Question

A current sheet \(\mathbf{K}=8 \mathbf{a}_{x} \mathrm{~A} / \mathrm{m}\) flows in the region \(-2

Short answer

Answer: The magnetic field intensity \(\mathbf{H}\) at point \(P(0,0,3)\) is \(\frac{32}{7}\,\text{A/m} \mathbf{a}_{x}\).

Step-by-step solution

Understand Ampere's Circuital Law and Biot-Savart's Law

Ampere's Circuital Law states that the circulation of the magnetic field intensity \(\mathbf{H}\) around a closed path is equal to the current enclosed by that path, i.e., \(\oint_{C} \mathbf{H} \cdot d\boldsymbol{\ell} = I_{enc}\). Biot-Savart's Law relates the infinitesimal magnetic field intensity \(d\mathbf{H}\) generated by an infinitesimal current element to the current and the position vectors.

Determine a suitable Amperian loop

For a current sheet with thickness located in the region \(-2<y<2\), choose a rectangular Amperian loop with edges parallel to the y-axis and z-axis, and with one side lying in the y-z plane. Let the loop have height \(h=3\) and width \(w\) (to be found). The loop is symmetric with respect to the y-axis, and hence the magnetic field at point \(P\) is along the \(x\)-axis, i.e., \(\mathbf{H} = H_x \mathbf{a_x}\), where \(H_x\) is the magnitude of \(\mathbf{H}\).

Apply Ampere's Circuital Law to the loop

Consider the current enclosed by the loop. As the current is uniformly distributed along the y-axis for \(-2<y<2\), we can write the total current enclosed as \(I_{enc} = K \cdot w\). Applying Ampere's Circuital Law, we get: $$ \oint_C \mathbf{H} \cdot d\boldsymbol{\ell} = H_x\int_{-w/2}^{w/2}dy + H_x\int_{0}^{3}dz = H_x(w+3) = K \cdot w $$

Solve for H_x

From the above equation, we can solve for the magnitude of the magnetic field intensity \(H_x\): $$ H_x = \frac{K \cdot w}{w + 3} $$ Now, substitute the given values of \(K=8 \,\text{A/m}\), and the width \(w = 4\) (from \(-2<y<2\)): $$ H_x = \frac{8 \cdot 4}{4 + 3} = \frac{32}{7}\,\text{A/m} $$

Write the final answer

We found the magnitude of the magnetic field intensity at point \(P\), which is along the \(x\)-axis. Therefore, the magnetic field intensity \(\mathbf{H}\) at \(P(0,0,3)\) is: $$ \mathbf{H} = H_x \mathbf{a}_{x} = \frac{32}{7}\,\text{A/m} \mathbf{a}_{x} $$

Key concepts

Ampere's Circuital Law

Ampere's Circuital Law is a fundamental principle in electromagnetism. It relates a magnetic field around a closed loop to the direct current passing through that loop. This law is crucial for calculating the magnetic fields in scenarios involving symmetrical current distributions. The mathematical expression of Ampere's Circuital Law is \( \oint_{C} \mathbf{H} \cdot d\boldsymbol{\ell} = I_{enc} \), where:

• \( \oint_{C} \) signifies integration around a closed path \( C \).
• \( \mathbf{H} \) is the magnetic field intensity.
• \( d\boldsymbol{\ell} \) is an infinitesimal vector element of the path.
• \( I_{enc} \) is the current enclosed within the path.

This law becomes particularly useful when dealing with problems where current flow is uniform, such as the current sheet in the given problem. By choosing the appropriate Amperian loop, we can simplify the calculations. Here, using a rectangular Amperian loop allows us to take advantage of symmetry to efficiently determine the magnetic field intensity at the desired point. The symmetry of the problem suggests that the magnetic field has only a component along the x-axis, denoted by \( H_x \). Keeping these details in mind helps simplify complex calculations and hone in on the symmetry present in physical situations.

Biot-Savart's Law

Biot-Savart's Law provides a detailed method for computing the magnetic field generated by a segment of current. This law is expressed mathematically in the form:

• \( d\mathbf{H} = \frac{I \, d\boldsymbol{\ell} \times \mathbf{r}}{4\pi r^3} \)

where:

• \( d\mathbf{H} \) is the infinitesimal magnetic field contribution from a small segment of current \( d\boldsymbol{\ell} \).
• \( I \) is the current through the segment.
• \( \mathbf{r} \) is the positional vector from the current segment to the observation point.
• \( r \) is the magnitude of \( \mathbf{r} \).

While this law can be very practical for calculating the magnetic field intensity generated by small sections of current-carrying wires, its direct use becomes cumbersome in problems involving continuous current distributions, like sheets. That’s when Ampere's Circuital Law steps in as a simpler method, especially when combined with symmetry considerations. In essence, Biot-Savart's Law lays the foundation understanding of magnetic field contributions by small segments, but for continuous currents, employing Ampere’s Law becomes mathematically efficient.

Magnetic Field Intensity

Magnetic Field Intensity, often represented as \( \mathbf{H} \), is a vector field that provides a measure of the distribution of forces a magnetic field exerts on moving charges. It's critical in understanding the influence of magnetic fields in space. In this context, we express \( \mathbf{H} \) through its magnitude and direction, typically written as \( \mathbf{H} = H_x \mathbf{a}_x \) for its components.When dealing with flat current sheets like in the exercise, the uniform direction of the current simplifies the magnetic field's formation in space, making calculations primarily rely on symmetry and direction strategies. The value of the magnetic field intensity at a specific point is directly influenced by the current distribution's characteristics and the point's position relative to the source. It is essential to employ strategies like symmetry to fully understand the attributions of the magnetic field at required locations.In solving the exercise, we calculated \( H_x \), the magnitude of the magnetic field intensity at point \( P \). By strategically selecting an Amperian loop that matches the symmetry of the current sheet, the challenge becomes straightforward, allowing us to focus on calculating the magnetic field based on the simple symmetry frame that was established. Ultimately, understanding \( \mathbf{H} \) deeply enhances the comprehension of how magnetic fields manifest in various configurations and guides practical approaches in areas spanning electromagnetism and engineering.