# Find magnetic flux density B circulation in closed contour

In summary, by using Ampere's law and the equation ∫B⋅dl=μ0⋅I, we can find the magnetic flux density B circulation in a closed contour consisting of a square with perimeter 4b and a square conductor with perimeter 4a. By taking the line integral over any one of the four current-carrying side sections, we get the circulation of B to be μ0j(a)^3. Adding this to the line integral over any section of the large perimeter with zero current, we get the final answer of μ0j(a)^3.

## Homework Statement

Find magnetic flux density B circulation in closed contour. The countour consists of square with a perimeter of 4b. It includes a square conductor with a perimeter of 4a. A homogenic current flows through a conductor with a current density of j.

∫B⋅dl=μ0⋅I[/B]

## The Attempt at a Solution

I get this equation ∫B⋅dl=μ0⋅j⋅a2 but the answer says that the integration contour is Γ. How do I get the Γ? ΓB⋅dl=μ0⋅j⋅a2

Well, since it's been nearly 120 hrs (5 days) with no other posters,I will suggest:
(1) line integral of H over ANY section of the large perimeter WITH ZERO CURRENT EVERYWHERE = 0. This includes the two "a" sections common to the large & small perimeters.
(2) line integral over any one of the four current-carrying section sides (by Ampere's law) = ja2(a) = ja3.
(3) Thus the circulation of H over the two common "a" sections = 2ja3 and of course B = μ0H.
(4) Add (3) to (1) and you get the answer of (3).

## 1. How do you calculate magnetic flux density B circulation in a closed contour?

To calculate magnetic flux density B circulation in a closed contour, you can use the formula B = μ₀I/2πr, where μ₀ is the permeability of free space, I is the current flowing through the contour, and r is the distance from the center of the contour. This formula is known as Ampere's law.

## 2. What is the unit of measurement for magnetic flux density B circulation?

The unit of measurement for magnetic flux density B circulation is Tesla (T). One Tesla is equal to one Newton per Ampere per meter (N/A·m).

## 3. Can magnetic flux density B circulation be negative?

Yes, magnetic flux density B circulation can be negative. This occurs when the direction of the current is opposite to the direction of the magnetic field, resulting in a negative value for B circulation. However, the magnitude of the B circulation will still be positive.

## 4. How is the direction of magnetic flux density B circulation determined?

The direction of magnetic flux density B circulation is determined by the right-hand rule. If you point your thumb in the direction of the current, the direction in which your fingers curl will indicate the direction of the magnetic field.

## 5. What is the significance of finding magnetic flux density B circulation in a closed contour?

Finding magnetic flux density B circulation in a closed contour allows us to determine the strength and direction of the magnetic field at a specific point. This is useful in understanding the behavior of magnetic fields and their effects on various materials and devices.

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