I understand that power is that rate at which work is done and that because of this the power is equal to d/dt (1/2Li^2). I also understand that the power is also equal to Li di/dt since Ldi/dt is v and v*i is power. I understand that since the power is equal to both of these equations that they are equal to each other. The part that I don't get is mathematically how to get from one to the other.
you have written the steps yourself, from what i can tell d/dt [ (1/2)Li^2 ]. L is constant, i d/dt [ q(t) ], not constant using the power rule for differentiation d/dt [ (1/2)Li^2 ] (2)(1/2)Li*(di/dt). implicit differentiation if that step did not make sense, review the calculus techniques from first semester calc,, i dq/dt v -dΦ/dt -d/dt ( BAcosθ ) || B || (μ0 / 4π) * i visualize the charge (q) running through wire approaching inductor, as it flows into the coil, a B-field is induced, which tries to resist the CHANGE in current, this is where the work occurs
An ideal inductor consumes no power. However an inductor stores energy E integral of (v)(i)dt, and since v Ldi/dt E integral of (Ldi/dt)(i)dt integral of L(i)di (1/2)L(i)^2 This energy is stored in the magnetic field.
