So, remember that a magnetic field at the center of a loop is given by this equation. If you have a single or a few loops, it's given by μ_{0}*I* divided by 2*r*. Now, if you have a solenoid, which is just a really, really long loop, the equation changes slightly to μ_{0}*IN* divided by *l*. These are the two equations we've seen so far for loops, but some of you also need to know about the toroidal solenoid, which is a special kind. All it is is a regular solenoid, which you may remember has a magnetic field through the center, and you can imagine it as a slinky that you turn into a doughnut shape. So imagine you do that and then run a current through it. The magnetic field is going to follow the center of this thing. A toroidal solenoid is a solenoid arranged in a doughnut shape.

Let's consider a scenario where a battery is connected to wire wrapped around this doughnut-shaped solenoid. Current will flow, and the wire goes around the doughnut, sometimes behind it where you can't see it before returning to the front. The direction of the magnetic field can be clockwise or counterclockwise, depending on the current's flow. You can figure out the direction by looking at the first motion of the current, which determines if your thumb points left or right, indicating the direction of the magnetic field within the doughnut.

The equation for a toroidal solenoid is μ_{0}*IN* divided by 2π*r*, where *r* is the distance from the center, distinct from other loop equations as it includes π and uses the smaller radius. The magnetic field only exists between the inner and outer radii of the toroid. These are significant because the magnetic field is zero outside of these bounds.

Finally, if you were to calculate the magnitude of the magnetic field at various positions within a 300-turn toroidal solenoid with inner and outer radii of 0.12 meters and 0.16 meters respectively, and a current of 5 amperes, the field would be zero at the center since there is no magnetic field there. For a position like 0.14 meters (within the inner and outer radii), you would use the equation to find the field's magnitude. However, at 0.20 meters (outside the specified radii), the field would also be zero. These distinctions are crucial for understanding where magnetic fields exist and how they behave in toroidal solenoids.