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Why is induced electric field always circular?

Whenever there is change in magnetic field, electric field is induced. The most common explanation that is found is: " If a closed loop circular conductor is placed in time varying magnetic field then current is induced as per the Faraday's Law and direction is given by Lenz's law. So, there must be electric field that cause charge carriers to move and this field is there even if there is no circular conductor. Furthermore, it's direction is always tangent to the loop of the conductor. Hence, circular electric field is induced due to changing magnetic field, field line being the circumference of the conducting loop."

Now that the position of the conductor is not unique, we can place conductor in one one position we get one field line and if we place same conductor in another position we get another field line following similar arguments. But as the position of the conductor is not unique, two field lines can be made to intersect, a contradiction.

Can this be explained in some other way?
If the electric field lines are concentric circles then what uniquely specifies their centres?

Thanks in advance.

[ Please try to explain this in the level of A-Level Physics i.e. with no knowledge of Maxwell's equations and vector calculus ]
That doesn't sound like a conventional explanation - there isn't a latent EMF that's still there when you take the conductor away, the EMF is generated by the interaction of charge carriers with a magnetic field... i.e. there is no EMF unless you are in some way dragging the charge carriers through the magnetic field.

maybe watch this series of four calculus free lectures and see if it makes any sense (in any case it shows you what you'll need to reproduce in A level exams so you might as well just learn it anyway)

https://www.youtube.com/watch?v=ejRJM-kCyWQ&list=PLX2gX-ftPVXV1z8nPIePOUk5WIGVYBzDy
Original post by Joinedup
That doesn't sound like a conventional explanation - there isn't a latent EMF that's still there when you take the conductor away, the EMF is generated by the interaction of charge carriers with a magnetic field... i.e. there is no EMF unless you are in some way dragging the charge carriers through the magnetic field.

maybe watch this series of four calculus free lectures and see if it makes any sense (in any case it shows you what you'll need to reproduce in A level exams so you might as well just learn it anyway)

https://www.youtube.com/watch?v=ejRJM-kCyWQ&list=PLX2gX-ftPVXV1z8nPIePOUk5WIGVYBzDy


Thanks for the reply. I watched those 4 lectures. I understand them. Could you please explain why the field lines of electric field produced by changing magnetic field are concentric circles?

I am referring to https://www.youtube.com/watch?v=2DH7ufrkeHM 0.00 to 2:10.
What would happen if he had square shaped closed conducting loop instead of the ring?
What would be the situation if magnetic field were passing throughout the plane of paper, not just the part shown in the video.

Thank you.
Original post by tangotangopapa2
Whenever there is change in magnetic field, electric field is induced. The most common explanation that is found is: " If a closed loop circular conductor is placed in time varying magnetic field then current is induced as per the Faraday's Law and direction is given by Lenz's law. So, there must be electric field that cause charge carriers to move and this field is there even if there is no circular conductor. Furthermore, it's direction is always tangent to the loop of the conductor. Hence, circular electric field is induced due to changing magnetic field, field line being the circumference of the conducting loop."

Now that the position of the conductor is not unique, we can place conductor in one one position we get one field line and if we place same conductor in another position we get another field line following similar arguments. But as the position of the conductor is not unique, two field lines can be made to intersect, a contradiction.

Can this be explained in some other way?
If the electric field lines are concentric circles then what uniquely specifies their centres?

Thanks in advance.

[ Please try to explain this in the level of A-Level Physics i.e. with no knowledge of Maxwell's equations and vector calculus ]


because ∇×E=−∂B∂t \nabla \times E = -\frac{\partial B}{\partial t}
Original post by langlitz
because ∇×E=−∂B∂t \nabla \times E = -\frac{\partial B}{\partial t}


Thank you so much for the reply.

I was looking for the explanation, if any, that does not involve curl of the vector field. Is there any purely intuitive explanation?

Thank you.
Original post by tangotangopapa2
Whenever there is change in magnetic field, electric field is induced. The most common explanation that is found is: " If a closed loop circular conductor is placed in time varying magnetic field then current is induced as per the Faraday's Law and direction is given by Lenz's law. So, there must be electric field that cause charge carriers to move and this field is there even if there is no circular conductor. Furthermore, it's direction is always tangent to the loop of the conductor. Hence, circular electric field is induced due to changing magnetic field, field line being the circumference of the conducting loop."

Now that the position of the conductor is not unique, we can place conductor in one one position we get one field line and if we place same conductor in another position we get another field line following similar arguments. But as the position of the conductor is not unique, two field lines can be made to intersect, a contradiction.

Can this be explained in some other way?
If the electric field lines are concentric circles then what uniquely specifies their centres?

Thanks in advance.

[ Please try to explain this in the level of A-Level Physics i.e. with no knowledge of Maxwell's equations and vector calculus ]


If you view Faraday's law in an integral form, you "may see the reason" of why it is a closed loop and a simple closed loop is a circle.

E=∮E⃗⋅ds⃗ \mathcal{E} =\oint \vec{E} \cdot d\vec{s} ---(1)

(An induced emf is the sum—via integration—of quantities E⃗⋅ds⃗ \vec{E} \cdot d\vec{s} around a closed path, where is the electric field E⃗ \vec{E} induced by a changing magnetic flux and ds⃗ d\vec{s} is a differential length vector along the path.)

and

E=−dΦBdx \mathcal{E} =-\frac{\mathrm{d} \Phi_{B}}{\mathrm{d} x} ---(2)

∮E⃗⋅ds⃗=−dΦBdx[br] \oint \vec{E} \cdot d\vec{s} = -\frac{\mathrm{d} \Phi_{B}} {\mathrm{d} x}[br] ---(3)

When you look at equation (3), we know that the magnetic flux is computed using magnetic flux density dot cross-sectional area or B⃗⋅A⃗ \vec{B} \cdot \vec{A} and the closed loop that is chosen for the integration in the LHS must be based on the area that you have chosen on the RHS.

If the magnetic flux is computed using a circular area, then the integration is performed on the circumference of the circle.

OR

Compare equation (1) with Ampere law

∮B⃗⋅ds⃗=μ0Ienc \oint \vec{B} \cdot d\vec{s} = \mu_{0}I_{\text{enc}} ---(4)

E=∮E⃗⋅ds⃗ \mathcal{E} =\oint \vec{E} \cdot d\vec{s} ---(1)

Based on equation 4, the magnetic field of a thin infinite long conducting wire flowing with current I, are the concentric circles - I hope you are not disputing it, so based on similar argument for E=∮E⃗⋅ds⃗ \mathcal{E} =\oint \vec{E} \cdot d\vec{s} the induced electric field associated with the induced emf should be also concentric circles.

I hope this two cents help.
Original post by Eimmanuel
If you view Faraday's law in an integral form, you "may see the reason" of why it is a closed loop and a simple closed loop is a circle.

E=∮E⃗⋅ds⃗ \mathcal{E} =\oint \vec{E} \cdot d\vec{s} ---(1)

(An induced emf is the sum—via integration—of quantities E⃗⋅ds⃗ \vec{E} \cdot d\vec{s} around a closed path, where is the electric field E⃗ \vec{E} induced by a changing magnetic flux and ds⃗ d\vec{s} is a differential length vector along the path.)

and

E=−dΦBdx \mathcal{E} =-\frac{\mathrm{d} \Phi_{B}}{\mathrm{d} x} ---(2)

∮E⃗⋅ds⃗=−dΦBdx[br] \oint \vec{E} \cdot d\vec{s} = -\frac{\mathrm{d} \Phi_{B}} {\mathrm{d} x}[br] ---(3)

When you look at equation (3), we know that the magnetic flux is computed using magnetic flux density dot cross-sectional area or B⃗⋅A⃗ \vec{B} \cdot \vec{A} and the closed loop that is chosen for the integration in the LHS must be based on the area that you have chosen on the RHS.

If the magnetic flux is computed using a circular area, then the integration is performed on the circumference of the circle.

OR

Compare equation (1) with Ampere law

∮B⃗⋅ds⃗=μ0Ienc \oint \vec{B} \cdot d\vec{s} = \mu_{0}I_{\text{enc}} ---(4)

E=∮E⃗⋅ds⃗ \mathcal{E} =\oint \vec{E} \cdot d\vec{s} ---(1)

Based on equation 4, the magnetic field of a thin infinite long conducting wire flowing with current I, are the concentric circles - I hope you are not disputing it, so based on similar argument for E=∮E⃗⋅ds⃗ \mathcal{E} =\oint \vec{E} \cdot d\vec{s} the induced electric field associated with the induced emf should be also concentric circles.

I hope this two cents help.


Thank you so much. Really appreciate it. Correct me if I am wrong:

1) Electric field lines induced due to single magnetic field line (varying in strength) is concentric circles, i.e. exactly analogous to Ampere's Circuital law for long straight conductor. (The centre of all circles being the field line)
2) If there are few magnetic field lines (varying in strength) then the induced electric field lines are concentric circles with line of circle being the line of symmetry. (i.e. the centroid line of the magnetic field lines)
3) If there is time varying magnetic field with field lines going into infinitely large plane then there is no unique electric field line as there is no unique axis of symmetry. (Are electric field lines still circular in this case? What about their centres?)

Thank you :smile:
Original post by tangotangopapa2
Thank you so much. Really appreciate it. Correct me if I am wrong:

1) Electric field lines induced due to single magnetic field line (varying in strength) is concentric circles, i.e. exactly analogous to Ampere's Circuital law for long straight conductor. (The centre of all circles being the field line)
2) If there are few magnetic field lines (varying in strength) then the induced electric field lines are concentric circles with line of circle being the line of symmetry. (i.e. the centroid line of the magnetic field lines)
3) If there is time varying magnetic field with field lines going into infinitely large plane then there is no unique electric field line as there is no unique axis of symmetry. (Are electric field lines still circular in this case? What about their centres?)

Thank you :smile:


You can't really have a single magnetic field line, there is no such thing as a magnetic field line. It's just a way of visualising magnetic fields

Consider this:
you have a thin disk of radius a with a varying magnetic field passing through it with B⃗=cos(wt)z⃗\vec{B}=cos(wt)\vec{z}

∮E⃗.dl⃗=−d/dt∫B⃗.dS⃗\displaystyle \oint \vec{E}.\vec{dl}= -d/dt \int \vec{B}.\vec{dS}
∮E⃗.dl⃗=ωsin(ωt)∫z⃗.dS⃗\displaystyle \oint \vec{E}.\vec{dl}= \omega sin(\omega t) \int \vec{z}.\vec{dS}
so for r<a
∮E⃗.dl⃗=ωsin(ωt)πr2\displaystyle \oint \vec{E}.\vec{dl}= \omega sin(\omega t) \pi r^2
Eϕ2πr=ωsin(ωt)πr2 E_{\phi} 2 \pi r= \omega sin(\omega t) \pi r^2
Eϕ=ωsin(ωt)r/2 E_{\phi}= \omega sin(\omega t)r/2

and for r>a
∮E⃗.dl⃗=ωsin(ωt)πa2\displaystyle \oint \vec{E}.\vec{dl}= \omega sin(\omega t) \pi a^2
Eϕ2πr=ωsin(ωt)πa2 E_{\phi} 2 \pi r= \omega sin(\omega t) \pi a^2
Eϕ=ωsin(ωt)a2/2r E_{\phi}= \omega sin(\omega t)a^2/2r

so these are concentric circles in the plane of the disk dying off as 1/r
(edited 7 years ago)
Original post by Eimmanuel
If you view Faraday's law in an integral form, you "may see the reason" of why it is a closed loop and a simple closed loop is a circle.

E=∮E⃗⋅ds⃗ \mathcal{E} =\oint \vec{E} \cdot d\vec{s} ---(1)

(An induced emf is the sum—via integration—of quantities E⃗⋅ds⃗ \vec{E} \cdot d\vec{s} around a closed path, where is the electric field E⃗ \vec{E} induced by a changing magnetic flux and ds⃗ d\vec{s} is a differential length vector along the path.)

and

E=−dΦBdx \mathcal{E} =-\frac{\mathrm{d} \Phi_{B}}{\mathrm{d} x} ---(2)

∮E⃗⋅ds⃗=−dΦBdx[br] \oint \vec{E} \cdot d\vec{s} = -\frac{\mathrm{d} \Phi_{B}} {\mathrm{d} x}[br] ---(3)

When you look at equation (3), we know that the magnetic flux is computed using magnetic flux density dot cross-sectional area or B⃗⋅A⃗ \vec{B} \cdot \vec{A} and the closed loop that is chosen for the integration in the LHS must be based on the area that you have chosen on the RHS.

If the magnetic flux is computed using a circular area, then the integration is performed on the circumference of the circle.

OR

Compare equation (1) with Ampere law

∮B⃗⋅ds⃗=μ0Ienc \oint \vec{B} \cdot d\vec{s} = \mu_{0}I_{\text{enc}} ---(4)

E=∮E⃗⋅ds⃗ \mathcal{E} =\oint \vec{E} \cdot d\vec{s} ---(1)

Based on equation 4, the magnetic field of a thin infinite long conducting wire flowing with current I, are the concentric circles - I hope you are not disputing it, so based on similar argument for E=∮E⃗⋅ds⃗ \mathcal{E} =\oint \vec{E} \cdot d\vec{s} the induced electric field associated with the induced emf should be also concentric circles.

I hope this two cents help.


I thought we integrate around a closed loop because the electric field is circular, not the other way around (ie the electric field isn't circular, because that's how we chose to integrate it.)
Also, where is the centre of the circles of the electric field? you could put a circular conductor anywhere but surely if there's nothing there, where will be centre be?

@tangotangopapa2
(edited 7 years ago)
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Avatar for mik1a
mik1a
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One possible source of confusion is that you're not considering physically-possible magnetic fields.

Remember that the magnetic field is a solenoidal field, i.e. it has the form "curl(B) = something" (Ampere's law), and that it has no divergence (Gauss's law for magnetism). That means the magnetic field lines must form closed loops. Your thought experiments are leaving you confused because none of them use such a field (e.g. single field line, or infinite and parallel field lines). These fields would indeed cause some problems in electromagnetism!

When the magnetic field lines have closed loops, it is much easier to visualise why the electric field that is generated is in concentric circles around a well-defined axis.

By the way, that definition at the start of this thread is slightly ropey. Faraday's law connects the induced electric field to the change in magnetic field, and says nothing about current. If a conductor of any shape happens to exist inside that induced electric field, then you'll probably get currents flowing, but not necessarily in circular paths.
(edited 7 years ago)
Original post by tangotangopapa2
Thank you so much. Really appreciate it. Correct me if I am wrong:

1) Electric field lines induced due to single magnetic field line (varying in strength) is concentric circles, i.e. exactly analogous to Ampere's Circuital law for long straight conductor. (The centre of all circles being the field line)
2) If there are few magnetic field lines (varying in strength) then the induced electric field lines are concentric circles with line of circle being the line of symmetry. (i.e. the centroid line of the magnetic field lines)
3) If there is time varying magnetic field with field lines going into infinitely large plane then there is no unique electric field line as there is no unique axis of symmetry. (Are electric field lines still circular in this case? What about their centres?)

Thank you :smile:


I think langlitz has explained the what is good and bad about your understanding.

I would like to add a few points to your point 1 and 3.
When field (vector field to be more precise in this example) is considered in Physics, it means a region of space ..... Can we have a magnetic field confined to a mathematical line region? I am not sure.

If the time-varying magnetic field extends into an infinite plane, the symmetry should be where you place your origin. In this case, (to me) one need to choose a coordinate system to make sense of it. The induced electric field should be still in the ϕ⃗ \vec{\phi} direction if the time-varying magnetic field point in the z⃗ \vec{z} direction. The induced electric field can be found using the differential form (given by langlitz) of Faraday's law ∇×E⃗=−∂B⃗∂t \nabla \times \vec{E}= -\frac{\partial \vec{B}}{\partial t} .

https://www.youtube.com/watch?v=OL3uBX68-CY

OR
Look at magnetostatics in the differential form and Biot-Savart law:

∇×B⃗=μ0J⃗ \nabla \times \vec{B}= \mu_{0} \vec{J} and ∇⋅B⃗=0 \nabla \cdot \vec{B}= 0

The solution of the magnetic field is Biot-Savart law.

B⃗=μ04π∫V(J⃗dV×r′⃗)∣r′⃗∣3 \vec{B}= \frac{\mu_{0}}{4 \pi} \int_{V} \frac{(\vec{J} dV \times \vec{r'})} { |\vec{r'}|^3}

Draw the analog

∇×E⃗=−∂B⃗∂t \nabla \times \vec{E}= -\frac{\partial \vec{B}}{\partial t} and ∇⋅E⃗=0 \nabla \cdot \vec{E}= 0

The solution of electric field is
E⃗=−14π∫V(∂B⃗∂t)×r⃗′∣r⃗′∣3dV \vec{E}= -\frac{1}{4 \pi}\int_V \frac{\left ( \frac{\partial \vec{B}}{\partial t} \right ) \times \vec{r}'}{\left | \vec{r}' \right |^3} dV
Original post by lawlieto
I thought we integrate around a closed loop because the electric field is circular, not the other way around (ie the electric field isn't circular, because that's how we chose to integrate it.)

@tangotangopapa2


A circular loop is usually chosen for easy computation - the magnitude of the induced electric field is constant. See langlitz example. Any closed loop can be chosen for the integration for the left-hand side as long as this closed loop is used for the integration for the right-hand side.

∮E⃗⋅dl⃗=−∂∂t∫B⃗(t)⋅dS⃗ \displaystyle \oint \vec{E} \cdot d\vec{l}= -\frac{\partial }{\partial t} \int \vec{B}(t) \cdot d\vec{S}

Original post by lawlieto

Also, where is the centre of the circles of the electric field? you could put a circular conductor anywhere but surely if there's nothing there, where will be centre be? @tangotangopapa2


I am sorry that I don't quite understand what do you mean.

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