Electromagnetic Induction

When moving an electrical wire inside a magnetic field and closing the circuit, we can measure a small current flowing through the wire. Depending on how fast we move it and how strong the magnetic field is, it might even be possible to light up a small LED. We interpret this behavior as a voltage that is induced by the variation of the magnetic field lines. In this section, we want to derive a formula to calculate this induction voltage. If a conductor is moved inside a magnetic field, the Lorentz force pushes the electrons to one of its ends, until the electric force compensates for the Lorentz force completely. This can be described with the help of the following equation: $$qE = qvB$$ By canceling out the charge $q$ and replacing $E$ with $U/l$, where $l$ is the length of the conductor, it follows for the induced voltage: $$U = lvB$$ We can therefore see that electromagnetic induction is an effect of the Lorentz force acting on the charge inside a wire. Now we can replace $v$ with $\mathrm{d}s/\mathrm{d}t$w which leads to $$U = l\frac{\mathrm{d}s}{\mathrm{d}t}B$$ Multiplying the length $l$ with the infinitesimal distance $\mathrm{d}s$ can be considered as the area that is covered by the movement of the conductor. This equation then turns into the derivative of the area with respect to time: $$U = \frac{\mathrm{d}A}{\mathrm{d}t}B$$ Now we can even go one step further and combine the area $A$ and the magnetic field $B$ to the magnetic flux $\Phi$ which then leads to the following Faraday's law of induction: $$U_\mathrm{ind} = \frac{\mathrm{d}\Phi}{\mathrm{d}t}$$ This equation states that the induced voltage equals the change of the magnetic flux in time. Since either $A$ or $B$ can change with respect to time, we can apply the product rule we learned in the differential calculus section which turns to $$U_\mathrm{ind} = A\frac{\mathrm{d}B}{\mathrm{d}t} + \frac{\mathrm{d}A}{\mathrm{d}t}B$$ It follows from this relation that if either the area or the magnetic field changes, an induced voltage can be observed.