# How to determine the voltage across an LR circuit?

Imagine a circuit with an inductor and a resistor rotating in a constant magnetic field. What is the relationship between the voltage induced across it's inductor ,resistor individually and it's generator voltage at a particular instance in time?and why?

• It is not clear what is rotating. A coil rotating in a constant magnetic field will produce an alternating voltage, so you have a voltage source in series with an inductor and a resistor. Sep 11 at 16:30
• yes, the coil is rotating
– A.G
Sep 11 at 16:44
• So what is the question? you have a AC generator and a resistor as "user" easier just to take a coil with some resistance. the smaller the resistor or the resistance of th coil the more force you need to rotate your coil. Sep 11 at 17:16
• i have added an edit, if it makes it any clearer
– A.G
Sep 11 at 17:24
– A.G
Sep 11 at 17:38

First a quick solution to the circuit with a sinusoidal applying voltage, $$V_g = V_0 \cos(\omega t)$$: \begin{align} V_R + V_L &= V_g.\\ i(t) R + L\frac{d i}{dt} &= V_0 \cos \left(\omega t\right). \tag{1}\\ \end{align}
Then assume a solution $$i(t) = i_0 \cos \left(\omega t - \theta\right)$$, the angle $$\theta$$ is called the phase delay. Plug this solution into Eq.(1), solving two unknown parameters $$i_0$$ and $$\theta$$. We have two equations, one for $$\cos\left(\omega t\right)$$, and the other for $$\sin\left(\omega t\right)$$ \begin{align} R i_0 \cos \left(\omega t - \theta\right) - L i_0 \omega \sin \left(\omega t - \theta\right) &= V_0 \cos \left(\omega t\right).\tag{2}\\ i_0 R\cos\theta + i_0 \omega L\sin\theta &= V_0; \tag{3}\\ i_0 R \sin\theta - i_0 \omega L\cos\theta &= 0. \tag{4} \end{align}
Parameter $$i_0$$ and $$\theta$$ can be solve using equations (3) and (4). And according to the linked figure, the equation (2) can be interpreted as $$V_R$$ in the $$\hat x$$ axis and $$V_L$$ in the $$\hat y$$ axis, their vector sum is always equal to the voltage of the generator, which is with a phase $$\theta$$ ahead of the current voltage. These three vectors are coherently rotate with angular frequency $$\omega$$ driven by the applying alternated voltage. This is called a phase diagram.