關於circular coil和solenoid公式的問題

2012-02-15 10:14 pm

The equations for magnetic flux intensity at the center of a circular coil and a solenoid are B=(permeability of air)(number of turns of the coil)(current) / 2(radius of the coil)and B=(permeability of air)(number of turns of the coil)(current) / (length of the solenoid) respectively.

I am very confused when thinking of a solenoid having a length as short as that of a circular coil of N turns:

According to the equation of solenoid, B=(m_0)NI/L. So, when L tends to 0, B tends to infinity. Also, when L tends to 0, a solenoid will be like a circular coil. However, it is inconsistent with the equation of circular coil when a solenoid is compressed into a small length and becomes a circular coil, which states that B=(m_0)NI/2R, and which therefore depends only on the radius of the coil.

?? What's wrong

回答 (1)

天同

2012-02-16 12:59 am

✔ 最佳答案

The equation for a solenoid B=(m_0)NI/L (with symbols defined in your question) in fact gives the magnetic flux density B at the central axis of an infinitely long solenoid. If the solenoid is of finite length, the equation doesn't hold and needs to be modified.

The following web-page gives a good description on the appropriate equation for a finite solenoid.

http://instruct.tri-c.edu/fgram/web/b-coil.htm

The equation involved for a finite solenoid is,

B = [(m_0)NI/2L].[cos(alpha) + cos(beta)]
where angles (alpha) and (beta) are angles subtented by a point, on the central axis of the solenoid, at the two ends respectively. If the point is right at the centre of the solenoid and the length L is small (such that the solenoid becomes a coil), then angle(alpha) = angle(beta) = (L/2)/R = L/2R

Hence, the solenoid equation reduces to
B = [(m_0)NI/2L].[L/2R + L/2R]
i.e. B = [(m_0)NI/2L].(L/R)
or B = [(m_0)NI/2R which is exactly the equation for a coil.

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