Help Solving a Problem with Cylindrical Wires: Magnetic Force Calculation

[tiagobt]

Could anyone please help me with the following problem?

A compact package contains n = 100 long straight wires, shaped like a cylinder with a radius of R = 0.500 cm. If each wire conducts i = 2.00 A, calculate the intensity and direction of the magnetic force per unit of length acting on a wire located r = 0.200 cm from the center of the package.

I tried to solve it as follows:

Complete cylinder with radius R
Current: $I_1 = n.i$
Area of the section: $A_1 = \pi R^2$

Cylinder with radius r
Current: $I_2$
Area of the section: $A_2 = \pi r^2$

$$\frac {I_1} {I_2} = \frac {A_1} {A_2}$$

$$I_2 = \frac {n i r^2} {R^2}$$

Using Ampère Law for a circle of radius r:

$$\oint \vec B \cdot d \vec s = \mu_0 I_2$$

$$B 2 \pi r = \frac {\mu_0 n i r^2} {R^2}$$

$$B = \frac {\mu_0} {2 \pi} \frac {n i r} {R^2} = 0.0032 T$$

Calculating the force that acts on the wire with distance r from the center:

$$F = i l B$$

$$\frac F l = iB = 0.0064 N/m = 6.4 mN/m$$

But I was supposed to find $\frac F l = 6.34 mN/m$. What did I do wrong?

Thanks,

Tiago

 

[tiagobt]

I forgot to say that the wires are all isolated. Does that change anything?

 

[tiagobt]

Does it appear to be right at least? I'm starting to think that my mistake was to consider the cylinder section having a uniform current distribution. Since the wires are isolated and don't "fit" perfectly in a cylinder (some gaps are left in between them), I may have used the wrong current in Ampère Law. Is there an easy way to fix this? My answer is close to the answer key, so it could be something like that.