Calculating Current Density And Enclosed Current In A Conducting Region

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Given the magnetic field intensity on plane Z = 0, H = y ax - x ay (A/m), determine (a) the current density J and (b) the enclosed current I_enc by taking the circulation of H around the edge of a rectangular path.

Introduction

In electromagnetics, understanding the relationship between magnetic fields and electric currents is crucial. This article delves into a specific scenario within a conducting region, focusing on determining the current density and enclosed current given the magnetic field intensity on a particular plane. We will explore the fundamental principles governing these relationships and apply them to solve a practical problem. This involves leveraging Maxwell's equations, specifically Ampère's law, to bridge the gap between magnetic fields and current distributions. The ability to accurately calculate current density and enclosed current is essential in various applications, ranging from designing electrical devices to analyzing electromagnetic interference.

Problem Statement

Consider a conducting region where the magnetic field intensity (H) on the plane Z = 0 is defined as:

H = y a_x - x a_y (A/m)

where a_x and a_y are the unit vectors in the x and y directions, respectively. Our objective is to determine:

(a) The current density, J.

(b) The enclosed current, I_enc, by taking the circulation of H around the edge of a rectangular path defined in the Z=0 plane.

(a) Determining the Current Density, J

The current density (J) represents the amount of current flowing per unit area and is a vector quantity. It's directly related to the magnetic field through Ampère's Law, one of Maxwell's fundamental equations. In its differential form, Ampère's Law states:

∇ × H = J

This equation tells us that the curl of the magnetic field intensity is equal to the current density. To find J, we need to calculate the curl of the given magnetic field H. In Cartesian coordinates, the curl of a vector field A = A_x a_x + A_y a_y + A_z a_z is given by:

∇ × A = (∂A_z/∂y - ∂A_y/∂z) a_x + (∂A_x/∂z - ∂A_z/∂x) a_y + (∂A_y/∂x - ∂A_x/∂y) a_z

In our case, H = y a_x - x a_y, so H_x = y, H_y = -x, and H_z = 0. Applying the curl formula:

∇ × H = (∂(0)/∂y - ∂(-x)/∂z) a_x + (∂(y)/∂z - ∂(0)/∂x) a_y + (∂(-x)/∂x - ∂(y)/∂y) a_z

∇ × H = (0 - 0) a_x + (0 - 0) a_y + (-1 - 1) a_z

∇ × H = -2 a_z (A/m²)

Therefore, the current density is:

J = -2 a_z (A/m²)

This result indicates that the current density is uniform and flows in the negative z-direction. The magnitude of the current density is 2 A/m², meaning that 2 Amperes of current are flowing through every square meter of the plane perpendicular to the z-axis.

(b) Determining the Enclosed Current, I_enc

The enclosed current (I_enc) is the total current passing through a given surface. We can determine I_enc by using Ampère's Circuital Law, which is the integral form of Ampère's Law. Ampère's Circuital Law states that the line integral of the magnetic field intensity H around a closed path is equal to the total current enclosed by that path:

∮ H ⋅ dl = I_enc

To calculate I_enc, we need to choose a closed path and evaluate the line integral of H around it. The problem specifies using a rectangular path in the Z = 0 plane. Let's consider a rectangular path with vertices at (0, 0), (a, 0), (a, b), and (0, b), where 'a' and 'b' are the lengths of the sides of the rectangle along the x and y axes, respectively.

We need to divide the integral into four parts, corresponding to the four sides of the rectangle:

  1. Path 1: From (0, 0) to (a, 0), dl = dx a_x, y = 0
  2. Path 2: From (a, 0) to (a, b), dl = dy a_y, x = a
  3. Path 3: From (a, b) to (0, b), dl = -dx a_x, y = b
  4. Path 4: From (0, b) to (0, 0), dl = -dy a_y, x = 0

Now, we calculate the line integral for each path:

  • Path 1: ∮ H ⋅ dl = ∫ (y a_x - x a_y) ⋅ (dx a_x) = ∫ (y dx) = ∫ (0 dx) = 0 (since y = 0)

  • Path 2: ∮ H ⋅ dl = ∫ (y a_x - x a_y) ⋅ (dy a_y) = ∫ (-x dy) = ∫ (-a dy) = -a ∫ dy = -ab (since x = a)

  • Path 3: ∮ H ⋅ dl = ∫ (y a_x - x a_y) ⋅ (-dx a_x) = ∫ (-y dx) = ∫ (-b dx) = -b ∫ dx = -b(0 - a) = ab (since y = b)

  • Path 4: ∮ H ⋅ dl = ∫ (y a_x - x a_y) ⋅ (-dy a_y) = ∫ (x dy) = ∫ (0 dy) = 0 (since x = 0)

The total circulation is the sum of the line integrals along each path:

∮ H ⋅ dl = 0 + (-ab) + ab + 0 = -2ab

Therefore, the enclosed current is:

I_enc = ∮ H ⋅ dl = -2ab (A)

This result shows that the enclosed current is proportional to the area of the rectangle (ab) and is negative, indicating that the current flows in a direction opposite to the direction of the path we chose for the circulation.

Alternatively, we can also calculate the enclosed current by integrating the current density over the area of the rectangle:

I_enc = ∬ J ⋅ dS

In this case, J = -2 a_z (A/m²) and dS = dx dy a_z. So,

I_enc = ∬ (-2 a_z) ⋅ (dx dy a_z) = -2 ∬ dx dy = -2 ∫₀ᵃ ∫₀ᵇ dy dx = -2ab (A)

This confirms our previous result obtained using Ampère's Circuital Law.

Conclusion

We have successfully determined the current density and enclosed current in a conducting region given the magnetic field intensity on the Z = 0 plane. By applying Ampère's Law in both its differential (curl form) and integral (circuital law) forms, we found that the current density is J = -2 a_z (A/m²) and the enclosed current for a rectangular path is I_enc = -2ab (A). This analysis highlights the fundamental relationship between magnetic fields and electric currents and demonstrates the power of Maxwell's equations in solving electromagnetic problems. Understanding these concepts is crucial for engineers and physicists working with electromagnetic phenomena.

This example underscores the importance of vector calculus in electromagnetics. The curl operation allows us to transition from the magnetic field to the current density, while line integrals enable the computation of enclosed currents. These techniques are broadly applicable in analyzing various electromagnetic systems, from simple circuits to complex antennas and waveguides. The ability to calculate current densities and enclosed currents is paramount in designing efficient and reliable electrical and electronic devices.

Furthermore, the negative sign of the enclosed current in this scenario indicates the direction of current flow relative to the chosen path of integration. This emphasizes the importance of considering the directionality of both the magnetic field and the current when applying Ampère's Law. A thorough understanding of these principles is essential for accurately predicting and controlling electromagnetic phenomena in a wide range of applications.

In summary, this exploration of current density and enclosed current within a conducting region provides a valuable insight into the fundamental principles of electromagnetics. By combining Ampère's Law with vector calculus techniques, we can effectively analyze the relationship between magnetic fields and current distributions, paving the way for advancements in various technological fields.