Why is the magnetic field inside an ideal solenoid uniform

[jayman16]

I understand that the magnetic field at points inside the solenoid is the vector sum of the B field due to each ring. How can the field inside be uniform then since if you consider just one loop in the solenoid the value of the B field is different at different points in that circle or loop.

 

[ModusPwnd]

But a solenoid is not one loop right? Ideally its an infinitely long line of loops. At each point your B field is the sum of the contribution from each of the infinite number of loops.

 

[jayman16]

ya but at different points in each of the loops let it be the center or a point that is off center, the B is different is it not?

 

[vanhees71]

Just use Ampere's Law in integral form. Due to symmetry of a very long coil, $\vec{H}$ must be along the coil's axis, and you can assume it's 0 outside.

For the closed line in the integral take a rectangle with one side (length $l$) along the axis, somewhere well inside the coil and the parallel side outside. Let there be $\lambda$ windings per unit length. Then you have, according to Ampere's Law (I neglect the signs here; you easily find the direction of the field, using the right-hand rule):
$$\int_{\partial A} \mathrm{d} \vec{r} \cdot \vec{H}=\frac{\lambda l}{c} I,$$
where $I$ is the current through the coil. This gives
$$|\vec{H}|=\frac{\lambda}{c},$$
independent of where you locate the rectangle's side within the coil. That's why $\vec{H}$ is uniform.

You can also argue with the differential form of Ampere's Law,
$$\vec{\nabla} \times \vec{H}=\frac{1}{c} \vec{j}.$$
In cylindrical coordinates, with the $z$ axis along the solenoid's axis and with the ansatz due to the symmetry of the problem $\vec{H}=\vec{e}_z H(r)$ you find, using the formulas for the curl in cylindrical coordinates $\vec{\nabla} \times \vec{H}=-H'(r)$. Since inside the coil there is no current density you get $H(r)=\text{const}$.

 

[sardar]

The uniformity of the magnetic field inside an ideal solenoid can be explained by the symmetry of the solenoid's construction. An ideal solenoid is a tightly wound coil of wire, with each turn of the coil parallel to the others. This results in a cylindrical shape with a constant cross-sectional area, and the magnetic field lines produced by the current flowing through the wire are also parallel and evenly spaced.

When we consider just one loop in the solenoid, it is true that the value of the magnetic field may vary at different points on that loop. However, as we move along the length of the solenoid, the magnetic field produced by each loop adds up and reinforces the field produced by the previous loop. This results in a constant and uniform magnetic field inside the solenoid.

Additionally, the magnetic field inside an ideal solenoid is also uniform because the magnetic field lines are confined within the solenoid. This confinement prevents the field lines from spreading out and becoming weaker, as they would in an open space. Instead, the field lines are forced to remain parallel and evenly spaced, leading to a uniform field throughout the solenoid.

In summary, the uniformity of the magnetic field inside an ideal solenoid can be attributed to the symmetry of its construction and the confinement of magnetic field lines within the solenoid. This uniformity is essential for many applications of solenoids, such as in electromagnets, electric motors, and particle accelerators.