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electrical_engineering_2:the_time-dependent_magnetic_field [2022/04/23 03:34] tfischer |
electrical_engineering_2:the_time-dependent_magnetic_field [2023/09/19 23:51] (aktuell) mexleadmin |
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| - | ====== 4. time-dependent magnetic Field ====== | + | ====== 4 Time-dependent magnetic Field ====== |
| + | < | ||
| < | < | ||
| - | This chapter is based on the Book ' | ||
| - | < | ||
| === Learning Objectives === | === Learning Objectives === | ||
| By the end of this section, you will be able to: | By the end of this section, you will be able to: | ||
| + | |||
| - determine the magnetic flux through a surface, knowing the strength of the magnetic field, the surface area, and the angle between the normal to the surface and the magnetic field | - determine the magnetic flux through a surface, knowing the strength of the magnetic field, the surface area, and the angle between the normal to the surface and the magnetic field | ||
| - use Faraday’s law to determine the magnitude of induced potential difference in a closed loop due to changing magnetic flux through the loop | - use Faraday’s law to determine the magnitude of induced potential difference in a closed loop due to changing magnetic flux through the loop | ||
| - use Lenz’s law to determine the direction of induced potential difference whenever a magnetic flux changes | - use Lenz’s law to determine the direction of induced potential difference whenever a magnetic flux changes | ||
| - | - use Faraday’s law with Lenz’s law to determine the induced potential difference in a coil and in a solenoid | + | - use Faraday’s law with Lenz’s law to determine the induced potential difference in a coil and a solenoid |
| </ | </ | ||
| - | We have been considering electric fields created by fixed charge distributions and magnetic fields produced by constant currents, but electromagnetic phenomena are not restricted to these stationary situations. Most of the interesting applications of electromagnetism are, in fact, time-dependent. To investigate some of these applications, | + | We have been considering electric fields created by fixed charge distributions and magnetic fields produced by constant currents, but electromagnetic phenomena are not restricted to these stationary situations. Most of the interesting applications of electromagnetism are, in fact, time-dependent. To investigate some of these applications, |
| - | <WRAP> | + | Lastly, we describe applications of these principles, such as the card reader shown in <imgref |
| - | < | + | |
| - | </ | + | < |
| - | {{drawio> | + | |
| - | </ | + | |
| ===== 4.1 Recap of magnetic Field ===== | ===== 4.1 Recap of magnetic Field ===== | ||
| - | The first productive experiments concerning the effects of time-varying magnetic fields were performed by Michael Faraday in 1831. One of his early experiments is represented in the simulation in <imgref ImgNr02> - in the tab '' | + | The first productive experiments concerning the effects of time-varying magnetic fields were performed by Michael Faraday in 1831. One of his early experiments is represented in the simulation in <imgref ImgNr02> - in the tab '' |
| - | < | + | < |
| - | < | + | |
| - | </ | + | |
| - | {{url> | + | {{url> |
| - | </ | + | |
| - | Faraday also discovered that a similar effect can be produced using two circuits: a changing current in one circuit induces a current in a second, nearby circuit. An example | + | Faraday also discovered that a similar effect can be produced using two circuits: a changing current in one circuit induces a current in a second, nearby circuit. An example |
| - | Faraday realized that in both experiments, | + | Faraday realized that in both experiments, |
| - | <callout icon=" | + | <callout icon=" |
| - | **Faraday' | + | |
| - | Any change in the magnetic field or change in orientation of the area of a coil with respect to the magnetic field induces | + | Any change in the magnetic field or change in the orientation of the area of a coil with respect to the magnetic field induces |
| - | The induced potential difference is the negative change of the so-called **magnetic flux** $\Phi_m$ per unit time. | + | |
| - | </ | + | |
| - | The magnetic flux is a measurement of the amount of magnetic field lines through a given surface area, as seen in <imgref ImgNr03> | + | The magnetic flux is a measurement of the amount of magnetic field lines through a given surface area, as seen in <imgref ImgNr03> |
| - | The magnetic flux is the amount of magnetic field lines cutting through a surface area defined by the surface vector $\vec{A}$. | + | |
| - | If the angle between the $\vec{A} = A\cdot \vec{n}$ and magnetic field vector | + | |
| - | < | + | < |
| - | < | + | |
| - | </ | + | |
| - | {{drawio> | + | |
| - | </ | + | |
| - | This definition | + | This definition |
| - | \begin{align*} | + | \begin{align*} \Phi_{\rm m} = \iint_A \vec{B} \cdot {\rm d} \vec{A} \end{align*} |
| - | \Phi_m = \iint_A \vec{B} \cdot 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*} | + | \begin{align*} |
| - | U_{ind} = -{{d \Phi_m}\over{dt}} = -{{d}\over{dt}}\iint_A \vec{B} \cdot d \vec{A} | + | |
| - | \end{align*} | + | |
| - | The negative sign describes the direction in which the induced potential difference drives current around a circuit. | + | 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. |
| - | However, that direction is most easily determined with a rule known as Lenz’s law, which we will discuss in the next subchapter. | + | |
| - | <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 alway zero: $\Phi_m = \iint_{A} \vec{B} \cdot d \vec{A} = 0$. \\ | + | Since the magnetic field is a source-free vortex field, the flux over a closed area is always |
| - | By this, it can be shown that any open surface bounded by the circuit in question can be used to evaluate $\Phi_m$ (similar to the [[the_stationary_electric_flow# | + | 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, $\Phi_m$ is the same for the various surfaces $S$, $S_1$,$S_2$,$S_3$ of the figure. | + | For example, $\Phi_{\rm m}$ is the same for the various surfaces $S$, $S_1$, $S_2$ of the figure. |
| - | < | + | < |
| - | < | + | |
| - | </ | + | |
| - | {{drawio> | + | |
| - | </ | + | |
| - | The SI unit for magnetic flux is the Weber (Wb), | + | The SI unit for magnetic flux is the Weber (Wb), \begin{align*} [\Phi_{\rm m}] = [B] \cdot [A] = 1 ~\rm |
| - | \begin{align*} | + | |
| - | [\Phi_m] = [B] \cdot [A] = 1 T \cdot m^2 = 1Wb | + | |
| - | \end{align*} | + | |
| - | Occasionally, | + | Occasionally, |
| In many practical applications, | In many practical applications, | ||
| - | Each turn experiences the same magnetic flux $\Phi_m$. | + | 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 | 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*} | + | \begin{align*} |
| - | U_{ind} = -{{d}\over{dt}}(N \cdot \Phi_m) = -N \cdot {{d \Phi_m}\over{dt}} | + | u_{\rm ind} = - { {\rm d} \over{{\rm d}t}} (N \cdot \Phi_{\rm m}) |
| + | | ||
| \end{align*} | \end{align*} | ||
| - | <panel type=" | + | <panel type=" |
| - | The square coil of <imgref ImgNr06> has sides $l=0.25m$ long and is tightly wound with $N=200$ turns of wire. | + | The square coil of <imgref ImgNr06> has sides $l=0.25~\rm m$ long and is tightly wound with $N=200$ turns of wire. The resistance of the coil is $R=5.0~\Omega$. The coil is placed in a spatially uniform magnetic field. The field is directed perpendicular to the face of the coil and whose magnitude is decreasing at a rate ${\rm d}B/{\rm d}t=−0.040~ \rm T/s$. |
| - | The resistance of the coil is $R=5.0\Omega$. | + | |
| - | The coil is placed in a spatially uniform magnetic field. The field is directed perpendicular to the face of the coil and whose magnitude is decreasing at a rate $dB/dt=−0.040T/s$. | + | - What is the magnitude of the potential difference induced in the coil? |
| - | - What is the magnitude of the potential difference induced in the coil? | + | |
| - What is the magnitude of the current circulating through the coil? | - What is the magnitude of the current circulating through the coil? | ||
| - | < | + | < |
| - | < | + | |
| - | </ | + | |
| - | {{drawio> | + | |
| - | </ | + | |
| - | **Strategy** | + | <button size=" |
| - | The surface $\vec{A}$ is perpendicular to area covering the loop. | + | The surface $\vec{A}$ is perpendicular to the area covering the loop. We will choose this to be pointing downward so that $\vec{B}$ is parallel to $\vec{A}$ and that the flux turns into the multiplication of magnetic field times area. The area of the loop is not changing in time, so it can be factored out of the time derivative, leaving the magnetic field as the only quantity varying in time. Lastly, we can apply Ohm’s law once we know the induced potential difference to find the current in the loop. </ |
| - | We will choose this to be pointing downward so that $\vec{B⃗}$ is parallel to $\vec{A}$ and that the flux turns into multiplication of magnetic field times area. | + | |
| - | The area of the loop is not changing in time, so it can be factored out of the time derivative, leaving the magnetic field as the only quantity varying in time. | + | |
| - | Lastly, we can apply Ohm’s law once we know the induced potential difference to find the current in the loop. | + | |
| - | **Solution** | + | <button size=" |
| The flux through one turn is | The flux through one turn is | ||
| - | \begin{align*} | + | \begin{align*} \Phi_{\rm m} = B \cdot A \end{align*} |
| - | \Phi_m = B \cdot A | + | |
| - | \end{align*} | + | |
| - | We can calculate the magnitude of the potential difference $|U_{ind}|$ from Faraday’s law: | + | We can calculate the magnitude of the potential difference $|U_{\rm ind}|$ from Faraday’s law: |
| - | \begin{align*} | + | \begin{align*} |
| - | |U_{ind}| &= |-{{d}\over{dt}}(N \cdot \Phi_m)| \\ | + | |U_{\rm ind}| &= |- {{\rm d}\over{{\rm d}t}}(N \cdot \Phi_{\rm 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) \\ | + | &= (200)(0.25 ~\rm m)^2(0.040 |
| - | &= 0.50 V | + | &= 0.50 ~V |
| \end{align*} | \end{align*} | ||
| The magnitude of the current induced in the coil is | The magnitude of the current induced in the coil is | ||
| - | \begin{align*} | + | \begin{align*} |
| - | |I| &= {{ |U_{ind}|}\over{R}} \\ | + | |I| &= {{ |U_{\rm ind}|}\over{R}} \\ |
| - | &= {{0.50V}\over{5.0\Omega}} = 0.10A \\ | + | &= {{0.50 ~\rm V}\over{5.0 |
| - | \end{align*} | + | \end{align*} |
| + | </ | ||
| - | </WRAP>< | + | <panel type=" |
| - | <panel type=" | + | A closely wound coil has a radius of $4.0 ~\rm cm$, $50$ turns, and a total resistance of $40~\Omega$. |
| - | A closely wound coil has a radius | + | At what rate must a magnetic field perpendicular to the face of the coil change in order to produce Joule heating in the coil at a rate of $2.0 ~\rm mW$? |
| - | At what rate must a magnetic field perpendicular to the face of the coil change in order to produce Joule heating in the coil at a rate of $2.0 mW$? | + | <button size=" |
| - | + | ||
| - | **Solution** | + | |
| - | + | ||
| - | $1.1 T/s$ | + | |
| - | </WRAP></WRAP></panel> | + | |
| + | $1.1 ~\rm T/s$ </ | ||
| ===== 4.2 Lenz Law ===== | ===== 4.2 Lenz Law ===== | ||
| - | The direction in which the induced potential difference drives current around a wire loop can be found through the negative sign. | + | The direction in which the induced potential difference drives current around a wire loop can be found through the negative sign. However, it is usually easier to determine this direction with Lenz’s law, named in honor of its discoverer, Heinrich Lenz (1804–1865). (Faraday also discovered this law, independently of Lenz.) We state Lenz’s law as follows: |
| - | However, it is usually easier to determine this direction with Lenz’s law, named in honor of its discoverer, Heinrich Lenz (1804–1865). | + | |
| - | (Faraday also discovered this law, independently of Lenz.) We state Lenz’s law as follows: | + | |
| - | <callout icon=" | + | <callout icon=" |
| - | **Lenz' | + | |
| - | The direction of the induced potential difference drives current around a wire loop to always oppose the change in magnetic flux that causes the potential difference. | + | The direction of the induced potential difference drives current around a wire loop to always oppose the change in magnetic flux that causes the potential difference. </ |
| - | </ | + | |
| - | Lenz’s law can also be considered in terms of conservation of energy. | + | Lenz’s law can also be considered in terms of the conservation of energy. If pushing a magnet into a coil causes current, the energy in that current must have come from somewhere. If the induced current causes a magnetic field opposing the increase in the field of the magnet we pushed in, then the situation is clear. We pushed a magnet against a field and did work on the system, and that showed up as current. If it were not the case that the induced field opposes the change in the flux, the magnet would be pulled in produce a current without anything having done work. Electric potential energy would have been created, violating the conservation of energy. |
| - | If pushing a magnet into a coil causes current, the energy in that current must have come from somewhere. | + | |
| - | If the induced current causes a magnetic field opposing the increase in field of the magnet we pushed in, then the situation is clear. | + | |
| - | We pushed a magnet against a field and did work on the system, and that showed up as current. | + | |
| - | If it were not the case that the induced field opposes the change in the flux, the magnet would be pulled in produce a current without anything having done work. | + | |
| - | Electric potential energy would have been created, violating the conservation of energy. | + | |
| - | To determine an induced potential difference $U_{ind}$, you first calculate the magnetic flux $\Phi_m$ and then obtain $d\Phi_m / dt$. | + | To determine an induced potential difference $u_{\rm ind}$, you first calculate the magnetic flux $\Phi_{\rm m}$ and then obtain ${\rm d}\Phi_{\rm m} / {\rm d}t$. The magnitude of $u_{\rm ind}$ is given by |
| - | The magnitude of $U_{ind}$ is given by | + | |
| - | \begin{align*} | + | \begin{align*} |u_{\rm ind}| & |
| - | |U_{ind}| &= |-{{d}\over{dt}}(\Phi_m)| | + | |
| - | \end{align*} | + | |
| - | Finally, you can apply Lenz’s law to determine the sense of $U_{ind}$. | + | Finally, you can apply Lenz’s law to determine the sense of $u_{\rm ind}$. This will be developed through examples that illustrate the following problem-solving strategy. |
| - | This will be developed through examples that illustrate the following problem-solving strategy. | + | |
| - | <callout icon=" | + | <callout icon=" |
| - | **Problem-Solving Strategy: Lenz’s Law** | + | |
| - | To use Lenz’s law to determine the directions of induced potential difference, currents and magnetic fields: | + | To use Lenz’s law to determine the directions of induced potential difference, currents, and magnetic fields: |
| - 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 $\vec{B_{ind}}$. 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 | + | - 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 |
| - | - Use right-hand rule to determine the direction of the induced current $I_{ind}$ that is responsible for the induced magnetic field $\vec{B_{ind}}$. | + | - Use the right-hand rule to determine the direction of the induced current $i_{\rm ind}$ that is responsible for the induced magnetic field $\vec{B}_{\rm ind}$. |
| - The direction (or polarity) of the induced potential difference can now drive a conventional current in this direction. | - The direction (or polarity) of the induced potential difference can now drive a conventional current in this direction. | ||
| + | |||
| </ | </ | ||
| - | Let’s apply Lenz’s law to the system of < | + | Let’s apply Lenz’s law to the system of < |
| - | We designate the “front” of the closed conducting loop as the region containing the approaching bar magnet, and the “back” of the loop as the other region. | + | |
| - | The north pole of the magnet moves toward the loop. | + | |
| - | Therefore, the flux through the loop due to the field of the magnet increases because the strength of field lines directed from the front to the back of the loop is increasing. | + | |
| - | A current is consequently induced in the loop. | + | |
| - | By Lenz’s law, the direction of the induced current must be such that its own magnetic field is directed in a way to oppose the changing flux caused by the field of the approaching magnet. | + | |
| - | Hence, the induced current circulates so that its magnetic field lines through the loop are directed from the back to the front of the loop. | + | |
| - | By the rigth hand rule, place your thumb pointing against the magnetic field lines, which is toward the bar magnet. | + | |
| - | Your fingers wrap in a counterclockwise direction as viewed from the bar magnet. | + | |
| - | Alternatively, | + | |
| - | This occurs when the induced current flows as shown, for then the face of the loop nearer the approaching magnet is also a north pole. | + | |
| - | < | + | < |
| - | < | + | |
| - | </ | + | |
| - | {{drawio> | + | |
| - | </ | + | |
| - | Part (b) of the figure shows the south pole of a magnet moving toward a conducting loop. | + | Part (b) of the figure shows the south pole of a magnet moving toward a conducting loop. In this case, the flux through the loop due to the field of the magnet increases because the number of field lines directed from the back to the front of the loop is increasing. To oppose this change, a current is induced in the loop whose field lines through the loop are directed from the front to the back. Equivalently, |
| - | In this case, the flux through the loop due to the field of the magnet increases because the number of field lines directed from the back to the front of the loop is increasing. | + | |
| - | To oppose this change, a current is induced in the loop whose field lines through the loop are directed from the front to the back. | + | |
| - | Equivalently, | + | |
| - | By the rigth hand rule, your thumb points away from the bar magnet. Your fingers wrap in a clockwise fashion, which is the direction of the induced current. | + | |
| - | ===== 4.3 motional Induction ===== | + | An animation of this situation can be seen [[https:// |
| - | Magnetic flux depends on three factors: | + | ===== 4.3 Motional Induction ===== |
| - | * the strength of the magnetic field, | + | |
| - | * the area through which the field lines pass, and | + | Magnetic flux depends on three factors: |
| - | * the orientation of the field with the surface area. | + | |
| - | If any of these quantities | + | * the strength of the magnetic field, |
| - | So far, we’ve only considered flux changes due to a changing field. Only the causing magnet was moving. This type of induction is called **static induction** (or stationary induction, in German: Ruheinduktion). | + | * the area through which the field lines pass, and |
| + | * the orientation of the field with the surface area. | ||
| + | |||
| + | If any of these quantities | ||
| Now we look at another possibility: | Now we look at another possibility: | ||
| - | Two examples of this type of flux change are represented in <imgref ImgNr07> | + | Two examples of this type of flux change are represented in <imgref ImgNr07> |
| - | * In part (a), the magnetic flux changes as a loop moves into a magnetic field. The flux through the rectangular loop increases as it moves into the magnetic field, | + | |
| + | * In part (a), the magnetic flux changes as a loop moves into a magnetic field. The flux through the rectangular loop increases as it moves into the magnetic field, | ||
| * In part (b), magnetic flux changes as a loop rotates in a magnetic field. The flux through the rotating coil varies with the angle $\varphi$. | * In part (b), magnetic flux changes as a loop rotates in a magnetic field. The flux through the rotating coil varies with the angle $\varphi$. | ||
| - | < | + | < |
| - | < | + | |
| - | </ | + | It’s interesting to note that what we perceive as the cause of a particular flux change actually depends on the frame of reference we choose. For example, if you are at rest relative to the moving coils of <imgref ImgNr07> (b), you would see the flux vary because of a changing magnetic field. In part (a), the field moves from left to right in your reference frame, and in part (b), the field is rotating. It is often possible to describe a flux change through a coil that is moving in one particular reference frame in terms of a changing magnetic field in a second frame, where the coil is stationary. However, reference-frame questions related to magnetic flux are beyond this introduction. We’ll avoid such complexities by always working in a frame at rest relative to the laboratory and explaining flux variations as due to either a changing field or a changing area. |
| - | {{drawio> | + | |
| - | </ | + | |
| - | It’s interesting to note that what we perceive as the cause of a particular flux change actually depends on the frame of reference we choose. | + | ==== Single Rod ==== |
| - | For example, if you are at rest relative to the moving coils of <imgref ImgNr07> (b), you would see the flux vary because of a changing magnetic field. | + | |
| - | In part (a), the field moves from left to right in your reference frame, and in part (b), the field is rotating. | + | |
| - | It is often possible to describe a flux change through a coil that is moving in one particular reference frame in terms of a changing magnetic field in a second frame, where the coil is stationary. | + | |
| - | However, reference-frame questions related to magnetic flux are beyond this introduction. | + | |
| - | We’ll avoid such complexities by always working in a frame at rest relative to the laboratory and explain flux variations as due to either a changing field or a changing area. | + | |
| - | ===== single | + | The first step to investigate the motional induction is shown in <imgref ImgNr09>: |
| - | The first step to investigate the motional induction is shown in <imgref ImgNr09>: | + | |
| - | | + | |
| * By this force, the positive charges move to one end of the rod and the negative to the other one. | * By this force, the positive charges move to one end of the rod and the negative to the other one. | ||
| - | * The separted | + | * The separated |
| - | * For a constant speed, the Lorentz force onto charges in the rod must have the same magnitude as the Coulomb force. | + | * For a constant speed, the Lorentz force onto charges in the rod must have the same magnitude as the Coulomb force. |
| - | < | + | < |
| - | < | + | |
| - | </ | + | |
| - | {{drawio> | + | |
| - | </ | + | |
| This leads to: | This leads to: | ||
| - | \begin{align*} | + | \begin{align*} |
| - | \vec{F}_C &= - \vec{F}_L \\ | + | \vec{F}_{\rm C} &= - \vec{F}_{\rm L} \\ Q \cdot \vec{E}_{\rm ind} |
| - | q \cdot \vec{E}_{ind} &= - q \cdot \vec{v} \times \vec{B} \\ | + | |
| - | \vec{E}_{ind} &= - \vec{v} \times \vec{B} \\ | + | \vec{E}_{\rm ind} &= - \vec{v} \times \vec{B} \\ |
| \end{align*} | \end{align*} | ||
| The induced potential difference in the rod will be: | The induced potential difference in the rod will be: | ||
| - | \begin{align*} | + | \begin{align*} |
| - | U_{ind} &= \int_l \vec{E}_{ind} \cdot d \vec{s} \\ | + | U_{\rm ind} & |
| - | &= - \int^0_1 \vec{v} \times \vec{B} \cdot d \vec{s} \\ | + | &= - \int^0_1 \vec{v} \times \vec{B} \cdot {\rm d} \vec{s} \\ |
| \end{align*} | \end{align*} | ||
| - | For constant $|\vec{v}|$ and $|\vec{B}|$ this leads to: | + | For constant $|\vec{v}|$ and $|\vec{B}|$ this leads to: |
| - | \begin{align*} | + | \begin{align*} |
| - | U_{ind} &= - v \cdot B \cdot l \\ | + | U_{\rm ind} &= - v \cdot B \cdot l \\ |
| \end{align*} | \end{align*} | ||
| - | ===== Rod in Circuit | + | ==== Rod in Circuit ==== |
| - | Now let’s look at the conducting rod pulled in a circuit, changing magnetic flux. | + | Now let’s look at the conducting rod pulled in a circuit, changing magnetic flux. The area enclosed by the circuit ' |
| - | The area enclosed by the circuit ' | + | |
| - | < | + | < |
| - | < | + | |
| - | </ | + | |
| - | {{drawio> | + | |
| - | </ | + | |
| - | The velocity of the rod is $v=dx/dt$. So the induced potential difference will get | + | The velocity of the rod is $v={\rm d}x/{\rm d}t$. So the induced potential difference will get |
| - | \begin{align*} | + | \begin{align*} |
| - | U_{ind} &= - v \cdot B \cdot l \\ | + | U_{\rm ind} &= - v \cdot B \cdot l \\ |
| - | &= - {{dx}\over{dt}} \cdot B \cdot l \\ | + | &= - {{{\rm d}x}\over{{\rm d}t}} \cdot B \cdot l \\ |
| - | &= - {{d}\over{dt}} \cdot B \cdot l \cdot x \\ | + | &= - {{B \cdot l \cdot {\rm d}x}\over{{\rm d}t}} \\ |
| - | &= - {{d}\over{dt}} \cdot B \cdot A \\ | + | &= - {{B \cdot {\rm d}A}\over{{\rm d}t}} \\ |
| - | &= - {{d}\over{dt}} \cdot \Phi_m \\ | + | &= - {{{\rm d}\Phi_{\rm m}}\over{{\rm d}t}} \\ |
| \end{align*} | \end{align*} | ||
| This is an alternative way to deduce Faraday' | This is an alternative way to deduce Faraday' | ||
| - | The current $I_{ind}$ induced in the given circuit is $U_{ind}$ divided by the resistance $R$ | + | The current $I_{\rm ind}$ induced in the given circuit is $U_{\rm ind}$ divided by the resistance $R$ |
| - | \begin{align*} | + | \begin{align*} |
| - | I_{ind} = {{v \cdot B \cdot l }\over{R}} | + | I_{\rm ind} = {{v \cdot B \cdot l }\over{R}} |
| \end{align*} | \end{align*} | ||
| 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. | + | 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 |
| - | 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> | + | |
| - | \begin{align*} | + | \begin{align*} |
| - | U_{ind} &= - {{d}\over{dt}} \cdot \Phi_m \\ | + | U_{\rm ind} &= - {{\rm d}\over{{\rm dt}}} \cdot \Phi_{\rm m} \\ |
| - | &= - B \cdot l \cdot {{dx}\over{dt}} \\ | + | &= - B \cdot l \cdot {{{\rm d}x}\over{{\rm d}t}} \\ |
| - | &= - B \cdot l \cdot v \\ | + | &= - B \cdot l \cdot v \\ |
| \end{align*} | \end{align*} | ||
| which is identical to the potential difference between the ends of the rod that we determined earlier. | which is identical to the potential difference between the ends of the rod that we determined earlier. | ||
| - | < | + | < |
| - | < | + | |
| - | </ | + | |
| - | {{drawio> | + | |
| - | </ | + | |
| - | <panel type=" | + | <panel type=" |
| - | Calculate the motional emf induced along a $20.0 km$ conductor moving at an orbital speed of $7.80 km/s$ perpendicular to Earth’s $5.00 \cdot 10^{-5}T$ magnetic field. | + | Calculate the potential difference motionally |
| - | **Strategy** | + | <button size=" |
| - | This is a great example of using the equation motional $U_{ind} = - B \cdot l \cdot v$ | + | This is a great example of using the equation motional $U_{\rm ind} = - B \cdot l \cdot v$ |
| - | **Solution** | + | Entering the given values into $U_{\rm ind} = - B \cdot l \cdot v$ gives |
| - | Entering the given values into $U_{ind} = - B \cdot l \cdot v$ gives | + | \begin{align*} |
| - | + | U_{ind} &= - {{{\rm d} \Phi_{\rm m}}\over{{\rm d}t}} \\ | |
| - | \begin{align*} | + | &= - B \cdot l \cdot v \\ |
| - | U_{ind} &= - {{d}\over{dt}} \cdot \Phi_m \\ | + | &= - (5.00 \cdot 10^{-5}~\rm T)(20.0 \cdot 10^{3} |
| - | &= - (5.00 \cdot 10^{-5}T)(20.0 \cdot 10^{3}m)(7.80\cdot 10^{3}m/ | + | |
| - | &= - 7.80 \cdot 10^3 V \\ | + | |
| \end{align*} | \end{align*} | ||
| + | |||
| + | </ | ||
| + | |||
| + | <button size=" | ||
| + | \begin{align*} U_{\rm ind} &= - 7.80 \cdot 10^3 ~\rm V \\ \end{align*} | ||
| + | |||
| + | </ | ||
| </ | </ | ||
| - | <panel type=" | + | <panel type=" |
| - | Part (a) of <imgref ImgNr11> shows a metal rod OS that is rotating in a horizontal plane around point O. | + | Part (a) of <imgref ImgNr11> shows a metal rod $\rm OS$ that is rotating in a horizontal plane around point $\rm O$. |
| - | The rod slides along a wire that forms a circular arc PST of radius $r$. | + | The rod slides along a wire that forms a circular arc $\rm PST$ of radius $r$. The system is in a constant magnetic field $\vec{B}$ that is directed out of the page. |
| - | The system is in a constant magnetic field $\vec{B}$ that is directed out of the page. | + | |
| - | If you rotate the rod at a constant angular velocity $\omega$, what is the current $I_{ind}$ in the closed loop OPSO? Assume that the resistor $R$ furnishes all of the resistance in the closed loop. | + | If you rotate the rod at a constant angular velocity $\omega$, what is the current $I_{\rm ind}$ in the closed loop $\rm OPSO$? |
| + | Assume that the resistor $R$ furnishes all of the resistance in the closed loop. | ||
| - | < | + | < |
| - | < | + | |
| - | </ | + | |
| - | {{drawio> | + | |
| - | </ | + | |
| - | **Strategy** | + | <button size=" |
| - | The magnetic flux is the magnetic field times the area of the quarter circle or $A = r^2 \varphi /2$. | + | The magnetic flux is the magnetic field times the area of the quarter circle or $A = r^2 \varphi /2$. When finding the potential difference |
| - | When finding the emf through Faraday’s law, all variables are constant in time but $\varphi$, with $\omega = d\varphi/dt$. | + | |
| - | To calculate the work per unit time, we know this is related to the torque times the angular velocity. | + | |
| - | The torque is calculated by knowing the force on a rod and integrating it over the length of the rod. | + | |
| - | **Solution** | + | <button size=" |
| - | From geometry, the area of the loop OPSO is $A=r^2\varphi /2$ . Hence, the magnetic flux through the loop is | + | From geometry, the area of the loop $\rm OPSO$ is $A=r^2\varphi /2$. Hence, the magnetic flux through the loop is |
| - | \begin{align*} | + | \begin{align*} |
| - | \Phi_m &= B\cdot A \\ | + | \Phi_{\rm m} &= B\cdot A \\ |
| - | | + | |
| \end{align*} | \end{align*} | ||
| Differentiating with respect to time and using $\omega = d\varphi/ | Differentiating with respect to time and using $\omega = d\varphi/ | ||
| - | \begin{align*} | + | \begin{align*} |
| - | U_{ind} &= |{{d}\over{dt}} \cdot \Phi_m | \\ | + | U_{\rm ind} &= |{{\rm d}\over{{\rm d}t}} \cdot \Phi_{\rm m} | \\ |
| - | &= B\cdot {{r^2\omega}\over{2}} | + | &= B\cdot {{r^2\omega}\over{2}} |
| \end{align*} | \end{align*} | ||
| - | When divided by the resistance $R$ of the loop, this yields | + | When divided by the resistance $R$ of the loop, this yields the magnitude of the induced current |
| - | \begin{align*} | + | \begin{align*} |
| - | I_{ind} &= {{|U_{ind}|}\over{R}} | + | I_{\rm ind} &= {{|U_{\rm ind}|}\over{R}} \\ |
| - | &= B\cdot {{r^2\omega}\over{2R}} | + | &= B\cdot {{r^2\omega}\over{2R}} |
| \end{align*} | \end{align*} | ||
| - | As $\varphi$ increases, so does the flux through the loop due to $\vec{B}$. | + | As $\varphi$ increases, so does the flux through the loop due to $\vec{B}$. To counteract this increase, the magnetic field due to the induced current must be directed into the page in the region enclosed by the loop. Therefore, as part (b) of <imgref ImgNr11> illustrates, |
| - | To counteract this increase, the magnetic field due to the induced current must be directed into the page in the region enclosed by the loop. | + | |
| - | Therefore, as part (b) of <imgref ImgNr11> illustrates, | + | |
| </ | </ | ||
| - | <panel type=" | + | <panel type=" |
| - | The following example is the basis for an electric generator: | + | The following example is the basis for an electric generator: A rectangular coil of area $A$ and $N$ turns is placed in a uniform magnetic field $\vec{B}$, as shown in <imgref ImgNr12> |
| - | A rectangular coil of area $A$ and $N$ turns is placed in a uniform magnetic field $\vec{B}$, as shown in <imgref ImgNr12> | + | The coil is rotated about the $z$-axis through its center at a constant angular velocity $\omega$. |
| - | The coil is rotated about the $z$-axis through its center at a constant angular velocity $\omega$. | + | |
| - | Obtain an expression for the induced potential difference $U_{ind}$ in the coil. | + | Obtain an expression for the induced potential difference $U_{\rm ind}$ in the coil. |
| - | < | + | < |
| - | < | + | |
| - | </ | + | |
| - | {{drawio> | + | |
| - | </ | + | |
| - | **Strategy** | + | <button size=" |
| - | According to the diagram, the angle between the surface vector $\vec{A}$ and the magnetic field $\vec{B}$ is $\varphi$. | + | According to the diagram, the angle between the surface vector $\vec{A}$ and the magnetic field $\vec{B}$ is $\varphi$. The dot product of $\vec{A} \cdot \vec{B}$ simplifies to only the $\cos \varphi$ component of the magnetic field times the area, namely where the magnetic field projects onto the unit surface vector $\vec{A}$. The magnitude of the magnetic field and the area of the loop are fixed over time, which makes the integration simplify |
| - | The dot product of $\vec{A} \cdot \vec{B}$ simplifies to only the $cos \varphi$ component of the magnetic field times the area, namely where the magnetic field projects onto the unit surface vector $\vec{A}$. | + | |
| - | The magnitude of the magnetic field and the area of the loop are fixed over time, which makes the integration simplify | + | |
| - | The induced potential difference is written out using Faraday’s law. | + | |
| - | **Solution** | + | <button size=" |
| - | When the coil is in a position such that its normal | + | When the coil is in a position such that its surface |
| - | \begin{align*} | + | \begin{align*} |
| - | \Phi_m | + | \Phi_{\rm m} &= \iint \vec{B} |
| - | &= BA\cdot cos \varphi \\ | + | |
| \end{align*} | \end{align*} | ||
| From Faraday’s law, the induced potential difference in the coil is | From Faraday’s law, the induced potential difference in the coil is | ||
| - | \begin{align*} | + | \begin{align*} |
| - | U_{ind} &= - N |{{d}\over{dt}} \Phi_m | + | u_{\rm ind} &= - N {{\rm d}\over{{\rm d}t}} \Phi_{\rm m} \\ |
| - | &= NBA \cdotsin | + | &= NBA \cdot \sin \varphi \cdot {{{\rm d}\varphi}\over{{\rm d}t}} |
| \end{align*} | \end{align*} | ||
| - | The constant angular velocity is $\omega = d\varphi/dt$. | + | The constant angular velocity is $\omega = {\rm d}\varphi/{\rm d}t$. The angle $\varphi$ represents the time evolution of the angular velocity or $\omega t$. |
| - | The angle $\varphi$ represents the time evolution of the angular velocity or $\omega t$. | + | This changes the function to time-space rather than $\varphi$. The induced potential difference, therefore, varies sinusoidally with time according to |
| - | This is changes the function to time space rather than $\varphi$. | + | |
| - | The induced potential difference therefore varies sinusoidally with time according to | + | |
| - | \begin{align*} | + | \begin{align*} |
| - | U_{ind} &= U_{ind,0} \cdot sin \omega t | + | |
| - | \end{align*} | + | |
| - | where $U_{ind,0} = NBA\omega$. | + | where $U_{\rm ind,0} = NBA\omega$. |
| </ | </ | ||
| - | <panel type=" | + | <panel type=" |
| - | The generator coil shown in <imgref ImgNr13> is rotated through one-fourth of a revolution (from θ=0o | + | 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-cm radius and is in a uniform 0.80-T magnetic field. | + | The $200$-turn circular coil has a $5.00 ~\rm cm$ radius and is in a uniform |
| - | What is the emf induced? | + | What is the value of the induced |
| - | < | + | < |
| - | < | + | |
| - | </ | + | |
| - | {{drawio> | + | |
| - | </ | + | |
| - | **Strategy** | + | <button size=" |
| - | Faraday’s law of induction is used to find the emf induced: | + | Faraday’s law of induction is used to find the potential difference |
| - | \begin{align*} | + | \begin{align*} |
| - | U_{ind} &= - N {{d \Phi_m}\over{dt}} | + | u_{\rm ind} &= - N {{{\rm d} \Phi_{\rm m}}\over{{\rm d}t}} |
| \end{align*} | \end{align*} | ||
| - | We recognize this situation as the same one in the task before. | + | We recognize this situation as the same one in the exercise |
| - | According to the diagram, the projection of the surface vector $\vec{A}$ to the magnetic field is initially ${A}\cdot cos \varphi$ and this is inserted by the definition of the dot product. | + | According to the diagram, the projection of the surface vector $\vec{A}$ to the magnetic field is initially ${A}\cdot |
| - | The magnitude of the magnetic field and area of the loop are fixed over time, which makes the integration simplify | + | The magnitude of the magnetic field and the area of the loop are fixed over time, which makes the integration simplify |
| - | The induced potential difference is written out using Faraday’s law: | + | |
| - | \begin{align*} | + | \begin{align*} |
| - | U_{ind} & | + | u_{\rm ind} &= N B \cdot \sin \varphi \cdot {{{\rm d} \varphi}\over{{\rm d}t}} |
| - | \end{align*} | + | \end{align*} |
| + | </ | ||
| - | **Solution** | + | <button size=" |
| - | We are given that $N=200$, | + | We are given that $N=200$, $B=0.80~\rm T$ , $\varphi = 90°$ , $d\varphi=90°=\pi/ |
| The area of the loop is | The area of the loop is | ||
| - | \begin{align*} | + | \begin{align*} |
| - | a = \pi r^2 = 3.14 \cdot (0.0500m)^2 = 7.85 \cdot 10^{-3} m^2 | + | A = \pi r^2 = 3.14 \cdot (0.0500~\rm m)^2 = 7.85 \cdot 10^{-3} |
| \end{align*} | \end{align*} | ||
| + | |||
| + | </ | ||
| Entering this value gives | Entering this value gives | ||
| - | \begin{align*} | + | \begin{align*} |
| - | U_{ind} & | + | U_{\rm ind} &= 200 \cdot 0.80 ~\rm T \cdot (7.85 \cdot 10^{-3} |
| \end{align*} | \end{align*} | ||
| + | </ | ||
| + | |||
| + | ===== 4.4 Self-Induction ===== | ||
| + | |||
| + | ==== Linked Flux ==== | ||
| + | |||
| + | When looking at the magnetic field in a coil multiple windings capture the passing flux, see <imgref ImgNr14> (a). | ||
| + | Each winding will create a potential difference. | ||
| + | It can also be interpreted in such a way that the flux is going through the closed surface of the circuit multiple times (in picture (b)). | ||
| + | |||
| + | < | ||
| + | |||
| + | The resulting electric voltage in such a situation is given by the sum of the induced potential differences in each winding. | ||
| + | |||
| + | \begin{align*} | ||
| + | u_{\rm ind} &= - {{{\rm d} \Phi_{\rm sum}}\over{{\rm d}t}} \\ | ||
| + | &= - \sum_{i=1}^n {{{\rm d} \Phi_{i}}\over{{\rm d}t}} \\ | ||
| \end{align*} | \end{align*} | ||
| + | |||
| + | The **linked flux** $\Psi$ is defined as the resulting flux given by the sum of the partial fluxes of the closed circuit. | ||
| + | |||
| + | \begin{align*} | ||
| + | \boxed{ \Psi = \sum_{i=1}^n \Phi_{i} } | ||
| + | \end{align*} | ||
| + | |||
| + | The linked flux simplifies the induced electric voltage of a coil to: | ||
| + | |||
| + | \begin{align*} | ||
| + | u_{\rm ind} &= - N \cdot {{{\rm d} \Phi}\over{{\rm d}t}} \\ | ||
| + | &= - {{\rm d}\over{{\rm d}t}} \Psi \\ | ||
| + | \end{align*} | ||
| + | |||
| + | ==== Self-Induction ==== | ||
| + | |||
| + | Up to now, we investigated the induction of electric voltages and currents based on the change of an external flux ${\rm d}\Psi / {\rm d}t$. | ||
| + | For the induced current $i_{\rm ind}$, we found that it counteracts the change of the external flux (Lenz law). | ||
| + | |||
| + | But what happens, when there is no external field - only a coil which creates the flux change itself (see <imgref ImgNr46> | ||
| + | |||
| + | < | ||
| + | |||
| + | To understand this, we will investigate the situation for a long coil (<imgref ImgNr15> | ||
| + | |||
| + | < | ||
| + | |||
| + | The created field density of the coil can be derived from Ampere' | ||
| + | |||
| + | \begin{align*} | ||
| + | \theta(t) &= \int & \vec{H}(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} | ||
| + | & | ||
| + | \end{align*} | ||
| + | |||
| + | With magnetic voltage $\theta(t) = N \cdot i$ this lead to the magnetic flux density $B(t)$ | ||
| + | |||
| + | \begin{align*} | ||
| + | N \cdot i &= {H}(t) \cdot l \\ | ||
| + | | ||
| + | | ||
| + | \end{align*} | ||
| + | |||
| + | Based on the magnetic flux density $B(t)$ it is possible to calculate the flux $\Phi(t)$: | ||
| + | |||
| + | \begin{align*} | ||
| + | \Phi(t) &= \iint_A \vec{B}(t) | ||
| + | &= \iint_A \mu_0 \mu_{\rm r} \cdot {{N \cdot i }\over {l}} \cdot {\rm d}A \\ | ||
| + | & | ||
| + | \end{align*} | ||
| + | |||
| + | The changing flux $\Phi$ is now creating an induced electric voltage and current, which counteracts the initial change of the current. | ||
| + | This effect is called **Self Induction**. The induced electric voltage $u_{\rm ind}$ is given by: | ||
| + | |||
| + | \begin{align*} | ||
| + | u_{\rm ind} &= - N \cdot {{{\rm d} | ||
| + | &= - N \cdot {{{\rm d} (\mu_0 \mu_{\rm r} \cdot {{N \cdot i }\over {l}} \cdot A)}\over{{\rm d}t}} \\ | ||
| + | &= - N \cdot \mu_0 \mu_{\rm r} \cdot {{N \cdot A }\over {l}} \cdot {{{\rm d}i}\over{{\rm d}t}} \\ | ||
| + | \end{align*} | ||
| + | |||
| + | \begin{align*} | ||
| + | \boxed{ u_{\rm ind} = - \mu_0 \mu_{\rm r} \cdot N^2 \cdot {{A }\over {l}} \cdot {{{\rm d}i}\over{{\rm d}t}} \\ } \\ | ||
| + | \text{for a long coil} | ||
| + | \end{align*} | ||
| + | |||
| + | The result means that the induced electric voltage $u_{\rm ind}$ is proportional to the change of the current ${{\rm d}\over{{\rm d}t}}i$. | ||
| + | The proportionality factor is also called **Self-inductance** | ||
| + | |||
| + | ===== 4.5 Inductance ===== | ||
| + | |||
| + | The inductance is another passive basic component of the electric circuit. | ||
| + | Besides the ohmic resistor $R$ and the capacitor $C$, the inductor $L$ is the lump component entailing the inductance. | ||
| + | |||
| + | Generally, the inductance is defined by: | ||
| + | \begin{align*} | ||
| + | \boxed{ L = \left|{{u_{\rm ind}}\over{{\rm d}i / {\rm d}t}}\right| \\ } | ||
| + | \end{align*} | ||
| + | |||
| + | The inductance $L$ can also be described differently based on Lenz law $u_{\rm ind} = - {{\rm d}\over{{\rm d}t}}\Psi(t)$ : | ||
| + | |||
| + | \begin{align*} | ||
| + | L &= \left|{{u_{\rm ind}}\over{{\rm d}i / {\rm d}t}}\right| \\ | ||
| + | & | ||
| + | \end{align*} | ||
| + | |||
| + | \begin{align*} \boxed{ L = {{ \Psi(t)}\over{i}} } \end{align*} | ||
| + | |||
| + | One can also consider an inductor a " | ||
| + | It reacts to any change in the current with a counteracting voltage since the current change leads to a changing flux and - therefore - an induced voltage. | ||
| + | The <imgref BildNr5> shows an inductor in series with a resistor and a switch (any real switch also behaves as a capacitor, when open). | ||
| + | Once the simulation is started, the inductor directly counteracts the current, which is why the current only slowly increases. | ||
| + | |||
| + | < | ||
| + | |||
| + | Mathematically the voltages can be described in the following way: | ||
| + | |||
| + | \begin{align*} | ||
| + | u_0 &= u_R &+ &u_L \\ | ||
| + | &= i \cdot R & + & | ||
| + | &= i \cdot R & + &L \cdot {{{\rm d}i}\over{{\rm d}t}} \\ | ||
| + | \end{align*} | ||
| + | |||
| + | ==== Inductance of different Components ==== | ||
| + | |||
| + | === Long Coil === | ||
| + | |||
| + | In the last sub-chapter, | ||
| + | By these, the inductance of a long coil is | ||
| + | |||
| + | \begin{align*} | ||
| + | \boxed{L_{\rm long \; coil} = \mu_0 \mu_{\rm r} \cdot N^2 \cdot {{A }\over {l}}} | ||
| + | \end{align*} | ||
| + | |||
| + | === Toroidal Coil === | ||
| + | |||
| + | The toroidal coil was analyzed in the last chapter(see [[: | ||
| + | Here, a rectangular intersection a assumed (see <imgref ImgNr16> | ||
| + | |||
| + | < | ||
| + | |||
| + | This leads to | ||
| + | |||
| + | \begin{align*} H(t) = {{N \cdot i}\over {l}} \end{align*} | ||
| + | |||
| + | with the mean magnetic path length (= length of the average field line) $l = \pi(r_{\rm o} + r_{\rm i})$: | ||
| + | |||
| + | \begin{align*} H(t) = {{N \cdot i}\over { \pi(r_{\rm o} + r_{\rm i})}} \end{align*} | ||
| + | |||
| + | The inductance $L$ can be calculated by | ||
| + | |||
| + | \begin{align*} | ||
| + | L_{\rm toroidal \; coil} & | ||
| + | & | ||
| + | \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 $A = h (r_{\rm o} - r_{\rm i})$, we get: | ||
| + | |||
| + | \begin{align*} | ||
| + | \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*} | ||
| + | |||
| + | \begin{align*} | ||
| + | \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*} | ||
| + | |||
| + | === Exercises === | ||
| + | |||
| + | <panel type=" | ||
| + | |||
| + | A change of magnetic flux is passing a coil with a single winding. The following pictures <imgref ImgNrEx01> | ||
| + | |||
| + | * Create for each $\Phi(t)$-diagram the corresponding $u_{\rm ind}(t)$-diagram! | ||
| + | * Write down each maximum value of $u_{\rm ind}(t)$ | ||
| + | |||
| + | < | ||
| + | |||
| + | <button size=" | ||
| + | |||
| + | For partwise linear $u_{\rm ind}$ one can derive: | ||
| + | \begin{align*} | ||
| + | u_{\rm ind} &= -{{{\rm d}\Phi}\over{{\rm d}t}} \\ | ||
| + | &= -{{\Delta \Phi}\over{\Delta t}} | ||
| + | \end{align*} | ||
| + | |||
| + | For diagram (a): | ||
| + | |||
| + | * $t= 0.0 ... 0.6 ~\rm s$: $u_{\rm ind} = -{{0 ~\rm Vs}\over{0.6 ~\rm s}}= 0$ | ||
| + | * $t= 0.6 ... 1.5 ~\rm s$: $u_{\rm ind} = -{{-3.75\cdot 10^{-3} ~\rm Vs}\over{0.9 ~\rm s}}= +4.17 ~\rm mV$ | ||
| + | * $t= 1.5 ... 2.1 ~\rm s$: $u_{\rm ind} = -{{0 ~\rm Vs}\over{0.6 ~\rm s}}= 0$ | ||
| + | |||
| + | </ | ||
| + | |||
| + | <button size=" | ||
| </ | </ | ||
| - | ===== 4.2 Induction ===== | + | <panel type=" |
| - | * thought experiment: Faraday experiment | + | A changing |
| - | * direction | + | The following pictures <imgref ImgNrEx02> |
| - | * | + | |
| - | * Bewegungsinduktion | + | |
| - | * difference to electrostatic: | + | |
| - | * Ruheinduktion | + | |
| - | ===== 4.3 Self-Induction ===== | + | * Create for each $u_{\rm ind}(t)$-diagram the corresponding $\Phi(t)$-diagram! |
| + | * Write down each maximum value of $\Phi(t)$ | ||
| - | Another example illustrating | + | Note the given start value $\Phi_0$ for each flux. |
| - | When the switch is opened, the decrease in current through the solenoid causes a decrease in magnetic | + | |
| - | This potential difference must oppose the change (the termination of the current) causing it. | + | |
| - | Consequently, | + | |
| - | This may generate an arc across the terminals of the switch as it is opened. | + | |
| - | < | + | < |
| - | < | + | |
| - | </ | + | |
| - | {{drawio> | + | |
| - | </ | + | |
| + | </ | ||
| - | * Linked flux (see chapter 3.) | + | <panel type=" |
| - | * self-induction | + | |
| - | * formulas of different inductor types | + | |
| + | A single winding is located in a homogenous magnetic field ($B = 0.5 ~\rm T$) between the pole pieces. | ||
| + | The winding has a length of $150 ~\rm mm$ and a distance between the conductors of $50 ~\rm mm$ (see <imgref ImgNrEx03> | ||
| - | < | + | * Determine the function $u_{\rm ind}(t)$, when the coil is rotating with $3000 ~\rm min^{-1}$. |
| - | < | + | * Given a current of $20 ~\rm A$ through the winding: What is the torque $M(\varphi)$ depending on the angle between the surface vector of the winding and the magnetic field? |
| - | </ | + | |
| - | {{drawio> | + | < |
| - | </ | + | |
| + | <button size=" | ||
| + | \begin{align*} | ||
| + | u_{\rm ind} &= - | ||
| + | &= - {{\rm d}\over{{\rm d}t}} B \cdot A \\ | ||
| + | &= - B \cdot {{\rm d}\over{{\rm d}t}} A\\ | ||
| + | &= - B \cdot {{\rm d}\over{{\rm d}t}} \left(l \cdot d \cdot \cos(\omega t) \right)\\ | ||
| + | &= + B \cdot l \cdot d \cdot \omega \cdot \sin(\omega t)\\ | ||
| + | \end{align*} | ||
| + | |||
| + | </ | ||
| + | |||
| + | <panel type=" | ||
| + | |||
| + | A rectangular coil is given by the sizes $a=10 ~\rm cm$, $b=4 ~\rm cm$, and the number of windings $N=200$. | ||
| + | This coil moves with a constant speed of $v=2 ~\rm m/s$ perpendicular to a homogeneous | ||
| + | The area of the coil is tilted with regard to the field in $\alpha=60°$ and enters the field from the left side (see <imgref ImgNrEx04>). | ||
| + | |||
| + | * Determine the function $u_{\rm ind}(t)$ on the coil along the given path. Sketch of the $u_{\rm ind}(t)$ diagram. | ||
| + | * What is the maximum induced voltage $u_{\rm ind, | ||
| + | |||
| + | < | ||
| + | |||
| + | <button size=" | ||
| + | |||
| + | Let assume, that $l$ is in the $x$-axis, $\vec{B}$ in the $y$-axis and $a$. | ||
| + | \\ \\ | ||
| + | |||
| + | **Step 1**: Calculate the effective area, perpendicular to the $\vec{B}$-field (independent from whether the area is in the $\vec{B}$-field or not). | ||
| + | |||
| + | For this $b$ has to be projected onto the plane perpendicular to the $\vec{B}$-field: | ||
| + | \begin{align*} | ||
| + | A_{\rm eff} &= a \cdot b \cdot \cos \alpha | ||
| + | \end{align*} | ||
| + | |||
| + | **Step 2**: Focus on entering and exiting the $\vec{B}$-field. \\ | ||
| + | Induction only occurs for ${{\rm d}\over{{\rm d}t}}(A\cdot B)\neq 0$, so here: when the area $A_{\rm eff}$ enters and leave the constant $\vec{B}$-field. | ||
| + | |||
| + | When entering the $\vec{B}$-field the area $A$ with $0< | ||
| + | The area moves with $v$. Therefore, after $\Delta t = b_{\rm eff} \cdot v$ the full $\vec{B}$-field is provided onto the area $A_{\rm eff}$: | ||
| + | \begin{align*} | ||
| + | u_{\rm ind} &= - {{{\rm d}\Psi}\over{{\rm d}t}} \\ | ||
| + | &= -N \cdot {{\rm d}\over{{\rm d}t}} B \cdot A \\ | ||
| + | &= -N \cdot | ||
| + | &= -N \cdot {{1}\over{b \cdot \cos \alpha \cdot v}} B \cdot a \cdot b \cdot \cos \alpha \\ | ||
| + | &= -N \cdot B \cdot {{a}\over{v}}\\ | ||
| + | \end{align*} | ||
| + | |||
| + | The following diagram shows ... | ||
| + | * ... how one can derive the effective width $b_{\rm eff}$, which is projected onto the plane perpendicular to the $\vec{B}$-field: | ||
| + | * ... what happens on the effective area $A_{\rm eff}$: there is a change of the field lines in the area only for entering and leaving the $\vec{B}$-field. | ||
| + | * ... how the $u_{\rm ind}(t)$ looks as a graph: the part of $A_{\rm eff}$, where the $\vec{B}$-field passes through increase linearly due to the constant speed $v$ | ||
| + | Be aware, that the task did not provide a clue for the direction of windings and therefore it provides no clue for the polarization of the induced voltage. \\ | ||
| + | So, the course of the voltage when entering or exiting is not uniquely given. | ||
| + | |||
| + | < | ||
| + | |||
| + | |||
| + | </ | ||
| + | |||
| + | <panel type=" | ||
| + | |||
| + | Calculate the inductance for the following settings | ||
| + | |||
| + | - cylindrical air coil with $N=400$, winding diameter $d=3 ~\rm cm$ and length $l=18 ~\rm cm$ | ||
| + | - similar coil geometry as explained in 1. , but with double the number of windings | ||
| + | - two coils as explained in 1. in series | ||
| + | - similar coil geometry and number of windings as explained in 1. , but with an iron core ($\mu_{\rm r}=1000$) | ||
| + | |||
| + | </ | ||
| + | |||
| + | <panel type=" | ||
| + | |||
| + | A cylindrical air coil (length $l=40 ~\rm cm$, diameter $d=5 ~\rm cm$, and a number of windings $N=300$) passes a current of $30 ~\rm A$. The current shall be reduced linearly in $2 ~\rm ms$ down to $0 ~\rm A$. | ||
| + | |||
| + | What is the amount of the induced voltage $u_{\rm ind}$? </ | ||
| + | |||
| + | <panel type=" | ||
| + | |||
| + | A coil with the inductance $L=20 ~\rm µH$ passes a current of $40 ~\rm A$. The current shall be reduced linearly in $5 ~\rm µs$ down to $0 ~\rm A$ (see <imgref ImgNrEx05> | ||
| + | |||
| + | * What is the amount of the induced voltage $u_{\rm ind}$? | ||
| + | * Sketch the course of $u_{\rm ind}(t)$! | ||
| + | |||
| + | < | ||
| + | |||
| + | </ | ||