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Buzz ♦
Buzz ♦
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But if I want to use that equation I have to caulculatecalculate couple of really tidioustedious integrals. So I found some online source where some dude used this suspicious equation $$U_{ind} = \int (\textbf{v} \times \textbf{B}) \cdot d\textbf{l} \; ,$$ it$$U_{ind} = \int (\textbf{v} \times \textbf{B}) \cdot d\textbf{l}\, .$$ It was said that the following equation can be always used.   

My question: isIs this equation equivalent to the ones above? Why or why not?

This other equation is for so-called motional EMFemf, which is the part of the total EMFemf induced in a conductor when this conductor moves in external magnetic field.

In the frame of the conducting pipe, the pipe does not move, so in this frame, the formula gives motional emf equal to zero -— $\mathbf v$ is zero. There is also the induced emf, due to induced electric field. In this frame, all EMFthe emf is due to induced EMFemf, and the familiar FaradayFaraday's law formula for a stationary loop applies.

In the frame of the magnet, the pipe is moving in the field of the stationary magnet, with variable velocity. So the formula does not give zero this time. It is the familiar Faraday formula that gives zero induced emf here. So the roles of the two mechanisms of EMF get reversed.

In both frames one gets the same value for the total EMFemf, so in a sense, it does not matter which frame is chosen. However, one frame may be better for calculations.

But if I want to use that equation I have to caulculate couple of really tidious integrals. So I found some online source where some dude used this suspicious equation $$U_{ind} = \int (\textbf{v} \times \textbf{B}) \cdot d\textbf{l} \; ,$$ it was said that the following equation can be always used.  My question is this equation equivalent to the ones above? Why or why not?

This other equation is for so-called motional EMF, which is part of total EMF induced in a conductor when this conductor moves in external magnetic field.

In the frame of the conducting pipe, the pipe does not move, so in this frame, the formula gives motional emf equal to zero - $\mathbf v$ is zero. There is also the induced emf, due to induced electric field. In this frame, all EMF is due to induced EMF and the familiar Faraday law formula for stationary loop applies.

In the frame of the magnet, the pipe is moving in the field of the stationary magnet, with variable velocity. So the formula does not give zero this time. It is the familiar Faraday formula that gives zero induced emf here. So the roles of the two mechanisms of EMF get reversed.

In both frames one gets the same value for total EMF, so in a sense, it does not matter which frame is chosen. However, one frame may be better for calculations.

But if I want to use that equation I have to calculate couple of really tedious integrals. So I found some online source where some dude used this suspicious equation $$U_{ind} = \int (\textbf{v} \times \textbf{B}) \cdot d\textbf{l}\, .$$ It was said that the following equation can be always used. 

My question: Is this equation equivalent to the ones above? Why or why not?

This other equation is for so-called motional emf, which is the part of the total emf induced in a conductor when this conductor moves in external magnetic field.

In the frame of the conducting pipe, the pipe does not move, so in this frame, the formula gives motional emf equal to zero — $\mathbf v$ is zero. There is also the induced emf, due to induced electric field. In this frame, all the emf is due to induced emf, and the familiar Faraday's law formula for a stationary loop applies.

In the frame of the magnet, the pipe is moving in the field of the stationary magnet, with variable velocity. So the formula does not give zero this time. It is the familiar Faraday formula that gives zero induced emf here. So the roles of the two mechanisms of EMF get reversed.

In both frames one gets the same value for the total emf, so in a sense, it does not matter which frame is chosen. However, one frame may be better for calculations.

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Ján Lalinský
Ján Lalinský
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But if I want to use that equation I have to caulculate couple of really tidious integrals. So I found some online source where some dude used this suspicious equation $$U_{ind} = \int (\textbf{v} \times \textbf{B}) \cdot d\textbf{l} \; ,$$ it was said that the following equation can be always used. My question is this equation equivalent to the ones above? Why or why not?

This other equation is for so-called motional EMF, which is part of total EMF induced in a conductor when this conductor moves in external magnetic field.

In the frame of the conducting pipe, the pipe does not move, so in this frame, the formula gives motional emf equal to zero - $\mathbf v$ is zero. There is also the induced emf, due to induced electric field. In this frame, all EMF is due to induced EMF and the familiar Faraday law formula for stationary loop applies.

In the frame of the magnet, the pipe is moving in the field of the stationary magnet, with variable velocity. So the formula does not give zero this time. It is the familiar Faraday formula that gives zero induced emf here. So the roles of the two mechanisms of EMF get reversed.

In both frames one gets the same value for total EMF, so in a sense, it does not matter which frame is chosen. However, one frame may be better for calculations.