# Ampere's Law: Constant Magnetic Field b/w Non-Coaxial Cylinders

• Merrank
PERE'S LAW states that the line integral of the magnetic field around a closed loop is equal to the enclosed current multiplied by the permeability of free space. By applying this principle to the two cylinders, it can be shown that the magnetic field inside the inner cylinder (radius "a") is constant. This is because the net current enclosed by the loop is zero, since the currents in the two cylinders cancel each other out. Therefore, the magnetic field inside the inner cylinder is only affected by the current in the larger cylinder, making it constant.
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)]

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• s Law Problem.bmp
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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)]
You have to use the principle of superposition. Work out the field for the large cylinder having a uniform current density (ie. uniform over its cross-sectional area) using Ampere's law; then work out the field for the small cylinder with a current flowing in the opposite direction with a cross-sectional density that that is the same magnitude (but opposite in direction) to that of the larger cylinder. Superimpose the two (add the field vectors).

AM

To show that the inner cylinder (radius "a") has a constant magnetic field, we can use Ampere's Law, which states that the line integral of the magnetic field around a closed loop is equal to the product of the enclosed current and the permeability of free space (μ0). In this case, we will consider a circular loop around the inner cylinder (radius "a") with a radius of r and a length of L, and use Ampere's Law to calculate the magnetic field at any point on the loop.

Step 1: Setting up the equation

We start by writing Ampere's Law in integral form:

∮B⃗ ⋅ dl⃗ = μ0Ienc

Where B⃗ is the magnetic field, dl⃗ is an infinitesimal element of the closed loop, and Ienc is the enclosed current. In this case, we will be considering a circular loop, so dl⃗ can be written as rdθ, where θ is the angle between the direction of the loop and the direction of the magnetic field.

Step 2: Simplifying the equation

We can simplify the equation by noting that the magnetic field is constant along the loop, and thus can be taken outside the integral. We can also substitute the value of dl⃗ with rdθ, and rewrite the enclosed current as the current density (J) multiplied by the area of the loop (πr2).

∮B⃗ ⋅ rdθ = μ0Jπr2

Step 3: Solving for the magnetic field

We can now solve for the magnetic field (B⃗) by rearranging the equation:

B⃗ = μ0Jπr2/∮rdθ

Step 4: Evaluating the integral

To evaluate the integral, we need to determine the limits of integration. Since the loop is a circle with a radius of r, the limits of integration will be from 0 to 2π, representing a full revolution around the loop. Substituting these values into the integral, we get:

B⃗ = μ0Jπr2/∮rdθ = μ0Jπr2/2π = μ0Jr/2

Step 5: Simplifying the equation

We can further simplify the equation by noting that the current density (J) can be written as the total current (

## 1. What is Ampere's Law?

Ampere's Law is a fundamental principle in electromagnetism that relates the magnetic field around a closed loop to the electric current passing through the loop. It states that the line integral of the magnetic field along a closed loop is equal to the permeability of free space times the total electric current passing through the loop.

## 2. How does Ampere's Law apply to a constant magnetic field between non-coaxial cylinders?

In the case of a constant magnetic field between non-coaxial cylinders, Ampere's Law can be used to determine the magnetic field strength at any point along the outer cylinder. The law states that the line integral of the magnetic field along a closed loop is directly proportional to the electric current passing through the loop. Therefore, by calculating the electric current passing through the loop and knowing the dimensions of the cylinders, the magnetic field strength can be determined.

## 3. What is a closed loop in the context of Ampere's Law?

A closed loop is any path that forms a complete circuit or loop. In the context of Ampere's Law, it is the path along which the line integral of the magnetic field is calculated. The loop must be closed in order for the law to be applied.

## 4. What is the significance of the permeability of free space in Ampere's Law?

The permeability of free space is a physical constant that represents the ability of a material to support the formation of a magnetic field. In Ampere's Law, it is used to relate the magnetic field strength to the electric current passing through a closed loop. It is an important factor in determining the strength and behavior of magnetic fields.

## 5. What are some real-world applications of Ampere's Law?

Ampere's Law has various real-world applications, including the design and operation of motors and generators, electromagnetic sensors, and the production of magnetic fields in medical imaging devices. It is also used in the study of plasma physics and fusion energy research.

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