Chapter 7: Problem 9
A current sheet \(\mathbf{K}=8 \mathbf{a}_{x} \mathrm{~A} / \mathrm{m}\) flows
in the region \(-2
Short Answer
Expert verified
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
01
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.
02
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 \(\mathbf{H}\).
03
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:
$$
\oint_C \mathbf{H} \cdot d\boldsymbol{\ell} = H_x\int_{-w/2}^{w/2}dy + H_x\int_{0}^{3}dz = H_x(w+3) = K \cdot w
$$
04
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 \(-2<y<2\)):
$$
H_x = \frac{8 \cdot 4}{4 + 3} = \frac{32}{7}\,\text{A/m}
$$
05
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:
$$
\mathbf{H} = H_x \mathbf{a}_{x} = \frac{32}{7}\,\text{A/m} \mathbf{a}_{x}
$$
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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:
- \( d\mathbf{H} = \frac{I \, d\boldsymbol{\ell} \times \mathbf{r}}{4\pi r^3} \)
- \( d\mathbf{H} \) 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 \( \mathbf{r} \).
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.