Question

A toroidal magnet has an inner radius of \(1.895 \mathrm{~m}\) and an outer radius of \(2.075 \mathrm{~m}\). When the wire carries a 33.45 - A current, the magnetic field at a distance of \(1.985 \mathrm{~m}\) from the center of the toroid is \(66.78 \mathrm{mT}\). How many turns of wire are there in the toroid?

Short answer

Answer: There are approximately 1907 turns of wire in the toroidal magnet.

Step-by-step solution

Understand the problem and write down the given information.

We are given: - Inner radius of the toroidal magnet: \(r_1 = 1.895\,\text{m}\) - Outer radius of the toroidal magnet: \(r_2 = 2.075\,\text{m}\) - Distance from the center to the point where the magnetic field is measured: \(r = 1.985\, \text{m}\) - Current through the wire: \(I = 33.45\, \text{A}\) - Magnetic field strength at the point \(r\): \(B = 66.78\times10^{-3}\,\text{T}\) We need to find the number of turns of wire, \(N\), in the toroid.

Use Ampere's Law to relate the magnetic field strength to the current.

Ampere's Law states that the line integral of the magnetic field, \(B\), around a closed loop is proportional to the current passing through the loop: \(\oint \vec{B} \cdot d\vec{l} = \mu_0 \cdot I\) Since the magnetic field is constant along a circle centered on the toroid's axis, we can write: \(B \oint d\vec{l} = \mu_0 \cdot NI\) \(B \cdot 2\pi r = \mu_0 \cdot NI\) Now we will solve for the number of turns, \(N\).

Solve for the number of turns of wire in the toroid using the given information and Ampere's Law.

We will use the given information and the formula derived in Step 2 to solve for the number of turns, \(N\): \(N=\frac{B \cdot 2\pi r}{\mu_0 \cdot I}\) Inserting the given values and the permeability constant \(\mu_0 = 4\pi\times10^{-7}\,\mathrm{T\cdot m/A}\): \(N=\frac{(66.78\times10^{-3}\,\mathrm{T})\cdot(2\pi \cdot 1.985\,\mathrm{m})}{(4\pi\times10^{-7}\,\mathrm{T\cdot m/A})\cdot(33.45\,\mathrm{A})}\) Now, perform the calculation: \(N \approx 1907\) The number of wire turns in the toroid is approximately 1907 turns.