# Why can't we consider Amperean loop inside solenoid just like a toroid?

According to my book, the magnetic field in any location inside the toroid (the empty region inside the toroid circumference) is zero because if we consider a circular loop passing through that point, the magnetic field would be zero as there is no current inside that loop. If we use this logic for inside the solenoid and take a small circle inside the solenoid with center at axis, we would get the same zero MF there, which is not true. If we apply this logic we will get a zero magnetic field wherever there is no current. I know this is not right and I am missing some point here.

The Amperian Loop centered at the axis of the solenoid encloses no current. This is true.

You note two rules.

1. By Ampere's Law, there is no magnetic field.
2. The solenoid has a magnetic field.

You think 1. and 2. contradict. However, realize that Ampere's Law only describes the net magnetic field along the Amperian Loop. It does not say anything about other magnetic fields. So let's modify the rules.

1. By Ampere's Law, there is no net magnetic field along the Amperian Loop.
2. The solenoid has a magnetic field.

To see why 1. and 2. are true and don't contradict, let's consider an ideal example, then generalize.

Ideal Example

The magnetic field lines within an infinitely long solenoid are perfectly parallel with the central axis of the solenoid. This is consistent with 2.

If you consider a circle centered at the solenoid's central axis, realize that all magnetic field lines are perpendicular to the circle.

So, there are no magnetic field lines that go along the Amperian Loop. This is consistent with 1.

General Example

In the real world, we can't have infinitely long solenoids. Real world solenoids deviate from the ideal solenoid.

A single wire loop has a circularly symmetric magnetic field. Similarly, a stack of single wire loops have a circularly symmetric magnetic field. This is consistent with rule 2.

Since the solenoid is made up of circles, note that any deviation from the ideal must be circularly symmetric.

By circular symmetry, there is no net magnetic field along the circular Amperian Loop. This is consistent with rule 1.

Conclusion

Rules 1 and 2 hold for all solenoids, and don't contradict each other.

(P.S. I'm pretty sure, but cannot prove, that real-world solenoids have magnetic field lines that deviate radially outwards from the central axis, radially perpendicular to the Amperian Loop. So, there is no single magnetic field line that contributes to the Amperian Loop, and we don't even need to consider circular symmetry.)

You have to be careful and understand what you are doing when evaluating the left hand side of the equation $\displaystyle \int_{\rm loop}\vec B \cdot d\vec l= \mu_0 I_{\rm enclosed}$.

For loops $X$ and $Z$ you can appeal to symmetry and say that the magnetic field $B$ has the same magnitude and is in the same direction as the loop at all points around the loop.

However for loop $Y$ because the integral $\displaystyle \int_{\rm loop}\vec B \cdot d\vec l$ is zero around the loop that does not mean that the magnetic field is zero at each point along the loop.
You will see from the diagram you will get positive and negative contributions to $\vec B \cdot d\vec l$ which will add up to zero.

• Actually my question was regarding loop z which is in the open area ( not inside the toroid). Why is the magnetic field zero at any point p inside, the explanation in the book is because there is that if we draw a loop , for example, z in this case ,since there is no enclosed current , MF is 0. Likewise, can we take loop anywhere outside the toroid and say there is no enclosed current hence the mf is zero or anywhere inside Solonoi.
– Raj
Jul 16 '17 at 9:09
• By symmetry the magnitude of the magnetic field must be the same at each point along the loop and the direction of the magnetic field must be at a tangent to the loop. This makes the integral $Bz$ where $z$ is the circumference of the loop. Since the enclosed current is zero the magnetic field $B$ must be zero. Jul 16 '17 at 9:19