11.1.7 | revision #20039 | Savchenko Solutions

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. $^{*)}$

For problem $11.1.7$ — For problem $$11.1.7$$

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:

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

Answer

$B = V /(a^2\omega )$