# Why is the torque exerted by the magnetic field in a moving coil galvanometer independent of the angle of rotation?

How does an iron core cause the magnetic field to be constant and always perpendicular to the loop's moment in a galvanometer?

The following image is from my textbook:

Image source: NCERT Physics Textbook for Class XII Part I - Page 164

Put another way, why is the torque independent of the angle?

• Related - If you look at the answer you will see a diagram of a radial magnetic field produced using curved pole pieces and a soft iron cylinder. Feb 29 '20 at 19:19
• While that answer clears up roughly how the iron core will act, my principal doubt, of how the field lines are perpendicular to the loop's moment, still remains. Feb 29 '20 at 19:31
• The edge of the coil moves in an arc of a circle and the magnetic field lines are radial. Feb 29 '20 at 21:03
• Any clue on how the field lines will be inside the core? Mar 1 '20 at 4:51
• Only as a suggestion you might look at the diagram here. Mar 1 '20 at 9:28

The cylindrical core of a galvanometer is mounted within a cylindrical gap between the N and S poles of a permanent magnet. The magnetic field lines cross a small gap between the poles and the core, going toward the center of the core as they enter from the N pole and away from the center as they cross to the S pole.

• Alright, I understand that, but why is the torque independent of the angle, is what I'm asking. Feb 29 '20 at 19:33

I hope that you have come across the following formula to determine the torque $$\tau$$ exerted by a magnetic field on a current carrying loop:

$$\vec\tau=ni(\vec A\times\vec B)$$

where, $$n$$ is the number of loops in the loop, $$i$$ is the current it carries, $$\vec A$$ is the area of the loop and $$\vec B$$ is the magnetic field.

The pole pieces are made cylindrical as shown in the following diagram:

You could see that the field lines are almost parallel to the plane of the loop. Or in other words, the magnetic field $$\vec B$$ is perpendicular to the area vector $$\vec A$$.

So torque exerted by the magnetic field on the coil is:

$$\tau=niAB\sin\theta$$

Put $$\theta=90^{\circ}$$:

$$\tau=niAB$$

Why is the torque independent of the angle?

The torque is independent of the angle because the magnetic poles are made cylindrical. If the poles were just flat (like an ordinary bar magnet), then the torque obviously depends upon the angle, but the case is different here.

• My doubt is, how is it that the field lines will always be parallel to the loop's plane? What kind of field line pattern inside the iron core will always be parallel to the loop's plane regardless of angle? How does the presence of the iron core help in this regard? Mar 1 '20 at 4:48
• @Green05 : The coil is wound over a cylindrical piece of soft iron core due to its ferromagnetic nature. The atomic magnetic moments inside the piece of iron will align in the direction of the magnetic field and reinforce the field, or in other words, the resultant magnetic field is larger when you wind the coil over a piece of iron. So the deflection is significant even for a small value of current. This is why we use iron core here. Field lines are made parallel to the loop by the cylindrical pole of the magnets. Mar 1 '20 at 4:58
• I do understand how the core strengthens the field, my doubt was about the field line pattern inside the core; I understand that it will be radial outside, but inside, when it matters, what pattern will ensure a constant magnitude and, is the equation exact or an approximation? Mar 1 '20 at 5:11

I think I got it. The iron core being ferromagnetic cause field lines to go through it when they might not have gone through that area in the core's absence. (This is due to the magnetic field created by the core's magnetisation being added to the magnet's field to make the new net field.)

This causes the following rough pattern of field lines inside the core:

Thus because of the core's ferromagnetism, the lines stay almost radial except at the centre, and at any angle of the loop, the field is radial for the most part, and so the equation used is a slight approximation. However, because of soft iron's strong ferromagnetism, the inaccuracy is tiny, as experiments have shown.

The key thing which solves the doubt is that two field lines together provide a field parallel to the loop's plane, and one alone cannot, as it change direction in the centre.

• I think you could very well have a moving coil galvanometer without a soft iron core. However, the amount of current required for a particular amount of deflection will be higher than the original setup. Mar 1 '20 at 5:11
• That's how the core increases the field, that's clear to me, my doubt was what field line pattern would ensure that the field is always along the loop's plane, if any will. Mar 1 '20 at 5:12