Skip to main content
Physics

Return to Question

added 255 characters in body
Source Link
psychgiraffe
psychgiraffe
  • 126
  • 6

I'm currently reading Bleaney & Bleaney 3rd Ed. On pg 136 they discuss the case of 2 coils. Their logic is as follows:

If a second coil is brought near to a coil carrying a current $I_1$ is passed, there will be a magnetic flux $\Phi_2$ through the second coil 1 thendue to the flux throughcurrent in the first coil 2. This is linearly proportional to $I_1$ so we can write $\Phi_2 = L_{21} I_{1}$ for a constant $L_{12}$$L_{21}$. IfThere will also be a flux $I_2$ is passed$\Phi_1$ through coil 2 then we havethe first circuit due to a current $\Phi_1 = L_{12} I_1$$I_2$ in the second circuit, given by $\Phi_1 = L_{12} I_{2}$ for a constant $L_{12}$. 

The potential energy of the system can be found from the flux of either coil due to the field of another: $U = -\Phi_1 I_1 = -L_{21} I_2 I_1 = -\Phi_2 I_2 = -L_{12} I_1 I_2$.

Hence $L_{12} = L_{21}$.

However I am struggling to understand the setup. Is there a current $I_1$ and $I_2$ flowing in both coils simultaneously? If so, why is the potential energy not $U = -\Phi_{11} I_1 - \Phi_{21} I_1 -\Phi_{22} I_2 - \Phi_{21} I_2$ where $\Phi_{ij}$ is the flux produced by coil i that threads coil j.

(If there is only a current through one coil at a time, I still dont understand their logic, as why should the potential energy when coil 1 has a current $I_1$ be equal to the potential energy when coil 2 has current $I_2$, given we haven't assumed that the coils are identical)

So I have 2 questions.

  1. In this derivation for $L_{12} = L_{21}$ is there a current through both coils simultaneously?
  2. Why can "The potential energy of the system can be found from the flux of either coil due to the field of another". Why can the flux produced by e.g. coil 1 which threads coil 1 be ignored?

I'm currently reading Bleaney & Bleaney 3rd Ed. On pg 136 they discuss the case of 2 coils. Their logic is as follows:

If $I_1$ is passed through coil 1 then the flux through coil 2 is $\Phi_2 = L_{21} I_{1}$ for a constant $L_{12}$. If $I_2$ is passed through coil 2 then we have $\Phi_1 = L_{12} I_1$. The potential energy of the system can be found from the flux of either coil due to the field of another: $U = -\Phi_1 I_1 = -L_{21} I_2 I_1 = -\Phi_2 I_2 = -L_{12} I_1 I_2$.

Hence $L_{12} = L_{21}$.

However I am struggling to understand the setup. Is there a current $I_1$ and $I_2$ flowing in both coils simultaneously? If so, why is the potential energy not $U = -\Phi_{11} I_1 - \Phi_{21} I_1 -\Phi_{22} I_2 - \Phi_{21} I_2$ where $\Phi_{ij}$ is the flux produced by coil i that threads coil j.

(If there is only a current through one coil at a time, I still dont understand their logic, as why should the potential energy when coil 1 has a current $I_1$ be equal to the potential energy when coil 2 has current $I_2$, given we haven't assumed that the coils are identical)

So I have 2 questions.

  1. In this derivation for $L_{12} = L_{21}$ is there a current through both coils simultaneously?
  2. Why can "The potential energy of the system can be found from the flux of either coil due to the field of another". Why can the flux produced by e.g. coil 1 which threads coil 1 be ignored?

I'm currently reading Bleaney & Bleaney 3rd Ed. On pg 136 they discuss the case of 2 coils. Their logic is as follows:

If a second coil is brought near to a coil carrying a current $I_1$, there will be a magnetic flux $\Phi_2$ through the second coil due to the current in the first coil. This is linearly proportional to $I_1$ so we can write $\Phi_2 = L_{21} I_{1}$ for a constant $L_{21}$. There will also be a flux $\Phi_1$ through the first circuit due to a current $I_2$ in the second circuit, given by $\Phi_1 = L_{12} I_{2}$ for a constant $L_{12}$. 

The potential energy of the system can be found from the flux of either coil due to the field of another: $U = -\Phi_1 I_1 = -L_{21} I_2 I_1 = -\Phi_2 I_2 = -L_{12} I_1 I_2$.

Hence $L_{12} = L_{21}$.

However I am struggling to understand the setup. Is there a current $I_1$ and $I_2$ flowing in both coils simultaneously? If so, why is the potential energy not $U = -\Phi_{11} I_1 - \Phi_{21} I_1 -\Phi_{22} I_2 - \Phi_{21} I_2$ where $\Phi_{ij}$ is the flux produced by coil i that threads coil j.

(If there is only a current through one coil at a time, I still dont understand their logic, as why should the potential energy when coil 1 has a current $I_1$ be equal to the potential energy when coil 2 has current $I_2$, given we haven't assumed that the coils are identical)

So I have 2 questions.

  1. In this derivation for $L_{12} = L_{21}$ is there a current through both coils simultaneously?
  2. Why can "The potential energy of the system can be found from the flux of either coil due to the field of another". Why can the flux produced by e.g. coil 1 which threads coil 1 be ignored?
Source Link
psychgiraffe
psychgiraffe
  • 126
  • 6

What logic is being used in Bleaney & Bleaney to prove the reciprocity theorem for mutual inductance?

I'm currently reading Bleaney & Bleaney 3rd Ed. On pg 136 they discuss the case of 2 coils. Their logic is as follows:

If $I_1$ is passed through coil 1 then the flux through coil 2 is $\Phi_2 = L_{21} I_{1}$ for a constant $L_{12}$. If $I_2$ is passed through coil 2 then we have $\Phi_1 = L_{12} I_1$. The potential energy of the system can be found from the flux of either coil due to the field of another: $U = -\Phi_1 I_1 = -L_{21} I_2 I_1 = -\Phi_2 I_2 = -L_{12} I_1 I_2$.

Hence $L_{12} = L_{21}$.

However I am struggling to understand the setup. Is there a current $I_1$ and $I_2$ flowing in both coils simultaneously? If so, why is the potential energy not $U = -\Phi_{11} I_1 - \Phi_{21} I_1 -\Phi_{22} I_2 - \Phi_{21} I_2$ where $\Phi_{ij}$ is the flux produced by coil i that threads coil j.

(If there is only a current through one coil at a time, I still dont understand their logic, as why should the potential energy when coil 1 has a current $I_1$ be equal to the potential energy when coil 2 has current $I_2$, given we haven't assumed that the coils are identical)

So I have 2 questions.

  1. In this derivation for $L_{12} = L_{21}$ is there a current through both coils simultaneously?
  2. Why can "The potential energy of the system can be found from the flux of either coil due to the field of another". Why can the flux produced by e.g. coil 1 which threads coil 1 be ignored?