Solution

Statement

$11.1.7.$ The induction of a constant magnetic field is measured using a square frame with dimensions $a \times a$, rotating with an angular velocity $\omega$. Its axis of rotation is perpendicular to the direction of the magnetic field. The amplitude of the electric voltage taken from the frame is $V$. Find the magnetic field induction.$^{*)}$

Solution

The magnetic flux passing through the square frame, rotated by an angle $\alpha=\omega t$, is:

$$\Phi = BS \cdot \cos\alpha = BS \cdot \cos \omega t$$

The induced EMF arising in the rotating square frame is:

$$\varepsilon_i = - \frac{d\Phi}{dt} = - {BS \cdot \frac{d}{dt}\left(\cos \omega t\right)}$$

From this, we find the induced EMF:

$$\varepsilon_i = BS\omega \cdot \sin \omega t$$

Thus, the amplitude of the induced EMF is:

$$V=BS\omega$$

Expressing the magnetic field induction:

$$B=\frac{V}{S\omega}=\frac{V}{a^2\omega}$$