# A current through a wire produces a magnetic field around it. Is the reverse possible?

If somehow a magnetic field around a wire can be made to exist that is identical to the magnetic field produced when a current passes through the wire, will a current be produced in the wire?
A thought experiment: if many very very small bar magnets are placed around a wire making concentric wells around it such that the "field lines" encircle the wire and the radial dependence of the field is made to follow Biot-Savart law, will it produce a current in the wire?

If this is incorrect, what effects will be observed if the thought experiment is performed?

If there is a current density in the wire, there needs to be an electric field. An electric field can be generated by a changing magnetic flux. This is Faraday's law.

So, in your thought experiment, while you are assembling your magnetic field there will be a transient current induced that produces an opposing magnetic field. Once you have established the magnetic field and it is static, there will be no current.

The reason for the asymmetry is that for there to be a static current you need to apply an electric field (an EMF) to the wire. This electric field is not present in your thought experiment. A static magnetic field does not exert a force on the electrons in the wire.

However, a different way to look at this is how exactly are you going to simulate the magnetic field due to a steady current in a wire? I would suggest that the only way you can actually do that accurately is to... have a steady current going through the wire. I don't think that any arrangement of permanent magnets can give you a curl free field outside the wire and a field with constant curl inside the wire. But this is a point of detail.

• Is it right to say current produces magnetic field? Because it sounds like current comes first which produces the field. The current and the field occurs simultaneously right? – Tea is life Apr 12 '15 at 15:23
• @scisyhp Yes. That is the meaning of my final paragraph, and Alfred's answer. – Rob Jeffries Apr 12 '15 at 15:53

If somehow a magnetic field around a wire can be made to exist identical to the magnetic field produced when a current passes through the wire, will current be produced in the wire?

If the magnetic field is identical, there is a current through the wire. Put another way, if there is no current through the wire, the magnetic field is not (nor can be made to be) identical.

Recall from Maxwell's equations, in the magnetostatic case (steady current),

$$\nabla \times \mathbf B = \mu_0 \mathbf J$$

where $\mathbf J$ is the current density. This leads to the well known result that the line integral of the magnetic field around a closed path encircling the wire is non-zero if and only if there is a current through the wire.

Thus, if there is no current through the wire, the closed line integral is zero while, if there is current through the wire, the closed line integral is non-zero.

In summary, the magnetic field cannot be made identical unless the current through the wire is identical.

• Is it right to say current produces magnetic field? Because it sounds like current comes first which produces the field. The current and the field occurs simultaneously right? – Tea is life Apr 12 '15 at 15:23

Yes and no.

On the one hand of you have a magnetic field that circulates just like it does around a steady current and the current is anything other than that steady current then the electric field will change:

$$\vec \nabla \times \vec B = \mu_0\vec J +\mu_0\epsilon_0\frac{\partial \vec E}{\partial t}$$ is the same as $$\frac{\partial \vec E}{\partial t}=\frac{1}{\mu_0\epsilon_0}\left(\vec \nabla \times \vec B -\mu_0\vec J\right)$$ so you can think of the imbalance as being associated with a changing electric field. But that electric field tends to make positive charges go that way and make negative charges go the opposite way. Which is exactly what you need to increase the current more to be what you want.

So if there isn't a wire there then the electric field just increases if there is a wire and the current is too large the electric field still changes but in a way to damp the current. And if there is too little current the electric field increases in a way to increase the current.

So having just the right current is the static solution the one that balances perfectly and let's things not change.

Now, since other answers claimed this wasn't possible I'd like to mention a specific way to create these fields.

In a wire you need a certain electric field to maintain a steady current dispute the fact that the wire is full of things inside the wire that the moving electrons bump into. You can change the current in a simple way without changing the magnetic fields that circulate around the wire.

Simply grab the wire and pull it in the opposite direction as the current. This moves a net positive charge in the right direction to decrease the current and if you keep pulling it down at a steady speed you now need a stronger electric field to have the previous current flow down the wire but that is what happens the electric field starts to increase until the right current (mobile electrons moving the same direction as you pulling the wire plus some more) is there to match the magnetic field.

You can always read Maxwell's equation as just telling the electromagnetic field how to change. The curl of the electric field telling the magnetic how to change and the the curl of the magnetic field telling the electric field how to change if it isn't absolutely perfectly matched by the current.

Yes ofcourse, it is a very famous phenomena named "Electromagnetic induction". and there is a famous law there named "Faraday's law of Electromagnetic induction".

and it doesn't work like the way that you explained. it's like this:

"if you have a time-varying magnetic field around the wire, it will produce a current inside the wire"

and it can be explained mathematically this way:

$$\mathscr E=-\frac{d\Phi_B}{dt}$$

actually Faraday's theory explained three ways that you can create an inducted current.You can read more here.

• I would request you to read the question again – Tea is life Apr 11 '15 at 19:13
• @scisyhp Yes I've read your question and still that's my answer. see the magnetic field should change in time to create a current in the wire. if you create a magnetic field around the wire with a constant amount it will not create any currents. – Mobin Apr 11 '15 at 19:16
• So thats what I am asking why isn't the process reversible – Tea is life Apr 11 '15 at 19:19
• @scisyhp Because there is a reason for happening of inducted current. when the magnetic field around the wire is getting lower, it will creates an opposite field in the wire so the wire would have a current that can cancel the changing of our main field. and if the main field gets higher over times, a current will be created due to the inducted field inside the wire. but if the magnetic field is constant it will not create any "Electric" field. it's a magnetic field, it can not make electrons move. did you get it? – Mobin Apr 11 '15 at 19:24
• What you are saying is all what you are taught. I am asking is a hypothetical one. Its like but not same as, if I somehow make a diverging electric field from a point will it create a charge at that point. – Tea is life Apr 11 '15 at 19:40

The answer to the thought experiment is No. Magnetic fields are produced only by charges in motion and electric current in a wire is an example of charges on electrons in motion. Current in a conductor can be produced by a magnetic field in motion and, as mentioned, a generator uses this principle. These two relations between magnetic and electric field both depend on motion without which neither field can influence the other. These relations were formulated mathematically by Maxwell around 1862 who thereby achieved the first successful unified theory; Maxwell's equations use mathematical calculus functions of integration and differentiation in 3 dimensions to quantify exactly the motion. http://en.wikipedia.org/wiki/Maxwell%27s_equations You should be warned that there is a great deal of disinformation on the web in various claims of "free energy" from systems of permanent magnets, and of supposed isolated magnetic poles, both of which violate the mainstream proven principles.

• "Magnetic fields are produced only by charges in motion...". Not true. – Rob Jeffries Apr 11 '15 at 22:41