Calculate the magnetic field at 2 points (checking my result)

[fluidistic]

Homework Statement

See the picture for a clear description of the situation. I must find the magnetic field at point $$P_1$$ and $$P_2$$. The picture represent a transversal cut of a wire carrying a current I going off the page to our direction. The density of the current is J.

Homework Equations

None given.

The Attempt at a Solution

I used Ampere's law. I know the figure doesn't seem enough symmetric to use Ampere's law, but I think it is. I'll explain why: I though the problem like if there were 2 wires. One of radius 2a and with a current I in our direction and another wire with radius a with current -I/4 (since the area of the "hole" of the wire is one fourth of the area of what would be a circular wire of radius 2a.)
Thanks to Ampere's law, $$B_1=\frac{\mu _0 I}{2 \pi d}$$.
$$B_1(P_1)=\frac{\mu _0 I}{2 \pi (2a+d)}$$ and $$B_1 (P_2)= \frac{\mu _0 I}{2 \pi (2a+d)}$$.

Now $$B_2 =-\frac{\mu _0 I}{8\pi d'}$$.
$$B_2(P_1)=-\frac{\mu _0 I}{8\pi (a+d)}$$ and $$B_2(P_2)=-\frac{\mu _0 I}{8 \pi (3a+d)}$$.
At last, $$B(P_1)=B_1(P_1)+B_2(P_1)$$ and $$B(P_2)=B_1(P_2)+B_2(P_2)$$.


Am I right?

 

[kuruman]

Thanks to Ampere's law, $$B_1=\frac{\mu _0 I}{2 \pi d}$$.
$$B_1(P_1)=\frac{\mu _0 I}{2 \pi (2a+d)}$$ and $$B_1 (P_2)= \frac{\mu _0 I}{2 \pi (2a+d)}$$.

Your approach is correct, but your use of Ampere's Law is not. It is valid for points external to the wire. You need the field for a point inside the wire. If you draw an Amperian loop passing through that point, the current enclosed by the loop is less than the total current I, is it not? So what does Ampere's Law say in this case?

 

[fluidistic]

Your approach is correct, but your use of Ampere's Law is not. It is valid for points external to the wire. You need the field for a point inside the wire. If you draw an Amperian loop passing through that point, the current enclosed by the loop is less than the total current I, is it not? So what does Ampere's Law say in this case?

You are absolutely right on the fact that my use of Ampere's law is valid only at exterior points of the wire. In fact $$P_1$$ and $$P_2$$ are exterior points.
The picture might be confusing now I realize. There's like a very small circle which says "I". In fact this circle only represent the sense of the current, the point in it means that it's going into our direction and from us to the sheet of paper.

Am I misunderstanding you?

 

[kuruman]

Sorry, I missed that the two points are external.

You need to watch your signs. If B1 is positive at P1, then it must be negative at P2. At each point, B2 must have opposite sign to B1 because the current is in the opposite direction.

 

[fluidistic]

Yes you are misunderstanding me. To calculate the field inside a solid wire of radius 2a, construct an Amperian loop or radius r (r<2a) concentric with the wire. Ampere's Law says that the line integral of B around the loop is equal to μ₀ times the current enclosed by the loop. I am saying that that enclosed current is not I but only a fraction of I. Naturally, the fraction depends on r - you enclose current I only when r = 2a.

Yes I know, but why would it help me? I don't have to calculate the magnetic field at any point inside the wire.
Anyway, $$I=\pi (2a)^2 J \Rightarrow J=\frac{I}{4\pi a^2}$$ and because J is constant in all the wire, I', the current enclosed by any wire of radius r is worth $$I'=\frac{\pi r^2 I}{4\pi a^2}$$. But I don't see the application. I guess I'm still not understanding you. (I'm sorry about it!)

 

[kuruman]

I edited my previous post. You did not misunderstand me, I misunderstood you.

 

[fluidistic]

Ok no problem! I thank you for helping me!

Sorry, I missed that the two points are external.

You need to watch your signs. If B1 is positive at P1, then it must be negative at P2. At each point, B2 must have opposite sign to B1 because the current is in the opposite direction.

My $$B_2$$ has opposed sign to $$B_1$$ in both $$P_1$$ and $$P_2$$. But I don't see why my $$B_1(P_1)$$ should be positive and not my $$B_1(P_2)$$.