# The rate of change of magnetic flux linkage with a rotating coil

When does the rate of change of magnetic flux linkage with a coil rotating in a uniform magnetic field equal zero?

I think the answer would be:

when the coil completes one revolution.

as the the the change in the magnetic flux in the first half of the revolution is in opposite direction to the change in the magnetic flux in the second half, or I am wrong? Because someone told me that it will be when the coil is parallel to the magnetic field lines, so what is correct?

The above picture holds the answer. So, why should maximum voltage occur when the coil is in-line with the lines of magnetic field (as shown).

Remember, the formula for induced voltage is proportional to rate of change of flux "cut" by the coil. This is an instantaneous quantity and not something that is "accumulated" over one rotation.

When the coil is in the position shown, there are no flux lines "cut" but one instant afterwards it is indeed "cutting" lines of flux so. the rate of change of flux lines cut is significant in this transitory area. Its rate rises from zero to some value dependant on the new angle of the coil and the speed of rotation.

Now consider what the rate of change of flux lines cut is when the coil is repositioned by 90 degrees (vertical to the picture shown). The maximum number of flux lines are passing through the coil but there are the same number of flux lines being cut slightly before and slightly after and therefore, the rate of change is actually zero.

A few further thoughts.

I can see how this might be confusing because if the magnetic field (from N to S) were produced from an open iron core and coil fed from an AC voltage, the maximum induced voltage to the rotatable coil would be when that coil is in the vertical position. This possibly seems contrary to the spinning coil and magnet shown in the picture.

• Dear Andy, I loved your answer, but I didn't understand when you said "the same number of flux lines being cut slightly before and slightly after" when you spoke about the vertical position, because I thought that the number of the flux lines must be fewer when the coil is not actually perpendicular to the field lines. Or I didn't interpret your words correctly? – Asmaa Jun 6 '17 at 18:53
• The flux lines are zero when the coil is flat but you immediately cut lines when moved away from flat. This represents a bigger change in flux than when the coil is vertical and rotated slightly. Looking at it from the north pole perspective, the flat coil goes from having no perceived presented area to having "some" area rather like rotating a piece of paper from its edge. One second you don't see it then it appears. That relative change isn't as apparent when the paper is face on and rotated a fraction. – Andy aka Jun 6 '17 at 19:04
• OK, I think I got it! – Asmaa Jun 6 '17 at 19:07
1. Write down an equation for the flux through the coil as a function of angle.

$$\Phi =BNA\sin \theta$$

where A is the area of the coil, N is the number of turns in the coil and theta is the angle between the B field and the plane that the coil lies in.

1. Differentiate the equation once with respect to angle. The result is the rate of change of flux through the coil as it rotates.

$$\frac{d\Phi}{d\theta} =BNA\cos \theta$$

1. Set the result equal to zero and solve for the angle.

$$0=BNA\cos \theta$$

$$\theta = 90, 270$$

• Sorry, I am poor at math, I will appreciate it so much if you show me how. Thanks in advance! – Asmaa Jun 6 '17 at 16:56
• I've edited the answer to include equations. – Supa Nova Jun 6 '17 at 17:28
• so the rate of change of magnetic flux linkage will be zero when the coil is parallel to the field lines, right? – Asmaa Jun 6 '17 at 18:01
• Yes, that's right. Just a formal way to arrive at the solution. – Supa Nova Jun 6 '17 at 18:45
• oops! $\cos \theta = 0$ when $\theta = 90, 270$, so the rate of change of flux is zero when the coil is perpendicular to the flux lines. – Supa Nova Jun 6 '17 at 18:56

'when the coil completes one revolution' is when the total change in flux from start to finish adds up to zero. This is not what the question was asking.

The rate of change of flux linkage is zero when there is no change in flux linkage with a small change in the angle of the coil. This occurs when the plane of the coil is normal to the field, the axis of the coil is parallel to the field, when the coil encloses the maximum possible flux.

• I really got confused, In my book there is a statement says: "The rate with which the coil intercepts the lines of magnetic field in the dynamo is maximum when the plane of the coil is parallel to the lines of the magnetic flux." is there a difference between "intercept" and the "flux linkage"? What is the difference between this phrase in the comment and the phrase in the question? – Asmaa Jun 6 '17 at 11:36
• Intercept is a tricky word. I can understand how it can be taken either way. I even less like 'the rate at which it intercepts...'. That's why I used the word 'encloses', which I think is clearer. – Neil_UK Jun 6 '17 at 13:09
• 'plane of the coil'... Don't you mean 'axis of the coil"? – user28910 Jun 6 '17 at 18:03
• @user28910 doh! I thought so hard before I typed that as well. You're the first person to spot that in 7 hours. I'm glad somebody has finally put up a diagram. – Neil_UK Jun 6 '17 at 19:38