The magnetic field of a cylindrical magnet that has a pole-face diameter of 3.3 cmcan be varied sinusoidally between 29.6 Tand 30.0 Tat a frequency of 15Hz. (The current in a wire wrapped around a permanent magnet is varied to give this variation in the net field.) At a radial distance of 1.6 cm, what is the amplitude of the electric field induced by the variation?

$r=1.6\times {10}^{-2}\mathrm{m}$

Diameter of cylindrical magnet $D=3.3\mathrm{cm}=3.3\times {10}^{-2}\mathrm{m}$

Frequency of sinusoidally varying magnetic field, f = 15.0 Hz

The rate of change of magnetic flux is related to the line integral of the electric field by Faraday’s law, so use Faraday’s law to evaluate the line integral,$\mathbf{\oint }\overrightarrow{\mathbf{E}}\mathbf{.}\mathit{d}\overrightarrow{\mathbf{s}}$. Using this, find the magnitude of the electric field inside the solenoid at distance rfrom the axis of the solenoid.

Faraday'slaw of electromagnetic inductionstates, Whenever a conductor is placed in a varying magnetic field, an electromotive force is induced in it.

Formulae are as follow:

$$\begin{gathered} {\ddot{o}}_{ind}=-\frac{d{\varphi }_B}{dt}=\oint \overrightarrow{E}.d\overrightarrow{s} \\ d{\varphi }_B=\overrightarrow{B}.d\overrightarrow{A} \\ E=\frac{r}{2}\frac{dB}{dt} \\ \omega =2\tau \tau f \end{gathered}$$

Where,${\mathbf{\Phi }}_{\mathbf{B}}$is magnetic flux, B is magnetic field, A is area,𝜀 is emf, E is electric field,𝜔 is angular velocity, f is frequency, r is radius.

As mentioned in the problem the magnetic field is varying sinusoidally, let’s assume that the following form of time is dependent on magnetic field,

$B=B_0\sin \left(\omega t\right)=B_0\sin \left(2\tau \tau ft\right)$

As the field is varying in between two values 30.0 T and 29.6 T

Taking difference and dividing it by 2, the amplitude of field,

$$\begin{gathered} B_0=\left(\frac{30.0-29.6}{2}\right)=\frac{04}{2}=0.2T \\ B=\left(0.2T\right)\sin \left(2\tau \tau \times 15\times t\right)=\left(0.2T\right)\sin \left(30\tau \tau t\right) \end{gathered}$$

Now, consider magnitude electric field inside solenoid at distance r from the axis of solenoid is,

$$\begin{gathered} E=\frac{r}{2}\frac{dB}{dt}=\frac{r}{2}.\frac{d}{dt}\left\{\left(02.T\right)\sin \left(30\tau \tau t\right)\right\}=0.2\times \frac{r}{2}\times \left(30\tau \tau \right)\cos \left(30\tau \tau t\right) \\ E=0.2\times \frac{1.6\times {10}^{-2}}{2}\times \left(30\tau \tau \right)\cos \left(30\tau \tau t\right) \end{gathered}$$

E will be maximum when cosine will have maximum value which is equal to one

$$\begin{gathered} E_{max}=0.2\times \frac{1.6\times {10}^{-2}}{2}\times \left(30\tau \tau \right) \\ {E}_{max}=15.072\times {10}^{-2} \\ {E}_{max}=0.15\frac{\mathrm{V}}{\mathrm{m}} \end{gathered}$$

Hence, amplitude of electric field induce is,$E_{max}=0.15\frac{\mathrm{V}}{\mathrm{m}}$

Therefore, value cosine function fluctuates between +1 and -1. The magnitude of cosine cannot exceed one, by considering this fact, the amplitude of electric field induced can be found.