Finding the magnetic field at a point

[subhradeep mahata]

Homework Statement

Find the magnetic field at point P.

(P is at the centre of two infinitely long wires, carrying current in opposite direction.

Homework Equations

The Attempt at a Solution

My guess is that since the current direction is opposite and distance between the point and the wires is same, net magnetic field at P is 0. Bust my book says that both the magnetic field direction is same, and hence net B will be the sum of B produced by the individual wires. Please help me out.

 

[Dick]

Homework Statement

Find the magnetic field at point P.
View attachment 233919
(P is at the centre of two infinitely long wires, carrying current in opposite direction.

Homework Equations

The Attempt at a Solution

My guess is that since the current direction is opposite and distance between the point and the wires is same, net magnetic field at P is 0. Bust my book says that both the magnetic field direction is same, and hence net B will be the sum of B produced by the individual wires. Please help me out.

You need to think about the direction of the magnetic fields, not just the magnitudes.

 

[subhradeep mahata]

Since, the direction of current is opposite, won't the B be equal and opposite and cancel out?

 

[Dick]

Since, the direction of current is opposite, won't the B be equal and opposite and cancel out?

You need to think a little harder about the field direction. It circles around each wire. The point is below one wire and above the other.

 

[subhradeep mahata]

Sir I think you did not get the diagram. The two wires are parallel to each other, and lying in the same plane, with the point P in between them.
The wires are not above or below.

 

[Dick]

Sir I think you did not get the diagram. The two wires are parallel to each other, and lying in the same plane, with the point P in between them.
The wires are not above or below.

I get the diagram. Just picture the plane as vertical. Do you know about Ampere's right hand rule for determining the field direction?

 

[subhradeep mahata]

Yes, I know it. But I want to find it out using Bio-Savart's law.

 

[Dick]

Yes, I know it. But I want to find it out using Bio-Savart's law.

That's fine. So $\vec {dL}$ and $\vec r$ have the same magnitude for each wire. How do their directions compare?

 

[subhradeep mahata]

One should be inside and other should be outside the plane of paper.

 

[Dick]

One should be inside and other should be outside the plane of paper.

That's not very clear. I think you would agree that that $\vec {dL_1} = -\vec {dL_2}$ (where the numbers indicate the two wires) just because the wires are parallel and the current flow is in opposite directions. What's the relation between $\vec r_1$ and $\vec r_2$?

 

[subhradeep mahata]

r₁ vector = -r₂ vector, i suppose

 

[Dick]

r₁ vector = -r₂ vector, i suppose

Right. That's what I meant by 'above' and 'below'. So what about the cross product and the magnetic field?

 

[subhradeep mahata]

So the B will be in the same direction.
I get it now.
Thanks.