Ampere's Law: Solving Parallel Cylinder Problem

[Merrank]

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

 

[learningphysics]

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.

 

[thepatientmental]

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