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Phy 122
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
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Diameter Question
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Set 2 Problem 3 states:
Under certain conditions of temperature and potential difference, a uniform wire with cross-sectional radius 2 cm carries a current of 1.3 amps. Under the same conditions, what will be the current of a second wire whose length is identical to that of the first, but which has diameter 3.7 cm?
Solution
Since the potential differences and the lengths of the wires are identical the potential gradients, or electric fields, will be identical. Thus the drift velocities will be equal.
Since the second wire has 3.7 / 2 = 1.85 the diameter of the first, it will have ( 3.7 / 2)^2 = ( 1.85) ^2 = 3.4225 the cross-sectional area of the first.
It will therefore have 3.4225 times as many electrons per unit length available to carry the current.
Thus the drift of electrons will carry 3.4225 times as many electrons past a given point every second.
The current in the second wire will therefore be 3.4225 * 1.3 amps = 4.44925 amps.
Generalized Solution
If wires having uniform circular cross-sections are of identical length and have the same potential difference from end to end, with one wire having diameter d1 and the other diameter d2, then the cross-sectional area of the second is (d2 / d1) ^ 2 times that of the first.
The second wire will therefore have (d2 / d1) ^ 2 times as many available charge carriers per unit length.
Since the drift velocity depends only on the potential gradient, the drift velocities are the same.
It follows that in any given time interval, the number of electrons drifting past a point in wire 2 is (d2 / d1) ^ 2 times as great as in wire 1.
The current in wire 2 is therefore (d2 / d1) ^ 2 times as great as in wire 1.
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Should the first line read ...cross-sectional DIAMETER 2 cm? If so all the subsequent math works. If not, I believe all the subsequent 2's should be 4's.
???Is that right???
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Thanks!
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You are correct.
Both the 2 cm and the 3.7 cm should describe the same dimension. That is both should be radius, or both should be diameter.
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