels out leaving meters/second from c as the unit of velocity for v.
So to test this I plugged in numbers in my calculator to try to make sense of these limits.
So for test values to see if the transmission speed is reasonable with the limits I used the gauge of telephone wire (24 gauge).
Telephone wire is a copper wire so it is a non-ferrous metal. Ferrous metal that is 24 gauge is...no suprises...about 0.024 inches thick. Non-ferrous metal that is 24 gauge is about .0201 inches thick.
24 gauge copper wire in metric is about 0.559 millimeters.
So lets say that the insulation of the telephone wire is half of that radius thick, which we will call 0.3 millimeters.
So then our r would be 0.6 mm and our R would be 0.6 mm + 0.3 mm = 0.9 mm.
So plugging those values into the equation including the negative sign, we get
( ( - 3.0 E 8) m/s ) ( ( 0.0006 m / 0.0009 m ) ^ 2 ) ( ln ( 0.0006 m / 0.0009 m ) )
= 54, 062, 014.41 m / s
To find out how fast this is compared to the speed of light, we divide by 3.0 E 8
This is .18 times as fast as the speed of light.
but what if we were allowed to destroy matter and we found a way with quantum physics to make a wire with no insulation but still having insulation properties such that R is less than r instead of greater than r? Then if R was less than r we would have exceeded the speed of light.
( ( - 3.0 E 8) m/s ) ( ( 0.0006 m / 0.0001 m ) ^ 2 ) ( ln ( 0.0006 m / 0.0001 m ) )
= 1.935 E 10 m/s = 64.5 times faster than the speed of light.
Is that why it is currently impossible for a wire to conduct faster than the speed of light because it is currently impossible to destroy matter creating an R that is less than r ?
but what if the wire had no insulation, such that r = R ?
Then the equation would suggest that ln ( r / R ) = ln (1) = 0, so the velocity is 0 when r = R.
That doesn't make any sense, because the electricity company uses bare wires on its electric poles and they sure have current flowing.
Having no velocity of current impulses would mean that no current is flowing.
