{{tag>induction coil induced_voltage exam_ee2_SS2021}}{{include_n>1200}}
 
#@TaskTitle_HTML@##@Lvl_HTML@#~~#@ee1_taskctr#~~ Coil in a magnetic Field \\
<fs medium>(written test, approx. 4 % of a 120-minute written test, SS2021)</fs> #@TaskText_HTML@#

A coil with $n = 300$ turns and a cross-sectional area $A = 600 ~\rm cm^2$ is located in a homogeneous magnetic field. \\
The rotation of the coil causes a sinusoidal change in the magnetic field in the coil with the frequency $f = 80~\rm Hz$. \\
The maximum value of the magnetic flux density in the coil is $\hat{B} = 2 \cdot  10^{-6} ~\rm {{Vs}\over{cm^2}}$.

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Derive the formula for the voltage induced in the coil and calculate the voltage amplitude.  

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The induced voltage $u_{\rm ind}$ is given by:

\begin{align*}
u_{\rm ind} &= - {{{\rm d}\Psi(t)}\over{{\rm d}t}} \\
            &= - n{{{\rm d}\Phi(t)}\over{{\rm d}t}} \\
\end{align*}

With $\Phi(t)= B(t) \cdot A$, where $A$ is the constant area of a single winding and $B(t)$ is the changing field through this winding. \\
Due to the rotation, the field changes as:

\begin{align*}
B(t) &= \hat{B} \cdot \sin(\omega t + \varphi) \\
     &= \hat{B} \cdot \sin(2\pi f \cdot t + \varphi) \\
\end{align*}

This leads to:
\begin{align*}
u_{\rm ind} &= - n{{{\rm d}}\over{{\rm d}t}}A \hat{B} \cdot \sin(2\pi f \cdot t + \varphi) \\
            &= - n \cdot A \hat{B} \cdot 2\pi f \cdot \cos(2\pi \cdot f t + \varphi) \\
\end{align*}

The absolute value of the factor in front of the $\cos$ is the maximum induced voltage $\hat{U}_{\rm ind}$:
\begin{align*}
\hat{U}_{\rm ind} &= n \cdot A \hat{B} \cdot 2\pi f  \\
                  &= 300 \cdot 0.06{~\rm m^2} \cdot 2 \cdot  10^{-2} ~\rm {{Vs}\over{m^2}} \cdot 2\pi \cdot 80 {{1}\over{\rm s}}  \\
                  &= 180.95... {~\rm m^2} \cdot {{\rm Vs}\over{\rm m^2}} \cdot {{1}\over{\rm s}}  \\
                  &= 180.95... {~\rm V}  \\
\end{align*}


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#@HiddenBegin_HTML~rDz03RSPbWUSy7wK_12,Result~@#
\begin{align*}
u_{\rm ind} = - 181 {~\rm V} \cdot \cos(503 {{1}\over{\rm s}} \cdot t ) \\
\end{align*}#@HiddenEnd_HTML~rDz03RSPbWUSy7wK_12,Result~@#

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