Correct usage of Ampère's law for calculating B-field outside parallel wires?

[papercace]

Homework Statement

The problem is basically about tow infinite parallel wires separated by a distance $d$ with equally strong but opposite currents. You have to calculate the B-field outside the wires (not the field in between them).

Homework Equations

Ampères law:
$\oint \mathbf B \cdot d\mathbf l = \mu_0 I_{enc}$

B-field outside one infinite wire with current I:
$B=\frac{\mu_0 I}{2\pi s}$
where $s$ is the distance from the wire.

The Attempt at a Solution

Using the second formula on each wire and adding the resulting fields, we get the right answer, which obviously is bigger than zero. If we instead use Ampère's law, where we enclose both wires by an amperian circular loop, we get that the enclosed current is zero, since they run in opposite directions, which in turn makes the B-field equal to zero, which is obviously the wrong answer. In what way am I using Ampères law wrongly?

 

[ehild]

Homework Statement

The problem is basically about tow infinite parallel wires separated by a distance $d$ with equally strong but opposite currents. You have to calculate the B-field outside the wires (not the field in between them).

Homework Equations

Ampères law:
$\oint \mathbf B \cdot d\mathbf l = \mu_0 I_{enc}$

B-field outside one infinite wire with current I:
$B=\frac{\mu_0 I}{2\pi s}$
where $s$ is the distance from the wire.

The Attempt at a Solution

Using the second formula on each wire and adding the resulting fields, we get the right answer, which obviously is bigger than zero. If we instead use Ampère's law, where we enclose both wires by an amperian circular loop, we get that the enclosed current is zero, since they run in opposite directions, which in turn makes the B-field equal to zero, which is obviously the wrong answer. In what way am I using Ampères law wrongly?

Yes, it is wrong. What does Ampère's law exactly state?

 

[papercace]

Yes, it is wrong. What does Ampère's law exactly state?

It states that the sum of the strength of the B-field in a tangential direction to the loop is proportional to the current enclosed by the loop.

If I may guess, we can only solve the integral analytically if the B-field in the tangential direction is assumed to be a constant over the entire loop, which is not the case with two parallel wires. Am I on going on the right track?

 

[ehild]

It states that the sum of the strength of the B-field in a tangential direction to the loop is proportional to the current enclosed by the loop.

If I may guess, we can only solve the integral analytically if the B-field in the tangential direction is assumed to be a constant over the entire loop, which is not the case with two parallel wires. Am I on going on the right track?

Yes. The tangential component of the B field is not constant along a loop enclosing both wires.

 

[papercace]

Yes. The tangential component of the B field is not constant along a loop enclosing both wires.

Now I understand. It's useful to take an extra hard look at the formulas when trying to use them. Thanks :)