# Induced EMF due to changing current in straight wire

I see a lot of problems on constant current inducing a current in another wire due to its associated magnetic field, but not so much a single wire's induced electric field. If the current along an infinitely long wire is flowing out of the page and inducing a counter-clockwise spinning magnetic field, what happens when the current continues to increase in magnitude with time in the same direction? Am I correct in thinking the changing magnetic field induces an emf pointing into the page? Or generates what is considered back/counter- emf outside of the wire?

When I go to back solve for the strength of the electric field in the x, y, and z directions after generically taking the curl of the electric field's components and setting it equal to the negative of the partial derivative with respect to time of a circular magnetic field (arbitrarily made the magnetic field change linearly with time to make the math easier), I find myself with a differential equation that I sort of just simplified with assumptions that I don't think are valid. Assuming the E-fields X and Y components are zero ensures the cross product on the left hand side ∇ x E still remains a valid solution to the partial derivative -dB/dt on the right hand side of the equation. However this is sort of just what I'm hoping will happen and it's not exactly a valid method of solving differential equations last time I checked haha.

Anyone want to shed some light on how they've done this problem in the past? I guess I am unsure whether I am just missing some first principle physics or if the lack of Z component in the magnetic field allows some assumptions about the X and Y direction of the of the E-field to be formed. Or if I just have a lot of dif eq to do... :)
Thanks in advance for any help!

## Answers and Replies

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Also I thought I'd mention that it may be impossible to have current flowing in an infinitely long wire and a short straight wire may be the only possible scenario to solve for. Either way I am just trying to consider a magnetic field circling only in the X and Y directions, where Z axis points out of the page.

You could use Biot Savart, or if ##dB/dt## is constant, use Faraday's law in polar coordinates to obtain E.

You could use Biot Savart, or if ##dB/dt## is constant, use Faraday's law in polar coordinates to obtain E.
Yeah I still end up with basically the same problem/ a PDE I can't solve, I need boundary conditions to solve for the result of Faraday's law in cylindrical coordinates (can't use polar coordinates in 3D correct?) or Cartesian coordinates as I originally did in the beginning. This is the result in cylindrical coordinates:
∂Er /∂z - ∂Ez /∂r = -μo⋅I / (2⋅π⋅r)

This is where I typically run into problems, any thoughts?

The question is really simple you are thinking it much than its required solve within the wire and you will get counter emf and differential of it with length will give you E vector. I am giving one example to show how to work for these problems.

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If your question is different than this wirte the question properly and i will solve that and reply it.

The question is really simple you are thinking it much than its required solve within the wire and you will get counter emf and differential of it with length will give you E vector. I am giving one example to show how to work for these problems.
I am looking for the electric field outside of the wire, not the emf inside of the wire. If current is changing ∂I/∂t ≠ 0 then there is a changing magnetic field outside the wire that must produce some non electrostatic electric field outside of the wire also.

Let me restate the question using more math and a coordinate system to avoid language barriers and keep our math similar:

A very long insulated wire oriented along the Z axis of a cylindrical coordinate system is carrying current that is defined by the following function: I(t) = t
(current increases linearly with time in the positive Z axis direction, 1 amp per second ). If r is the distance from the Z axis, what is the resulting electric field outside of the wire assuming the current could increase indefinitely and ignoring any effect that the end of the wires may have?

My attempt at the solution thus far:
since I(t)= t

B = (μo ⋅ t / (2⋅π⋅r) )

faraday's law states that:

∂B/∂t = (μo / (2⋅π⋅r) )

thus after taking the curl of a generic electric field E in cylindrical coordinates Faraday's law becomes:
∂Er /∂z - ∂Ez /∂r = -μo / (2⋅π⋅r)

You will notice only the
direction remains which is because the magnetic field is only changing in one direction, the other parts of ∇ x E dissappear because they are equal to zero. Maybe these parts are useful for solving this differential equation? I don't know but this is what I was hoping to receive some help on.

You're implying that the current increases linearly with time. ##I=I_0+kt##, where ##k## is in units of amperes per second.

##B_\phi = \frac{\mu_0 (I_0+kt) }{2 \pi r}##, the magnetic field is only in the ##\hat{\phi}## direction.

I may have mislead you above. You need to apply Ampere's law in the absence of current.

##\nabla \times B = \mu_0 \epsilon_0 \frac{\partial E}{\partial t}##

Knowing that the magnetic field varies only in the ##r## direction, and using the curl in cylindrical coordinates,

##\frac{1}{r} \frac{\partial (r B_{\phi})}{\partial r} = \mu_0 \epsilon_0 \frac{\partial E_z}{\partial t}##.

I think that's about right. Is this enough that you can complete the rest? You need to plug ##B_\phi## into the last equation. A lot of stuff cancels. Take the derivative with respect to ##r## on the left hand side then integrate over time.

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TheJfactors
You're implying that the current increases linearly with time. ##I=I_0+kt##, where ##k## is in units of amperes per second.

##B_\phi = \frac{\mu_0 (I_0+kt) }{2 \pi r}##, the magnetic field is only in the ##\hat{\phi}## direction.

I may have mislead you above. You need to apply Ampere's law in the absence of current.

##\nabla \times B = \mu_0 \epsilon_0 \frac{\partial E}{\partial t}##

Knowing that the magnetic field varies only in the ##r## direction, and using the curl in cylindrical coordinates,

##\frac{1}{r} \frac{\partial (r B_{\phi})}{\partial r} = \mu_0 \epsilon_0 \frac{\partial E_z}{\partial t}##.

I think that's about right. Is this enough that you can complete the rest? You need to plug ##B_\phi## into the last equation. A lot of stuff cancels. Take the derivative with respect to ##r## on the left hand side then integrate over time.
Interesting method, didn't expect to use polarization current.

So the answer is zero?
Because if you multiply the magnetic field by r there is no longer an r component in the equation Bϕ0(I0+kt)/(2π) , and the derivative with respect to r is just zero then? Or I may just be a bit rusty on my calculus. Also how do you switch from the derivative of the electric field with respect to time to the electric field only in the z direction (I don't quite understand how that subscript just appears)? Where in the math does that occur?

Also is there any reason why Faraday's law doesn't work for this?

Zero seems like a somewhat non intuitive answer, when I visually try and combine differential area segments with changing magnetic field strength it seems that there should be a resultant electric field created.

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So the answer is zero?
No. scrap it all and start over. You can't use Bϕ0(I0+kt)/(2π). This is ampere's law where, implicitly, the change in the electric field is set to zero, so it gives a goofy result.

No. scrap it all and start over. You can't use Bϕ0(I0+kt)/(2π). This is ampere's law where, implicitly, the change in the electric field is set to zero, so it gives a goofy result.
I appreciate your efforts, any new direction you think might be worth taking? The Faraday's law approach still exists but there needs to be a few boundary conditions if the differential equation is to be solved.

I took a second look at it.

Biot-Savat says is how to solve for the magnetic field due to current contributions located at various positions, but it is only valid for static fields. For dynamic currents and charge, I used Jefimenk's equations to find the retarded vector potential field.

##A(r,t) = \hat{z} \frac{\mu_0}{4 \pi} \int_{-\infty}^{\infty} \frac{I_r}{d}dz##

##I_r## is the retarded current. It is the current sensed at position ## (r,t) ## at a distance ##d## away, delayed by the speed of light.

##I_r=k(t-d/c)##

To simplify, I assumed the unretarded current was zero at ##t=0##.

##A(r,t)=\frac{\mu_0 k}{4 \pi}\frac{2ct}{r}## [edited]

The magnetic field strength, ##B## is the curl of ##A##.

I get ##B = \hat{\phi} \frac{-\mu_0 k t }{2 \pi r^2}##.

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I took a second look at it.

Biot-Savat says is how to solve for the magnetic field due to current contributions located at various positions, but it is only valid for static fields. For dynamic currents and charge, I used Jefimenk's equations to find the retarded vector potential field.

##A(r,t) = \hat{z} \frac{\mu_0}{4 \pi} \int_{-\infty}^{\infty} \frac{I_r}{d}dz##

##I_r## is the retarded current. It is the current sensed at position ## (r,t) ## at a distance ##d## away, delayed by the speed of light.

##I_r=k(t-d/c)##

To simplify, I assumed the unretarded current was zero at ##t=0##.

##A(r,t)=\frac{\mu_0 k}{4 \pi}\frac{2ct}{r}## [edited]

The magnetic field strength, ##B## is the curl of ##A##.

I get ##B = \hat{\phi} \frac{-\mu_0 k t }{2 \pi r^2}##.
Ok I'm going to do a lot of reading cause it is clear to me that I have a lot to learn haha, I will let you know if I find anything strange or different from what you posted. But in the mean time if the B-feild is as you have written it, are you plugging that back into the equation you worked out at the end of post 8? The resulting E-field seems to be much more intuitive but I'm not entirely sure what interval of time I am supposed to be integrating over.

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But in the mean time if the B-feild is as you have written it, are you plugging that back into the equation you worked out at the end of post 8?
This will only get us ##\partial E / \partial t## so it's not enough. This method is really ugly mathematics. It just works. To find E, we need to find the retarded electric potential, ##\phi## as well as the vector potential, I just haven't done it. I'm learning too. It's a good problem.

Here are a few web sites.

From the land of lidigation: http://isites.harvard.edu/fs/docs/icb.topic970148.files/Spherical_coord.pdf

For Jefimenko's equation there is http://www.physicspages.com/2014/11/21/retarded-potentials/

For the related solution for a straight wire where the current is zero for t<0, and I=kt for t>0, the last web site links to http://www.physicspages.com/2014/11/21/retarded-potentials/, which is good for seeing how to apply the electric potential and vector potential developed in the parent web page, though I think there is an error in the upper limit of integration, but this shouldn't effect your particular problem setup.

If you have a copy of Jackson's Classical Electrodynamics, you could read a little about this.

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This will only get us ##\partial E / \partial t## so it's not enough. This method is really ugly mathematics. It just works. To find E, we need to find the retarded electric potential, ##\phi## as well as the vector potential, I just haven't done it. I'm learning too. It's a good problem.

Here are a few web sites.

From the land of lidigation: http://isites.harvard.edu/fs/docs/icb.topic970148.files/Spherical_coord.pdf

For Jefimenko's equation there is http://www.physicspages.com/2014/11/21/retarded-potentials/

For the related solution for a straight wire where the current is zero for t<0, and I=kt for t>0, the last web site links to http://www.physicspages.com/2014/11/21/retarded-potentials/, which is good for seeing how to apply the electric potential and vector potential developed in the parent web page, though I think there is an error in the upper limit of integration, but this shouldn't effect your particular problem setup.

If you have a copy of Jackson's Classical Electrodynamics, you could read a little about this.
Are you certain about Ir=k(t−d/c) ?
I feel like it should be Ir=k⋅t−d/c otherwise the delay scales with the current ramp rate which doesn't have to do anything with the position it is being sampled from? Idk, this is far from my forte haha. Not entirely sure how to actually evaluate the triple integral correctly for J(r',t) due to lack of definitions outside of recognizing that its basically the double integral for current density to current with one integral left over from the only remaining direction along the direction of current flow as you wrote it.

Think you lost me on the evaluation of the last remaining integral too. bright side is the E field seems rather close to being found after we do the same integral for the electric potential also assumidly taking on some sort of delay

So I guess now some clarifying questions arise. Namely if u= c*t/s in the previous links i posted where s is the distance from the z axis, does the E field continue to increase with time as long as K (the current ramp rate) is the same? Also what happens in discrete cases when the current goes from say 0 to 100 amps in .001 seconds and then stays at 100 amps after .001 seconds? Does the field just disappear causing circular magnetic field lines to be produced which result in oscillations in current?

Any idea how to figure out the Max K a wire can support? I don't actually know what the physical constraints are when a wire is shorted out for instance that would limit the current ramp rate.

Sorry for the long pause. I've been spending the time double checking my work.

So, after all the false starts, in the problem as you gave it in post 1, I = kt, the integral unfortunately diverges, which means the field values go to infinity. This can be traced back to the d term in the denominator. As a check, I tried it for s/d = cos(theta) and wrote the integral to sum over theta. Too bad, but it's been educational.

For the case, where I=0 for t < s/c, and the current is switched on at t=0 and ramps at rate k, the integral is finite, and the E and B field values are found here, equations 24 and 25. s/c is the time delay for the current change at the closest part of the wire to be seen at a distance, s from the wire.

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Sorry for the long pause. I've been spending the time double checking my work.

So, after all the false starts, in the problem as you gave it in post 1, I = kt, the integral unfortunately diverges, which means the field values go to infinity. This can be traced back to the d term in the denominator. As a check, I tried it for s/d = cos(theta) and wrote the integral to sum over theta. Too bad, but it's been educational.

For the case, where I=0 for t < s/c, and the current is switched on at t=0 and ramps at rate k, the integral is finite, and the E and B field values are found here, equations 24 and 25. s/c is the time delay for the current change at the closest part of the wire to be seen at a distance, s from the wire.
I think equation 24 and 25 are for a current impulse function, I believe 10 and 11 are for the ramp function. Equations 24 and 25 seem to make sense because the E and B field decay as a function of time as one would expect. From equation 10 and 11 it seems that the E field grows in strength as a function of time.

The interesting bit is that the field disappears at some radius, that still sort of messes with my head lol

You can get a quick picture of what the curves look like by plugging them into the wolfram alpha website. For eq 10 for instance "-ln(t + sqrt(t^2-1))", where r, c and other terms are normalized to 1.

You can get a quick picture of what the curves look like by plugging them into the wolfram alpha website. For eq 10 for instance "-ln(t + sqrt(t^2-1))", where r, c and other terms are normalized to 1.
Can we calculate self inductance from this by anychance? I am under the impression if the E field exists pointing in the opposite direction of the current and a high rate of current change produces a proportionally larger E field then this E field must oppose movement of charges such that there is a fundamental limit to the rate at which current can increase. Essentially if there is a max current limit then increasing the ramp rate for the current (or decreasing t) decreases the radius where the E field disappears at however the E field gets stronger within that radius which should oppose current flow an even greater amount? Wondering if this where the inductance in a straight wire comes from that must be accounted for in some electrical engineering applications.
I know there are straight wire inductance calculators like this:
http://www.daycounter.com/Calculators/Straight-Wire-Inductor-Calculator.phtml
But I'm not sure if that holds for all voltages, currents, and thus induced E-fields.
The most interesting case that comes to mind is the case where the electric field ends up being entirely inside the radius of the wire. Or possibly even a stranger case would be where the electric field only occurs at a radius even smaller than the wires radius. Seems like cases like these might result in interesting information if they are even possible. Though I'm not sure if there's enough information in the jefimenko E-field equation alone to find some sort of physical limit.

Well, inductance can be found from v(t) = L di/dt. In the infinite straight wire problem, we've been pretending there isn't a voltage drop along the wire. We would have to include an electric field in the z direction to drive the current, I think.

Perhaps you might look at sinusoidal current flow in the wire, but it might also be divergent. The is some indication this could be the case, because Griffiths does't cover it.