Infinitely Long Wire with Loop (Magnetic Field)

[Hitchslaps]

Homework Statement

The wires below are infinitely long and some of them are with loops and semi loops. The current I is constant. What is the magnetic field in point O, when R is the distance from it?

Questions:
In F, E and H, the magnetic field exerted by the terminal infinitely long wire is 0 because it is 180 degrees?
If the loop is a semi-circle, I use B/2? If the loop is a quarter of a circle, then B/4 and so on?

How to solve for G?

Homework Equations

B= μI/2R
B=2μI/4piR

The Attempt at a Solution

I included in my calculations the magnetic fields of both the infinite wire and the loop.

A: μI/2R Outwards
B: μI/2R + 2μI/4piR Outwards
C: μI/2R - 2μI/4piR Outwards
D: μI/2R - μI/2R +2μI/8piR = 2μI/8piR Inwards
E: 2μI/8piR Inwards
F: μI/2R + 2μI/16piR Inwards
G: ?
H: μI/2R + 2μI/8piR + μI/2R Inwards

 

[Simon Bridge]

You are asked to find the field at point O right?
Then yes: EF and H have sections of wire that do not contribute.
If B is the field of a complete loop, then an 1/nth of a loop contributes B/n.

You have equations:
BL = μI/2R <---<<< for the field center of a current loop radius R
BW =2μI/4piR=μI/(2piR) <---<<< the field radial distance R from infinite straight wire

This makes for a nice way to summarize your answers to make double-checking easy:

If we make BW=B, then BL=pi.B

Using + to indicate "into the page"
Your answers are:

A: -B
B: -(pi+1)B (what is the total current at the place the loop and the wire meet?)
C: -(pi-1)B (there is another difference between C and B)
D: +(pi-pi+1/2)B = B/2 (are you saying the two wires cancel out? Check by RH rule.)
E: +B/2 (which is half that due to a wire - does that make sense?)
F: +(pi+1/4)B
G: ?
H: +(pi + 1/2 +pi)

... I think one of us got confused between the contributions of the wire and that of the loop-section.

Checking:
http://hyperphysics.phy-astr.gsu.edu/HBASE/magnetic/curloo.html#c2
http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magcur.html

$$B_L=\frac{\mu_0 I}{2R}=\pi\frac{\mu_0 I}{2\pi R}=\pi B_W$$

For some of them, only half the wire contributes to the field.
Did you check that the final direction you gave is that for the net field - i.e. a negative magnitude into the page is a positive magnitude out of the page - so the net direction is "outwards".

For G - it's the same as the others: what's the problem?

 

[Hitchslaps]

Thank you for your help!
I confused the two equations, sorry.

My answers now:
A: -B
B: -(pi+1)B The current is NI, which is 2I? So the loop’s magnetic field is μ2I/2R= μI/R.
C: +(+pi-1)B The wires don’t meet, so the current stays the same?
D: +(2+1/2pi)B
E: +pi.B/2
F: +(1/4pi+1)B
G: +(2+1/2pi)B
H: +(1 +1/2pi)Byes, these are all net directions, assuming I'm correct.

 

[Simon Bridge]

It's interesting about the gap in C isn't it?
It may just be that the gap is supposed to be invisibly small and shown large for emphasis.
I think you have the right of it: one subtracts the wire while the other adds it.

Sometimes you have to account for gaps by subtracting the field that would otherwise be present.

You seem to have a handle on what's required enough to check your own math - well done.

 

[HalfLostAndSearching]

I would first clarify the assumptions and conditions of the problem. Is the wire assumed to be infinitely thin? Are the loops and semi-loops assumed to be perfect circles or semi-circles? Is the current assumed to be flowing in the same direction in all wires?

Based on the equations provided, it seems that the magnetic field at point O would depend on the distance R and the current I, as well as the geometry of the wires and loops. It is also important to note that the magnetic field at point O would be a vector quantity, with both magnitude and direction.

To solve for G, we would need more information about the geometry and current direction of the wires and loops. Without this information, it is not possible to accurately calculate the magnetic field at point O.

In general, the magnetic field at point O would be the vector sum of the magnetic fields produced by all the wires and loops. Each wire or loop would contribute a magnetic field that follows the right-hand rule, with the direction of the field determined by the direction of the current and the distance from the wire/loop to point O.

In conclusion, to accurately solve for the magnetic field at point O, we would need more information about the specific geometry and conditions of the wires and loops.