Help Solving a Problem with Cylindrical Wires: Magnetic Force Calculation

In summary, the conversation discusses a problem involving a compact package containing 100 long straight wires shaped like a cylinder, each conducting 2.00 A. The question asks for the intensity and direction of the magnetic force acting on a wire located 0.200 cm from the center of the package. The solution involves using Ampère Law to calculate the magnetic field and then using the equation for force to find the force per unit of length. A mistake may have been made in considering the wires to have a uniform current distribution due to some gaps between them.
  • #1
tiagobt
31
0
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: [itex]I_1 = n.i[/itex]
Area of the section: [itex]A_1 = \pi R^2[/itex]

Cylinder with radius r
Current: [itex]I_2[/itex]
Area of the section: [itex]A_2 = \pi r^2[/itex]

[tex]\frac {I_1} {I_2} = \frac {A_1} {A_2}[/tex]

[tex]I_2 = \frac {n i r^2} {R^2}[/tex]

Using Ampère Law for a circle of radius r:

[tex]\oint \vec B \cdot d \vec s = \mu_0 I_2[/tex]

[tex]B 2 \pi r = \frac {\mu_0 n i r^2} {R^2}[/tex]

[tex] B = \frac {\mu_0} {2 \pi} \frac {n i r} {R^2} = 0.0032 T[/tex]

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

[tex]F = i l B[/tex]

[tex]\frac F l = iB = 0.0064 N/m = 6.4 mN/m[/tex]

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

Thanks,

Tiago
 
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  • #2
I forgot to say that the wires are all isolated. Does that change anything?
 
  • #3
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.
 
Last edited:

1. How do I calculate the magnetic force on a cylindrical wire?

To calculate the magnetic force on a cylindrical wire, you can use the equation F = I x L x B, where F is the magnetic force, I is the current running through the wire, L is the length of the wire, and B is the magnetic field strength. This equation is known as the Lorentz force law and is used to calculate the force on a charged particle in a magnetic field.

2. What is the direction of the magnetic force on a cylindrical wire?

The direction of the magnetic force on a cylindrical wire can be determined using the right-hand rule. If you point your thumb in the direction of the current in the wire, and your fingers in the direction of the magnetic field, then the direction your palm is facing will be the direction of the magnetic force.

3. How do I determine the magnetic field strength for a cylindrical wire?

The magnetic field strength for a cylindrical wire can be calculated using the equation B = μ0 x I / (2πr), where μ0 is the permeability of free space, I is the current in the wire, and r is the distance from the wire. Alternatively, you can use a Gauss meter to measure the magnetic field strength directly at a certain distance from the wire.

4. Can I use the same equation to calculate the magnetic force on a non-cylindrical wire?

No, the equation F = I x L x B is specifically for calculating the magnetic force on a cylindrical wire. For a non-cylindrical wire, you will need to use a different equation, such as the Biot-Savart law, which takes into account the shape of the wire.

5. What factors can affect the accuracy of my magnetic force calculations for a cylindrical wire?

Some factors that can affect the accuracy of your magnetic force calculations for a cylindrical wire include variations in the magnetic field strength, variations in the current running through the wire, and imperfections in the shape and size of the wire. It is important to take these factors into account and use precise measurements to ensure accurate calculations.

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