Cylindrical Conductors Carrying a Current I -- Formula (?)

requied · Jun 23, 2020

Geometric diagram of two overlapping circular cross-sections centred at A and B, each of radius a, inside a larger circle of radius r, with point O at their contact and point P2 outside on the right, showing field vectors B1, B2, B3 and B2', B3', (B2'+B3')cosθ and angle θ with labelled lengths r and a.

Typeset text: The current through the cylindrical conductor is I. Its cross-sectional area is πa². The two cavities act as conductors carrying current I' into the plane of the paper, where I' = I/(πa²) · π(a/2)² = I/4.

How can I' be the formula above? Is there any formula to get this same

$Handwritten expression: 1 divided by (pi a squared) times pi times (a/2) squared, with the fraction 1/(pi a^2) multiplied by pi(a/2)^2.$

Simon Bridge · Jun 23, 2020

The suggested path is to treat the situation as if you had a full current in the entire big conductor without the cavities, and then add in the effect of having an additional current going the other way where the cavities should be.

Did you try that?

IRL the current would be on the outer surface of the conductor ... it looks like you are supposed to model the current as uniform through the conductor (check).

Note: you can use TeX markup for equations ...

$$\frac{1}{\pi a^2}\pi \left(\frac{a}{2}\right)^2 = \frac{1}{4}$$

The code fopr that was

Code:

$$\frac{1}{\pi a^2}\pi \left(\frac{a}{2}\right)^2 = \frac{1}{4}$$

Cylindrical Conductors Carrying a Current I -- Formula (?)

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