Non symmetric case of Ampere's law

[ubergewehr273]

When we use Ampere's law, the most basic case that of an infinite current carrying wire is taken whose magnetic field is evaluated at a distance r from the wire. However there's nothing wrong in using the law for non symmetric scenarios. If this is the case how do you explain the B field at a distance r from a "finite" current carrying wire using Ampere's law ?

 

[Dale]

You cannot have a finite current carrying wire. It would violate the continuity equation at the ends.

You could have a finite current carrying loop. Perhaps that is what you meant.

 

[ubergewehr273]

You cannot have a finite current carrying wire. It would violate the continuity equation at the ends.

But in my senior class, we did derivations based on finding B field at a distance r from a finite section of a wire whose ends subtend angles $\theta_{1}$ and $\theta_{2}$ at the point. Is it correct to find the B field due to a finite portion of an infinite current carrying wire?

 

[ZapperZ]

But in my senior class, we did derivations based on finding B field at a distance r from a finite section of a wire whose ends subtend angles $\theta_{1}$ and $\theta_{2}$ at the point. Is it correct to find the B field due to a finite portion of an infinite current carrying wire?

But what did you use? Ampere's law, or Biot-Savart Law?

Zz.

 

[ubergewehr273]

But what did you use? Ampere's law, or Biot-Savart Law?

Zz.

We used Biot savart law, but how do I do it using Ampere's law?

 

[ZapperZ]

We used Biot savart law, but how do I do it using Ampere's law?

Not easily. You need to learn numerical analysis, because you have to solve it numerically.

Again, refer back to Gauss's law. Did you ever solve for the E-field of a finite line charge using Gauss's law?

The reason why Gauss's law and Ampere's law are used in highly-symmetric situations is because you have to do very minimal calculus due to the geometry of the field. Otherwise, you do not have an analytical solution and you will have to solve it numerically.

If you continue in physics and end up doing advanced E&M using something like the Jackson's text, you WILL solve not only non-symmetric sources, but also solving for "off-axis" locations. Try solving the field for a circular loop of current, but NOT just along the axis of the loop. It is not easy, and you'll end up with an infinite series.

So my advice at the moment is, enjoy and appreciate these simpler situations while you can. It gets real ugly real soon enough.

Zz.

 

[ubergewehr273]

Not easily. You need to learn numerical analysis, because you have to solve it numerically.

Again, refer back to Gauss's law. Did you ever solve for the E-field of a finite line charge using Gauss's law?

The reason why Gauss's law and Ampere's law are used in highly-symmetric situations is because you have to do very minimal calculus due to the geometry of the field. Otherwise, you do not have an analytical solution and you will have to solve it numerically.

If you continue in physics and end up doing advanced E&M using something like the Jackson's text, you WILL solve not only non-symmetric sources, but also solving for "off-axis" locations. Try solving the field for a circular loop of current, but NOT just along the axis of the loop. It is not easy, and you'll end up with an infinite series.

So my advice at the moment is, enjoy and appreciate these simpler situations while you can. It gets real ugly real soon enough.

Zz.

Thanks for the advice but just out of curiosity does finding for such a non symmetric case involve the second term of ampere's law ? Because well let's just say that charge accumulation takes place at the ends of the finite wire and this would induce a changing electric field till $\vec E$ becomes zero. Is this right?

 

[ZapperZ]

Thanks for the advice but just out of curiosity does finding for such a non symmetric case involve the second term of ampere's law ? Because well let's just say that charge accumulation takes place at the ends of the finite wire and this would induce a changing electric field till $\vec E$ becomes zero. Is this right?

Actually, no. That is the displacement current, and it has more to do with the transient effects in space when a capacitor is charging or discharging. If you are curious on why that term has to be there, read ahead on the topic of displacement current.

Zz.

 

[ubergewehr273]

Actually, no. That is the displacement current, and it has more to do with the transient effects in space when a capacitor is charging or discharging. If you are curious on why that term has to be there, read ahead on the topic of displacement current.

Zz.

Isn't the case that I described above a scenario of transient effect ?

 

[ZapperZ]

Isn't the case that I described above a scenario of transient effect ?

Not from the way you described it. While it is due to a changing E-field, it has more to do with what happens at a break in the circuit during the transient period, i.e. if the surface bounded by the Amperian loop isn't crossed by the real current in the wire. The changing E-field can eventually be related to this "displacement current" if you wish, but the displacement current is the extra term in Ampere's law.

Zz.

 

[Dale]

We used Biot savart law, but how do I do it using Ampere's law?

In addition to what @ZapperZ said, it actually cannot be done with Ampere’s law. The violation of the continuity equation causes a serious problem.

Recall that you define a loop, then define a surface whose boundary is the loop, and that it can be any surface with the boundary. If you have a finite wire and a loop around the wire, then you can choose a surface crossing the wire or a surface going around the wire. Both surfaces are valid, but the current is different.

This can be fixed by having charge accumulate at the ends and create a dipole E field, and adding the displacement term to Amperes law.

 

[ubergewehr273]

In addition to what @ZapperZ said, it actually cannot be done with Ampere’s law. The violation of the continuity equation causes a serious problem.

While calculating the B field using biot savart law, we never considered violation of the continuity equation but yet the approach was right. Or was it?

or a surface going around the wire. Both surfaces are valid, but the current is different.

Could you elaborate on the point of surface going around the wire. Also how is the current different in both cases ?

 

[Dale]

While calculating the B field using biot savart law, we never considered violation of the continuity equation but yet the approach was right. Or was it?

The Biot Savart law doesn’t have the same mathematical issue; the results are mathematically well defined but non-physical because of the violation of the continuity equation.

Could you elaborate on the point of surface going around the wire. Also how is the current different in both cases ?

For example

The current is different through the two surfaces, $S_1$ and $S_2$

 

[ubergewehr273]

The Biot Savart law doesn’t have the same mathematical issue; the results are mathematically well defined but non-physical because of the violation of the continuity equation.

For exampleView attachment 218563

The current is different through the two surfaces, $S_1$ and $S_2$

Thanks for the details.
P.S. Which textbook is the image from?

 

[ubergewehr273]

The Biot Savart law doesn’t have the same mathematical issue; the results are mathematically well defined but non-physical because of the violation of the continuity equation.

Why aren't the results of Ampere's law "physical" and otherwise for Biot savart law ? Isn't the former derived from the latter ?

 

[Dale]

Why aren't the results of Ampere's law "physical" and otherwise for Biot savart law ?

They are non-physical because you are modeling a non-physical scenario. You cannot have a current which just stops, that is non physical. So any model which takes that non physical scenario as an input will produce non physical output.

 

[ubergewehr273]

They are non-physical because you are modeling a non-physical scenario. You cannot have a current which just stops, that is non physical. So any model which takes that non physical scenario as an input will produce non physical output.

Mathematically speaking shouldn't Ampere's law provide the same result as Biot savart law?

 

[Dale]

Mathematically speaking shouldn't Ampere's law provide the same result as Biot savart law?

They are not mathematically identical, so no. Under specific conditions they are equivalent, but those circumstances do not include this non-physical scenario.