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

In summary: Now I understand. It's useful to take an extra hard look at the formulas when trying to use them. Thanks :)
  • #1
papercace
13
4

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?
 
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  • #2
papercace said:

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?
 
  • #3
ehild said:
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?
 
  • #4
papercace said:
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.
 
  • #5
ehild said:
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 :)
 

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

1. What is Ampère's law?

Ampère's law is a fundamental law of electromagnetism that relates the magnetic field around a closed loop to the electric current passing through the loop. It was first formulated by French physicist André-Marie Ampère in the early 19th century.

2. How is Ampère's law used in calculations?

Ampère's law can be used to calculate the magnetic field due to a current-carrying wire or a solenoid. It can also be used to solve for the current flowing through a closed loop if the magnetic field and loop dimensions are known. It is an important tool in many applications, such as designing electromagnets and understanding the behavior of electric motors.

3. What is the mathematical form of Ampère's law?

Ampère's law can be written in its integral form as ∮CB·dl = μ0I, where ∮C represents the line integral around a closed loop, B is the magnetic field, dl is a small element of the loop, μ0 is the permeability of free space, and I is the current passing through the loop. In its differential form, it is written as ∇×B = μ0J, where ∇× represents the curl operator, B is the magnetic field, μ0 is the permeability of free space, and J is the current density.

4. What are the limitations of Ampère's law?

Ampère's law is only valid in situations where the electric current is steady and the magnetic field is constant. It also assumes that there are no changing electric fields present. In cases where these conditions are not met, the more general form of Maxwell's equations must be used.

5. How does Ampère's law relate to other laws of electromagnetism?

Ampère's law is one of the four Maxwell's equations that describe the relationship between electric and magnetic fields. It is closely related to Gauss's law for magnetism and Faraday's law of induction. Together, these laws form the foundation of classical electromagnetism and are essential for understanding the behavior of electric and magnetic fields.

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