Unterschiede
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electrical_engineering_2:the_time-dependent_magnetic_field [2023/03/17 09:14] mexleadmin |
electrical_engineering_2:the_time-dependent_magnetic_field [2023/09/19 23:51] (aktuell) mexleadmin |
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| Zeile 1: | Zeile 1: | ||
| - | ====== 4. time-dependent magnetic Field ====== | + | ====== 4 Time-dependent magnetic Field ====== |
| < | < | ||
| Zeile 44: | Zeile 44: | ||
| This definition leads to a magnetic flux similar to the electric flux studied earlier: | This definition leads to a magnetic flux similar to the electric flux studied earlier: | ||
| - | \begin{align*} \Phi_m = \iint_A \vec{B} \cdot d \vec{A} \end{align*} | + | \begin{align*} \Phi_{\rm m} = \iint_A \vec{B} \cdot {\rm d} \vec{A} \end{align*} |
| Therefore, the induced potential difference generated by a conductor or coil moving in a magnetic field is | Therefore, the induced potential difference generated by a conductor or coil moving in a magnetic field is | ||
| - | \begin{align*} \boxed{ U_{ind} = -{{d \Phi_m}\over{dt}} = -{{d}\over{dt}}\iint_A \vec{B} \cdot d \vec{A} } \end{align*} | + | \begin{align*} \boxed{ U_{\rm ind} = -{{{\rm d} \Phi_{\rm m}}\over{{\rm d}t}} = -{{\rm d}\over{{\rm d}t}}\iint_A \vec{B} \cdot {\rm d} \vec{A} } \end{align*} |
| The negative sign describes the direction in which the induced potential difference drives current around a circuit. However, that direction is most easily determined with a rule known as Lenz’s law, which we will discuss in the next subchapter. | The negative sign describes the direction in which the induced potential difference drives current around a circuit. However, that direction is most easily determined with a rule known as Lenz’s law, which we will discuss in the next subchapter. | ||
| Zeile 54: | Zeile 54: | ||
| <imgref ImgNr05> depicts a circuit and an arbitrary surface $S$ that it bounds. Notice that $S$ is an open surface: The planar area bounded by the circuit is not part of the surface, so it is not fully enclosing a volume. | <imgref ImgNr05> depicts a circuit and an arbitrary surface $S$ that it bounds. Notice that $S$ is an open surface: The planar area bounded by the circuit is not part of the surface, so it is not fully enclosing a volume. | ||
| - | Since the magnetic field is a source-free vortex field, the flux over a closed area is always zero: $\Phi_{\rm m} = \iint_{A} \vec{B} \cdot {\rm d} \vec{A} = 0$. \\ By this, it can be shown that any open surface bounded by the circuit in question can be used to evaluate $\Phi_{\rm m}$ (similar to the [[: | + | Since the magnetic field is a source-free vortex field, the flux over a closed area is always zero: $\Phi_{\rm m} = {\rlap{\Large \rlap{\int} \int} \, \LARGE \circ}_{A} \vec{B} \cdot {\rm d} \vec{A} = 0$. \\ |
| + | By this, it can be shown that any open surface bounded by the circuit in question can be used to evaluate $\Phi_{\rm m}$ (similar to the [[: | ||
| + | For example, $\Phi_{\rm m}$ is the same for the various surfaces $S$, $S_1$, $S_2$ of the figure. | ||
| < | < | ||
| - | The SI unit for magnetic flux is the Weber (Wb), \begin{align*} [\Phi_m] = [B] \cdot [A] = 1 T \cdot m^2 = 1Wb \end{align*} | + | The SI unit for magnetic flux is the Weber (Wb), \begin{align*} [\Phi_{\rm m}] = [B] \cdot [A] = 1 ~\rm |
| - | Occasionally, | + | Occasionally, |
| + | In many practical applications, | ||
| + | Each turn experiences the same magnetic flux $\Phi_{\rm m}$. | ||
| + | Therefore, the net magnetic flux through the circuits is $N$ times the flux through one turn, and Faraday’s law is written as | ||
| - | \begin{align*} u_{ind} = -{{d}\over{dt}}(N \cdot \Phi_m) = -N \cdot {{d \Phi_m}\over{dt}} \end{align*} | + | \begin{align*} |
| + | u_{\rm ind} = - { {\rm d} \over{{\rm d}t}} (N \cdot \Phi_{\rm m}) | ||
| + | | ||
| + | \end{align*} | ||
| <panel type=" | <panel type=" | ||
| Zeile 81: | Zeile 89: | ||
| The flux through one turn is | The flux through one turn is | ||
| - | \begin{align*} \Phi_m = B \cdot A \end{align*} | + | \begin{align*} \Phi_{\rm m} = B \cdot A \end{align*} |
| We can calculate the magnitude of the potential difference $|U_{\rm ind}|$ from Faraday’s law: | We can calculate the magnitude of the potential difference $|U_{\rm ind}|$ from Faraday’s law: | ||
| - | \begin{align*} |U_{ind}| &= |-{{d}\over{dt}}(N \cdot \Phi_m)| \\ &= -N \cdot {{d}\over{dt}} (B \cdot A) \\ &= -N \cdot l^2 \cdot {{dB}\over{dt}} \\ &= (200)(0.25m)^2(0.040 T/s) \\ &= 0.50 V \end{align*} | + | \begin{align*} |
| + | |U_{\rm ind}| &= |- {{\rm d}\over{{\rm d}t}}(N \cdot \Phi_{\rm m})| \\ | ||
| + | | ||
| + | | ||
| + | | ||
| + | | ||
| + | \end{align*} | ||
| The magnitude of the current induced in the coil is | The magnitude of the current induced in the coil is | ||
| - | \begin{align*} |I| &= {{ |U_{ind}|}\over{R}} \\ &= {{0.50V}\over{5.0\Omega}} = 0.10A \\ \end{align*} </ | + | \begin{align*} |
| + | |I| &= {{ |U_{\rm ind}|}\over{R}} \\ | ||
| + | | ||
| + | \end{align*} | ||
| + | </ | ||
| <panel type=" | <panel type=" | ||
| Zeile 123: | Zeile 141: | ||
| - Make a sketch of the situation for use in visualizing and recording directions. | - Make a sketch of the situation for use in visualizing and recording directions. | ||
| - Determine the direction of the applied magnetic field $\vec{B}$. | - Determine the direction of the applied magnetic field $\vec{B}$. | ||
| - | - Determine whether its magnetic flux is increasing or decreasing. | + | - Determine whether |
| - Now determine the direction of the induced magnetic field $\vec{B_{\rm ind}}$. The induced magnetic field tries to reinforce a magnetic flux that is decreasing or opposes a magnetic flux that is increasing. Therefore, the induced magnetic field adds or subtracts from the applied magnetic field, depending on the change in magnetic flux. | - Now determine the direction of the induced magnetic field $\vec{B_{\rm ind}}$. The induced magnetic field tries to reinforce a magnetic flux that is decreasing or opposes a magnetic flux that is increasing. Therefore, the induced magnetic field adds or subtracts from the applied magnetic field, depending on the change in magnetic flux. | ||
| - Use the right-hand rule to determine the direction of the induced current $i_{\rm ind}$ that is responsible for the induced magnetic field $\vec{B}_{\rm ind}$. | - Use the right-hand rule to determine the direction of the induced current $i_{\rm ind}$ that is responsible for the induced magnetic field $\vec{B}_{\rm ind}$. | ||
| Zeile 215: | Zeile 233: | ||
| Furthermore, | Furthermore, | ||
| - | The situation of the single rod can be interpreted in the following way: We can calculate a motionally induced potential difference with Faraday’s law even when an actual closed circuit is not present. We simply imagine an enclosed area whose boundary includes the moving conductor, calculate $\Phi_m$, and then find the potential difference from Faraday’s law. For example, we can let the moving rod of <imgref ImgNr10> be one side of the imaginary rectangular area represented by the dashed lines. The area of the rectangle is $A = l \cdot x$, so the magnetic flux through it is $\Phi= B\cdot l \cdot x$. Differentiating this equation, we obtain | + | The situation of the single rod can be interpreted in the following way: We can calculate a motionally induced potential difference with Faraday’s law even when an actual closed circuit is not present. We simply imagine an enclosed area whose boundary includes the moving conductor, calculate $\Phi_{\rm m}$, and then find the potential difference from Faraday’s law. For example, we can let the moving rod of <imgref ImgNr10> be one side of the imaginary rectangular area represented by the dashed lines. The area of the rectangle is $A = l \cdot x$, so the magnetic flux through it is $\Phi= B\cdot l \cdot x$. Differentiating this equation, we obtain |
| \begin{align*} | \begin{align*} | ||
| Zeile 325: | Zeile 343: | ||
| This changes the function to time-space rather than $\varphi$. The induced potential difference, therefore, varies sinusoidally with time according to | This changes the function to time-space rather than $\varphi$. The induced potential difference, therefore, varies sinusoidally with time according to | ||
| - | \begin{align*} u_{ind} &= U_{ind,0} \cdot sin \omega t \end{align*} | + | \begin{align*} u_{ind} &= U_{ind,0} \cdot \sin \omega t \end{align*} |
| where $U_{\rm ind,0} = NBA\omega$. </ | where $U_{\rm ind,0} = NBA\omega$. </ | ||
| Zeile 333: | Zeile 351: | ||
| <panel type=" | <panel type=" | ||
| - | The generator coil shown in <imgref ImgNr13> is rotated through one-fourth of a revolution (from $\phi_0=0°$ to $\phi_1=90°$) in $5.0 ~\rm ms$. | + | The generator coil shown in <imgref ImgNr13> is rotated through one-fourth of a revolution (from $\varphi_0=0°$ to $\varphi_1=90°$) in $5.0 ~\rm ms$. |
| The $200$-turn circular coil has a $5.00 ~\rm cm$ radius and is in a uniform $0.80 ~\rm T$ magnetic field. | The $200$-turn circular coil has a $5.00 ~\rm cm$ radius and is in a uniform $0.80 ~\rm T$ magnetic field. | ||
| Zeile 364: | Zeile 382: | ||
| \begin{align*} | \begin{align*} | ||
| - | A = \pi r^2 = 3.14 \cdot (0.0500~\r, m)^2 = 7.85 \cdot 10^{-3} ~\rm m^2 | + | A = \pi r^2 = 3.14 \cdot (0.0500~\rm m)^2 = 7.85 \cdot 10^{-3} ~\rm m^2 |
| \end{align*} | \end{align*} | ||
| Zeile 424: | Zeile 442: | ||
| \begin{align*} | \begin{align*} | ||
| \theta(t) &= \int & \vec{H}(t) \cdot {\rm d}\vec{s} \\ | \theta(t) &= \int & \vec{H}(t) \cdot {\rm d}\vec{s} \\ | ||
| - | &= \int & \vec{H}_{inner}(t) \cdot {\rm d}\vec{s} & + & \int \vec{H}_{outer}(t) \cdot {\rm d} \vec{s} \\ | + | &= \int & \vec{H}_{\rm inner}(t) \cdot {\rm d}\vec{s} & + & \int \vec{H}_{\rm outer}(t) \cdot {\rm d} \vec{s} \\ |
| - | &= \int & \vec{H}(t) \cdot {\rm d}\vec{s} | + | &= \int & \vec{H}(t) \cdot {\rm d}\vec{s} |
| & | & | ||
| \end{align*} | \end{align*} | ||
| Zeile 433: | Zeile 451: | ||
| \begin{align*} | \begin{align*} | ||
| N \cdot i &= {H}(t) \cdot l \\ | N \cdot i &= {H}(t) \cdot l \\ | ||
| - | | + | |
| - | | + | |
| \end{align*} | \end{align*} | ||
| Zeile 518: | Zeile 536: | ||
| \begin{align*} H(t) = {{N \cdot i}\over {l}} \end{align*} | \begin{align*} H(t) = {{N \cdot i}\over {l}} \end{align*} | ||
| - | with the mean magnetic path length (= length of the average field line) $l = \pi(r_o + r_i)$: | + | with the mean magnetic path length (= length of the average field line) $l = \pi(r_{\rm o} + r_{\rm i})$: |
| - | \begin{align*} H(t) = {{N \cdot i}\over { \pi(r_o + r_i)}} \end{align*} | + | \begin{align*} H(t) = {{N \cdot i}\over { \pi(r_{\rm o} + r_{\rm i})}} \end{align*} |
| The inductance $L$ can be calculated by | The inductance $L$ can be calculated by | ||
| Zeile 529: | Zeile 547: | ||
| \end{align*} | \end{align*} | ||
| - | With the magnetic flux density $B(t) = \mu_0 \mu_{\rm r} H(t) = \mu_0 \mu_{\rm r} {{i \cdot N }\over {l}}$ and the cross section $A = h (r_o - r_i)$, we get: | + | With the magnetic flux density $B(t) = \mu_0 \mu_{\rm r} H(t) = \mu_0 \mu_{\rm r} {{i \cdot N }\over {l}}$ and the cross section $A = h (r_{\rm o} - r_{\rm i})$, we get: |
| \begin{align*} | \begin{align*} | ||
| - | \quad \quad L_{\rm toroidal \; coil} &= {{ N \cdot \mu_0 \mu_{\rm r} {{i \cdot N } \over { \pi(r_o + r_i)}} \cdot h(r_o - r_i)}\over{i}} \\ | + | \quad \quad L_{\rm toroidal \; coil} &= {{ N \cdot \mu_0 \mu_{\rm r} {{i \cdot N } \over { \pi(r_{\rm o} + r_{\rm i})}} \cdot h(r_{\rm o} - r_{\rm i})}\over{i}} \\ |
| - | & | + | & |
| \end{align*} | \end{align*} | ||
| \begin{align*} | \begin{align*} | ||
| - | \boxed{ L_{\rm toroidal \; coil} = \mu_0 \mu_{\rm r} \cdot N^2 \cdot {{ h(r_o - r_i)}\over{ \pi(r_o + r_i)}} } | + | \boxed{ L_{\rm toroidal \; coil} = \mu_0 \mu_{\rm r} \cdot N^2 \cdot {{ h(r_{\rm o} - r_{\rm i})}\over{ \pi(r_{\rm o} + r_{\rm i})}} } |
| \end{align*} | \end{align*} | ||
| Zeile 649: | Zeile 667: | ||
| So, the course of the voltage when entering or exiting is not uniquely given. | So, the course of the voltage when entering or exiting is not uniquely given. | ||
| - | < | + | < |