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ee2:task_rdz03rspbwusy7wk_with_calculation [2024/07/03 08:50]
mexleadmin angelegt 		ee2:task_rdz03rspbwusy7wk_with_calculation [2024/07/03 10:26] (aktuell)
mexleadmin
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-{{tag>induction flux induced_voltage exam_ee2_SS2021}}{{include_n>1200}}+{{tag>induction coil induced_voltage exam_ee2_SS2021}}{{include_n>1200}}
    
 #@TaskTitle_HTML@##@Lvl_HTML@#~~#@ee1_taskctr#~~ Coil in a magnetic Field \\ #@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@# <fs medium>(written test, approx. 4 % of a 120-minute written test, SS2021)</fs> #@TaskText_HTML@#
  
-A coil with a number of turns $n = 300$ and a cross-sectional area $A = 600 ~\rm cm^2$ is located in a homogeneous magnetic field. \\+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 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 $\hat{B} = 2 ⋅ 10-6 ~\rm {{Vs}\over{cm^2}}$.+The maximum value of the magnetic flux density in the coil is $\hat{B} = 2 \cdot  10^{-6} ~\rm {{Vs}\over{cm^2}}$.
  
 {{drawio>ee2:rDz03RSPbWUSy7wK_question1.svg}} {{drawio>ee2:rDz03RSPbWUSy7wK_question1.svg}}
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 #@HiddenBegin_HTML~rDz03RSPbWUSy7wK_11,Path~@# #@HiddenBegin_HTML~rDz03RSPbWUSy7wK_11,Path~@#
  
-The induced voltage is given by:+The induced voltage $u_{\rm ind}$ is given by:
  
 \begin{align*} \begin{align*}
-U_{\rm ind} &= - {{{\rm d}\Psi}\over{{\rm d}t}} \\ +u_{\rm ind} &= - {{{\rm d}\Psi(t)}\over{{\rm d}t}} \\ 
-            &= - n{{{\rm d}\Phi}\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*} \end{align*}
  
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 #@HiddenBegin_HTML~rDz03RSPbWUSy7wK_12,Result~@# #@HiddenBegin_HTML~rDz03RSPbWUSy7wK_12,Result~@#
-{{drawio>ee2:LUdzWiUhjxITZ85B_solution1.svg}} +\begin{align*} 
-#@HiddenEnd_HTML~rDz03RSPbWUSy7wK_12,Result~@#+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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