Electromagnetic induction > Mutual inductance. Inductance of conductors. Transformers > Problem 11.3.21

Statement

$11.3.21.$

Why does a loaded transformer hum? What is the fundamental frequency of
sound if the transformer is connected to an industrial network?

Solution

The transformer hums due to magnetostriction: the ferromagnetic core slightly contracts and expands when magnetized by the alternating current.
The deformation$ \epsilon $is proportional to the square of the magnetic induction B

$\epsilon(t) \propto B(t)^2 = B_0^2 \sin^2(\omega t) = \frac{B_0^2}{2}\bigl[1 - \cos(2\omega t)\bigr], \qquad \omega = 2\pi f_{\text{grid}}$

The oscillatory component has twice the frequency of the mains.
For an industrial mains frequency of $f_{\text{grid}} = 50\ \text{Hz}$

$\boxed{\nu = 2 \times 50\ \text{Hz} = 100\ \text{Hz}}$

Answer

$\boxed{\nu = 2 \times 50\ \text{Hz} = 100\ \text{Hz}}$