Maximum Magnetic Field B/w 2 Parallel Wires

[UMD_UG_ME]

Two infinitely long wires are in the negative x direction. The wires are separated by distance 2a. They are parallele to one another. They lie along the x-axis with x = o being in the middle.Both wires are at z = 0.
(a) Sketch the magnetic field pattern in the yz plane.
(b) At what distance d along the z axis is the magnetic field a maximum?


Relevant equations
B= [(4*pi*10^-7*I)/(2*pi*R^2)]*r B=magnetic field, I = current, R = radius of the wire, r = distance from the wire

B(total) = B(1) + B(2) + ... Superposition

The attempt at a solution
(a) Using the right hand rule I figured that the magnetic field pattern would be clockwise. However, with superposition I am not sure how I draw the field lines when they 'hit' each other. I am guessing that they will 'hit' each other at the midpoint if I let r = a. And then at that point the Magnetic field is a maximum at d = 0?

(b) Would the magnetic field maximum be at the center of the two wires? Thus when r = a, the magnetic field is a maximum. Since the fields just touch at r = a, z = o = d?

 

[gneill]

What does it mean for a wire to be in the negative x direction? They're infinitely long.

What is the direction of the current flowing in each wire? Are the currents of equal magnitude?

 

[UMD_UG_ME]

Umm, the diagram has the current flowing from the positive x-axis into the - x axis. The problem does not state the magnitudes of the currents but I will make an assumption that they are and make a note of that on mt paper.

 

[gneill]

The field lines add like vectors where they intersect. At z = 0 they will have equal magnitudes and opposite directions.

 

[UMD_UG_ME]

Thank You Mr./Mrs. Gneill. Okay so at z=o if r=a The wire on the left would have a vector pointed in the direction of the negative z axis, the wire on theright would have a vector equal in magnitude but pointed in the positive z axis and thus the magnetic field would be zero there. Now what you said honestly makes sense to me. However now I am confused as to answering the question "At what disatnce d along the z axis is the magnetic field a maximum?" At every distance along the z axis, wouldn't the magnetic vield line vectors cancel each other?

 

[blueboy01]

no they don't cancel along the z axis (except at z=0). the z component of the magnetic field cancels along that axis but the y component of both actually add. It's a balancing act between the angle that they are at (and hence the y component of the magnetic field) and the distance from the wire...I think I am probably in ur class because I am doing the exact same question and I have no idea how to do it either lol.

 

[gneill]

Thank You Mr./Mrs. Gneill. Okay so at z=o if r=a The wire on the left would have a vector pointed in the direction of the negative z axis, the wire on theright would have a vector equal in magnitude but pointed in the positive z axis and thus the magnetic field would be zero there. Now what you said honestly makes sense to me. However now I am confused as to answering the question "At what disatnce d along the z axis is the magnetic field a maximum?" At every distance along the z axis, wouldn't the magnetic vield line vectors cancel each other?

You're welcome. And it's Mr.

Along the z axis the field lines will meet at different angles; at very large z they will be directly aligned (horizontally) whereas they were directly opposed at z = 0 (vertically). In between they'll meet at other angles. But significantly, the field strength will drop with distance.

So, the question becomes, where along the z-axis will the effects result in a maximum net field strength?

Here's a diagram to help you out. The view is towards the negative x-axis (going into the page).

 

[blueboy01]

I notice that directly above each wire you have the magnetic field completely horizontal. is it ok to ignore the slight downward or upward effect that the other wire will have on the original wire

 

[UMD_UG_ME]

You're welcome. And it's Mr.

Along the z axis the field lines will meet at different angles; at very large z they will be directly aligned (horizontally) whereas they were directly opposed at z = 0 (vertically). In between they'll meet at other angles. But significantly, the field strength will drop with distance.

So, the question becomes, where along the z-axis will the effects result in a maximum net field strength?

Here's a diagram to help you out. The view is towards the negative x-axis (going into the page).

:) Thank you sir. If this were yahoo answers I would give you the +10 thing! One thing I love about PhysicsF(even before I had an account), You truley learn the process/thinking. That is evident for me here, thank you!

 

[gneill]

I notice that directly above each wire you have the magnetic field completely horizontal. is it ok to ignore the slight downward or upward effect that the other wire will have on the original wire

The fields depicted are those produced by the individual wires. Use superposition to find the net field at any given point.

 

[blueboy01]

yeah and the maximum occurs when z=a

 

[otcunanan]

So a year later I am doing the same assignment and I have followed everything up until solving for d. I don't understand how to go about maximizing B.