Calculating Magnetic Field at Point P

[cuppy]

1. Homework Statement

What is the magnitude of the magnetic field at point P if a=R and b=2R (see attached image)

2. Homework Equations
Ampere's Law where integral of B ds over a closed surface = μ0I where μ0= 4πx10-7


3. The Attempt at a Solution

I used the formula and found after integrating (ds becomes 2*Pi*2R the circumference of the circle) that for the outer circle (radius 2R) B= μ0I/4*Pi*R

I then thought that the current through the inner circle would be in the same ratio as that of the areas of the two circles.
So inner current/outer current = Pi*(2R)^2/Pi*R^2
so inner current = 4I

i then subsituted this into my integral for the inner circle using Ampere's Law and found B for the inner circle to be equal to μ0*4I/2*Pi*R which simplifies to μ0*2I/Pi*R

the next step is what I'm having problems with as I'm not 100% certain as to what to do with these two values of B i have found. current flows in the same direction in both so i thought id have to add one B to the other but none of these answers show up in the five multiple choice options so i must have gone wrong somewhere. have i perhaps missed a crucial step along the way or was my working out in the first place off the mark? any advice would be great

cheers

 

[Doc Al]

I can't quite follow the path that you are integrating over. Rather that attempt to use Ampere's law, why not use the Biot_Savart law to calculate the magnetic field at the center of a current loop. Unless you happen to the know the formula already.

 

[m2003]

As a scientist, it is important to first understand the problem and the given information. From the attached image, we can see that there are two concentric circles with radii a and b, and a current I flowing through the outer circle. The question is asking for the magnitude of the magnetic field at point P, which lies on the axis between the two circles at a distance R from the center.

To solve this problem, we can use Ampere's Law, which states that the integral of the magnetic field B over a closed path is equal to μ0 times the current enclosed by that path. In this case, the closed path is a circle of radius R, centered at point P. Therefore, we can write:

∫Bds = μ0I

Since the magnetic field is constant along the path, we can take it out of the integral and integrate over the path length, which is 2πR. This gives us:

B(2πR) = μ0I

Solving for B, we get:

B = μ0I/2πR

This is the magnetic field at point P due to the current I flowing through the outer circle. However, we also need to take into account the contribution from the current flowing through the inner circle. From the given information, we know that the current through the inner circle is 4I, and its radius is a=R. Using the same method as before, we can find the magnetic field at point P due to this current to be:

B = μ0(4I)/2πR = 2μ0I/πR

Now, we can combine the contributions from both currents by adding their respective magnetic fields:

B = μ0I/2πR + 2μ0I/πR = (3μ0I)/2πR

Therefore, the total magnetic field at point P is given by:

B = (3μ0I)/2πR

It is important to note that the magnetic fields from the two currents are in the same direction, and thus we can simply add them together. This is because Ampere's Law only considers the net current enclosed by the path, regardless of its direction.

In conclusion, the magnitude of the magnetic field at point P is given by (3μ0I)/2πR, where μ0 is the permeability of free space, I is the current flowing through the outer circle, and R is