A rectangular toroid has 2000 windings around its core. And the core has a height of 0.10 meters. The toroid has a self-inductance of 0.040 henries. What is the current through the toroid when the energy in its magnetic field is 2.0 times 10 to the negative sixth joules?
We can call the self-inductance of the toroid, 0.040 henries, capital 𝐿. And we’ll call the energy in the toroid’s magnetic field, 2.0 times 10 to the negative sixth joules, 𝐸. We want to solve for the current that runs through the toroid under these conditions. We’ll call this current 𝐼.
So we have a rectangular toroid. And we want to know the current 𝐼 that runs through the 2000 windings of the toroid. To solve for this current, we can recall a relationship for the energy of a magnetic field. This energy, 𝐸 sub 𝐵, is equal to one-half the inductance 𝐿 multiplied by the current 𝐼 squared.
In our case, 𝐸 is equal to one-half 𝐿𝐼 squared. Or rearranging to solve for 𝐼, we find it’s equal to the square root of two times 𝐸 over 𝐿. And since both the energy 𝐸 and the inductance 𝐿 are given in our problem statement, we’re ready to plug in and solve for 𝐼. When we do and enter these values on our calculator, we find that 𝐼 is 0.010 amps. And that’s the current that runs through the toroid.