Ampere's Law: Solving Parallel Cylinder Problem

In summary, using superposition, we can show that the inner cylinder (radius "a") has a constant magnetic field, regardless of the point chosen, by finding the vector sum of the magnetic fields created by the outer and inner cylinders. This can be solved using Ampere's Law and the fact that the total current density in the area of the inner cylinder is 0.
  • #1
Merrank
3
0
The question is:

Between two long parallel cylinders of radius "a" and "b" (non-coaxial) and an axal separation of "c", a steady current of "I" flows. (See attachment below) Show that the inner cylinder (radius "a") has a constant magnetic field. Use Ampere's Law. Indicate all steps clearly. [Hint: 0 = 1 + (-1)]

Could someone please show me step by step on how to do this I have no idea where to start.

Thank You
 

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  • #2
Merrank said:
The question is:

Between two long parallel cylinders of radius "a" and "b" (non-coaxial) and an axal separation of "c", a steady current of "I" flows. (See attachment below) Show that the inner cylinder (radius "a") has a constant magnetic field. Use Ampere's Law. Indicate all steps clearly. [Hint: 0 = 1 + (-1)]

Could someone please show me step by step on how to do this I have no idea where to start.

Thank You

Use superposition. I'll let J=current density (I/area between the two cylinders).

The situation you described above is equivalent to a current density of J going through the entire outer cylinder + a current density of -J going through the inner cylinder (so in the area of the inner cylinder: the total current density is J + -J =0)

Find the magnetic field vector a created by the outer cylinder with a current density of J, at an arbitrary point inside the inner cylinder.

Find the magnetic field vector b created by the inner cylinder with a current density of -J, at the same point.

You should find that the vector sum of the two magnetic fields is independent of the point chosen (both magnitude and direction are independent of the point chosen)

Hint: The triangle created by the centers of the inner and outer cylinders, and the point where the magnetic field is calculated... is "similar" to the triangle formed by the vectors a, b and the sum.

It's kind of tricky. Hope this helps.
 
  • #3


To solve this problem using Ampere's Law, we first need to understand the concept of Ampere's Law. It states that the line integral of the magnetic field B around a closed loop is equal to the product of the current enclosed by the loop and the permeability of free space (μ0). In other words, it relates the magnetic field around a closed loop to the current passing through the loop.

In this problem, we have two long parallel cylinders with a steady current flowing through them. We want to find the magnetic field at a point inside the inner cylinder (radius a). To do this, we will draw a closed loop around the inner cylinder and apply Ampere's Law.

Step 1: Choosing a closed loop
To apply Ampere's Law, we need to choose a closed loop that encloses the current passing through it. In this case, we will choose a circular loop with a radius r, where r < a. This will ensure that the loop encloses the current passing through the inner cylinder.

Step 2: Determining the magnetic field
Next, we need to determine the magnetic field at a point inside the inner cylinder. According to the Biot-Savart Law, the magnetic field at a point P inside a current-carrying wire is given by:

B = μ0I/2πr

where μ0 is the permeability of free space, I is the current passing through the wire, and r is the distance from the wire to the point P.

In our problem, the current passing through the inner cylinder is I, and the distance from the wire to the point P is r. So, the magnetic field at point P inside the inner cylinder is given by:

B = μ0I/2πr

Step 3: Applying Ampere's Law
Now, we can apply Ampere's Law to the closed loop we chose in Step 1. The line integral of the magnetic field B around the closed loop is given by:

∫B·dl = μ0Ienclosed

where μ0 is the permeability of free space, Ienclosed is the current passing through the loop, and ∫dl is the line integral around the loop.

Since the magnetic field B is constant along the loop, we can take it outside the integral. Also, since the loop encloses the current I, Ienclosed = I. So, the equation becomes:

B∫dl = μ0
 

Related to Ampere's Law: Solving Parallel Cylinder Problem

1. What is Ampere's Law and how is it used in the context of solving parallel cylinder problems?

Ampere's Law is a fundamental law in electromagnetism that relates the magnetic field around a closed loop to the electric current passing through that loop. In the context of solving parallel cylinder problems, Ampere's Law can be used to calculate the magnetic field between two parallel cylinders carrying electric currents, by considering the current enclosed by the loop formed by the two cylinders.

2. How do you determine the direction of the magnetic field using Ampere's Law for parallel cylinder problems?

The direction of the magnetic field can be determined by using the right-hand rule. Point your thumb in the direction of the current and curl your fingers around the loop formed by the cylinders. The direction of your fingers will indicate the direction of the magnetic field.

3. Can Ampere's Law be used for non-parallel cylinder problems?

Yes, Ampere's Law can be used for any closed loop, not just for parallel cylinders. It can also be used for non-cylindrical shapes, as long as the current is enclosed by the loop.

4. What are the assumptions made when using Ampere's Law for solving parallel cylinder problems?

Some of the key assumptions made when using Ampere's Law for parallel cylinder problems include: the cylinders are infinitely long and parallel to each other, the current is constant along the length of the cylinders, and the magnetic field is uniform between the cylinders.

5. Are there any limitations to using Ampere's Law for solving parallel cylinder problems?

Yes, there are some limitations when using Ampere's Law for parallel cylinder problems. It may not be accurate for very small distances or when the cylinders are not perfectly parallel. Additionally, it does not take into account any external magnetic fields that may affect the system.

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