If a wire of length $l$ is moved through a homogeneous and constant magnetic field of strength $B$ in a direction perpendicular to the field lines, an electrical voltage can be measured at both ends of the wire. This voltage results from the Lorentz force acting on the electrons in the wire, which shifts the charges until they are compensated by the resulting electric field.
Analogous to the Hall effect, one can write:
$$
q\vec{E} = -q\vec{v}\times\vec{B}
$$
The voltage is obtained by taking the scalar product of the field strength with the vector of the conductor length $\vec{l}$:
$$
U_\mathrm{ind} = \vec{E}\cdot \vec{l} = -\vec{l}\cdot\left(\vec{v}\times\vec{B}\right)
$$
If the wire moves perpendicular to the field lines, the magnitude is:
$$
\boxed{U_\mathrm{ind} = -Blv}
$$
Example:
A conductor of length 10 cm moves with a velocity of 1 cm/s in a homogeneous magnetic field of strength 200 mT. The induced voltage is 200 μV.
Multiplying the conductor length $l$ by the displacement $\mathrm{d} s$, which follows from the definition $v = \mathrm{d} s/\mathrm{d} t$, gives the area $\mathrm{d} A$. Thus, the induced voltage can be written more generally as the time derivative of the magnetic flux through an area:
$$
U_\mathrm{ind} = -B\frac{\mathrm{d} A}{\mathrm{d} t}
$$
Often, the product of $B$ and $A$ is summarized as the magnetic flux $\Phi$:
$$
\boxed{\Phi = BA}
$$
This is the reason why $B$ is historically called flux density rather than field strength. With this, Faraday’s law of induction, discovered experimentally by Michael Faraday in 1830, becomes:
$$
\boxed{U_\mathrm{ind} = -\frac{\mathrm{d}\Phi}{\mathrm{d} t}}
$$
Thus, the induced voltage is proportional to the time variation of the magnetic flux.
Applying the product rule of differentiation yields:
$$
U_\mathrm{ind} = -\left(B\frac{\mathrm{d} A}{\mathrm{d} t} + \frac{\mathrm{d} B}{\mathrm{d} t}A\right)
$$
From this, we see that not only the motion of a conductor in a magnetic field but also a changing magnetic field itself can induce a voltage. In this case, the term $\mathrm{d}d A/\mathrm{d}d t$ disappears, leaving:
$$
U_\mathrm{ind} = -\frac{\mathrm{d} B}{\mathrm{d} t}A
$$
Both formulas are special cases of Faraday’s general law of induction.
Example:
A square loop of side length 10 cm is placed in a magnetic field whose flux density increases at a rate of 25 mT/s. The induced voltage is 250 μV.
The concept of induction can be generalized further by considering a closed loop in a magnetic field. Since the electric field is equal to the integral over the voltage and we must integrate along the closed loop, we obtain:
$$
\oint \vec{E}\cdot \mathrm{d} \vec{s} = -\int \frac{\mathrm{d} \vec{B}}{\mathrm{d} t} \,\mathrm{d} \vec{A}
$$
This relation expresses that the line integral of the electric field around a closed path equals the surface integral of the time-varying magnetic field across the enclosed area. This principle is known as Stokes’ theorem.
Using the curl operator, we write:
$$
\int \mathrm{rot}\vec{E}\cdot\mathrm{d} \vec{A} = -\int \frac{\mathrm{d} \vec{B}}{\mathrm{d} t}\,\mathrm{d}\vec{A}
$$
Since the surfaces on both sides are ide