Toroidal Solenoids aka Toroids - Online Tutor, Practice Problems & Exam Prep

The magnetic field at the center of a loop is calculated using the equation $B=\frac{\mu I}{2r}$. For a toroidal solenoid, the magnetic field is given by $B=\frac{\mu IN}{2\pi r}$, where r is the distance from the center. The magnetic field exists only between the inner and outer radii, with zero field at the center and outside the toroid.

Toroidal Solenoids aka Toroids

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 μ₀I divided by 2r. Now, if you have a solenoid, which is just a really, really long loop, the equation changes slightly to μ₀IN 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 μ₀IN divided by 2πr, 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.

Here’s what students ask on this topic:

What is a toroidal solenoid and how does it differ from a regular solenoid?

A toroidal solenoid, or toroid, is a solenoid that is bent into a doughnut shape. Unlike a regular solenoid, which is a long coil of wire, a toroidal solenoid forms a closed loop. The magnetic field in a toroid is confined within the core of the doughnut shape, whereas in a regular solenoid, the magnetic field extends outside the coil. The magnetic field in a toroid is given by the equation:

$B=\mu IN/2\pi r$

where $r$ is the distance from the center of the toroid.

How do you calculate the magnetic field inside a toroidal solenoid?

The magnetic field inside a toroidal solenoid is calculated using the equation:

$B=\mu IN/2\pi r$

where $\mu$ is the permeability of free space, $I$ is the current, $N$ is the number of turns, and $r$ is the distance from the center of the toroid. The magnetic field only exists between the inner and outer radii of the toroid.

Why is the magnetic field zero at the center of a toroidal solenoid?

The magnetic field is zero at the center of a toroidal solenoid because the field is confined within the core of the doughnut shape. The field exists only between the inner and outer radii of the toroid. At the center, the distance from the center is zero, and since the field is only present between the inner and outer radii, it does not exist at the center.

What is the significance of the mean radius in a toroidal solenoid?

The mean radius in a toroidal solenoid is the average of the inner and outer radii. It is given by:

$r$_mean=(r₁+r₂)/2

where $r$₁ is the inner radius and $r$₂ is the outer radius. The mean radius is used to approximate the distance from the center when calculating the magnetic field inside the toroid.

How does the direction of the magnetic field in a toroidal solenoid depend on the current?

The direction of the magnetic field in a toroidal solenoid depends on the direction of the current flowing through the wire. Using the right-hand rule, if you curl your fingers in the direction of the current, your thumb will point in the direction of the magnetic field. The field can be either clockwise or counterclockwise, depending on the current's direction.