Magnetic vector potential for antiparallel currents

In summary, the conversation discusses using Ampere's law to find the magnetic field and vector potential for two infinitely long wires carrying opposite currents. The potential vector is found by integrating the magnetic field expression with respect to two different distances. The conversation also mentions using the superposition principle to find the expression for a single wire and applying it to the current scenario.
  • #1
teastation
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Two infinitely long wires separated by distance d. Currents: I1 = -I2. Find potential vector as a function of r1 and r2 at a point P (r1 and r2 distances to P from wire one and wire two).
Del cross A= B
B = (mu I)/(2pi r)



Using Ampere's, I get an expression for the magnetic field that involves two different distances, r1and r2. I see that integrating this expression with respect to distance will give me the vector potential. But with two distances to take into account, I don't know how to solve this.
 
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  • #2
Oh I'm using cylindrical coordinates. With the wires oriented along the z axis, the only term that survives del cross A is the azimuthal component.
-partial dA/ds = (mu I)/2pi [(1/r1) - (1/r2)] in the phi direction.
 
  • #3
Hello.

Can you find an expression for A for a single infinitely long straight wire carrying current I ? If so, then the superposition principle will get you the answer fairly easily.
 
  • #4
Thanks TSny. Duh... the problem asks me to get the vector as a function of two different distances. I don't need to find a way to relate them. QuiteEasilyDone
 
  • #5
Can you provide any insight?

The magnetic vector potential for antiparallel currents in this scenario can be found by using the Biot-Savart law, which states that the magnetic field at a point is equal to the cross product of the current and the displacement vector from the current element to the point, divided by the distance between the two. In this case, we have two infinitely long wires separated by distance d, with currents I1 and I2 flowing in opposite directions.

Using this law, we can find the magnetic field at a point P located at a distance r1 from wire one and a distance r2 from wire two. The direction of the magnetic field will be perpendicular to the plane formed by the two wires, and its magnitude will be given by B = (mu I)/(2pi r), where mu is the permeability of the medium, I is the current, and r is the distance from the wire.

Next, we can use the relationship between the magnetic field and the magnetic vector potential, which is given by Del cross A = B. This means that the curl of the magnetic vector potential is equal to the magnetic field at that point. By solving this equation, we can find the vector potential A as a function of r1 and r2 at point P.

However, as you mentioned, taking into account two different distances can make this calculation more complex. One approach to solving this problem would be to use the superposition principle, which states that the total potential at a point due to multiple sources is equal to the sum of the potentials due to each individual source. In this case, we can consider the potential due to each wire separately and then add them together to get the total potential at point P.

Another approach would be to use vector calculus and integrate the expression for the magnetic field with respect to both r1 and r2, and then solve for the vector potential. This method may be more time-consuming, but it would give a more accurate solution.

In conclusion, finding the magnetic vector potential for antiparallel currents in this scenario involves using the Biot-Savart law, the relationship between the magnetic field and the vector potential, and possibly the superposition principle or vector calculus. It may also be helpful to draw a diagram and use the right-hand rule to determine the direction of the magnetic field and the resulting vector potential.
 

1. What is the magnetic vector potential for antiparallel currents?

The magnetic vector potential for antiparallel currents is a mathematical concept used to describe the magnetic field created by two parallel currents flowing in opposite directions. It is a vector field that is defined as the line integral of the magnetic field around a closed loop in space.

2. How is the magnetic vector potential calculated?

The magnetic vector potential can be calculated using the Biot-Savart law, which states that the magnetic field at a point in space is directly proportional to the current flowing through a wire and inversely proportional to the distance from the wire. It can also be calculated using Maxwell's equations, specifically the curl of the magnetic field equation.

3. What is the significance of antiparallel currents in relation to the magnetic vector potential?

Antiparallel currents are important because they create a magnetic field that is perpendicular to the direction of the currents. This perpendicular field is what is described by the magnetic vector potential, and it is used to calculate the total magnetic field at any point in space.

4. Can the magnetic vector potential be used to calculate the magnetic field for non-parallel currents?

Yes, the magnetic vector potential can be used to calculate the magnetic field for any configuration of currents, not just antiparallel ones. However, for more complex configurations, the calculation may become more difficult and require the use of numerical methods.

5. How is the magnetic vector potential related to the magnetic flux?

The magnetic vector potential and the magnetic flux are related through Faraday's law of induction. This law states that the induced electromotive force in a closed loop is equal to the negative rate of change of the magnetic flux through the loop. The magnetic vector potential is a key component in calculating the magnetic flux and understanding the behavior of magnetic fields.

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