magnetic field generated by circular current loop with N turns and solenoid

[PG1995]

Hi

I understand it's a long post and perhaps I should have made the queries in separate posts. But all three queries are related so I thought it would be a better idea to combine them in one post. It would be really kind of you if you could help me. Thank you for your time and help.

Magnetic field of an infinite straight current carrying conductor is given as: [latex]B=\frac{\mu _{0}I}{2\pi r}[/latex].

Likewise, electric field of an infinite line of charge is also given by a similar formula: [latex]E=\frac{\lambda }{2\pi \epsilon _{0}R}[/latex], "λ" is linear charge density, λ=Q/2a.

Magnetic field at the center of N circular loops is given as: [latex]B=\frac{\mu _{0}NI}{2a}[/latex], where "N" is number of loops and "a" is radius.

Q: Suppose we have a loop with an infinitesimal radius dr. It would be reasonable to assume that field over the cross section of the loop is constant, at least for the radius dr. The circumference of the loop is: 2π(dr). Assume that this circumference can be subdivided into infinitesimally small 10 segments, dl's, i.e. 10(dl)=2π(dr). When radius is double, the circumference gets doubled, i.e. circumference for 2dr radius=2π(2dr)=4π(dr). This means that dl segments carved out of the circumference also get doubled, i.e. 2(10)=20. This might lead one to erroneously conclude that magnetic field in the cross section of the loop is still constant because even though radius has been doubled but there have also been twice more segments to contribute to the field. But here one should consider the area of a circle which does have linear relationship with radius: area=πr^2. It means for each increase in radius area would increase more as compared to a simple linear relation. But still don't you think that the magnetic field at the very center of the loop should remain constant?

You can find Q1 and Q2 included in **broken link removed** attachment. By the way, in Q2, I mention a picture at the very beginning, you can find it **broken link removed**.

Regards
PG

 

[MrAl]

Hi,

If i understand your question correctly...

You've 'calculated' a 'dr' but you havent related B to that dr. You're assuming that when you use dr you can use B but really you would have to use dB not B if you choose to use dr. You'll find the relationship between dr and dB to be different than the relationship between r and B, because dr implies a change in circumference dc and that implies a current element not a whole wire. A current element here would be a very short length of wire that is so short that any distance away from it no matter how small would be considered very very far away from the wire.
See if you can come up with an expression for dB such as dB/dc and see what you get. It should explain what is really happening here.

 

[MrAl]

Hi again,


Here's an interesting way to prove what the B at the center of the loop is...

First calculate the value of B at a distance R from the center of a finite wire segment of length L using Biot-Savart, call that B1.
Next, since we know that the circumference of a circle is 2*pi*r and the loop follows this path, we place one length L of a wire above right at some point on this circle. The wire length is to be placed on the circle at one point on the circle and tangent to the circle at that point.
Now since we know what B1 is from above and we know the length L of the short wire placed on the circle circumference, we know the value of B at the center right now is B1 also. That's using one wire segment.
But the circle is not complete yet because we've only used one short length of wire, so what we have to do next is place more wire segments of the same length next to each other so that they form a circle, a crude circle but the wire segments follow the circumference and enclose the same area as the circle once we have enough wire segments.
To express this process mathematically, all we have to do is make each length equal to the circumference divided by some number N where N will be the number of segments used to approximate the circle:
L=2*pi*r/N

So now we have the length of each segment, but since we want to go all the way around the circle that means we have to place N such segments around the circle. This means when we do this we also multiply the B of one segment by N and we can do this because of superposition. What we get at the center then is B1*N.

So what we would end up with using Biot-Savart is a function in R and L where B=F(R,L), F being this function for one segment and L each length, and L is defined above. Then we multiply this function by N (because of the N segments) and then take the limit as we let N approach infinity, and what we get is:
B=u0*I/(2*R)

If you would like to try this yourself as an exercise you'll have to start with Biot-Savart, but in case that's a little too much work here is a function you can use:
B=L*I*u0/(2*pi*R*sqrt(4*R^2+L^2))

[LATEX]B=\frac{L I \mu_0}{2 \pi R \sqrt{4 R^2+L^2}}[/LATEX]

That is a function that describes the field at the mid point of the current segment wire length where the wire length is L and the distance from the wire is R. That can be used in this proof and was derived directly from Biot-Savart. See attached diagram.

When you are done you might want to multiply by N but this N is the number of turns in the coil. The N above is the number of wire segments.