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H I аф Σπρ Idz az x [pap - zaz] 4π[p² +2213/2 - 10.² -a

H I аф Σπρ Idz az x [pap - zaz] 4π[p² +2213/2 - 10.² -a

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
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Need help simplifying this integral please :)

### Expression for Magnetic Field \(\mathbf{H}\) The magnetic field \(\mathbf{H}\) can be expressed as the following equation: \[ \mathbf{H} = \frac{I}{2 \pi \rho} \mathbf{a}_\phi - \int_{-a}^{a} \frac{I \, dz \, \mathbf{a}_z \times \left[ \rho \mathbf{a}_\rho - z \mathbf{a}_z \right]}{4 \pi [\rho^2 + z^2]^{3/2}} \] #### Explanation of Terms: - **\(\mathbf{H}\)**: Represents the magnetic field vector. - **\(I\)**: Current, a scalar quantity. - **\(\rho\)**: Radial distance from the wire. - **\(\mathbf{a}_\phi\)**, **\(\mathbf{a}_\rho\)**, and **\(\mathbf{a}_z\)**: Unit vectors in cylindrical coordinates, representing the azimuthal, radial, and axial directions, respectively. - **\(a\)**: Upper and lower limits of integration. - **\(z\)**: Axial position variable. #### Integral Term: This portion of the expression accounts for the contribution to the magnetic field from a differential current element along the z-axis. The integral evaluates the contribution from \(z = -a\) to \(z = a\). It uses cross-product and vector subtraction within the integrand to reflect the vector nature of the magnetic field due to differential current elements. This expression commonly arises in the context of electromagnetic theory, especially in applications involving the calculation of magnetic fields around conductors with current.
Transcribed Image Text:### Expression for Magnetic Field \(\mathbf{H}\) The magnetic field \(\mathbf{H}\) can be expressed as the following equation: \[ \mathbf{H} = \frac{I}{2 \pi \rho} \mathbf{a}_\phi - \int_{-a}^{a} \frac{I \, dz \, \mathbf{a}_z \times \left[ \rho \mathbf{a}_\rho - z \mathbf{a}_z \right]}{4 \pi [\rho^2 + z^2]^{3/2}} \] #### Explanation of Terms: - **\(\mathbf{H}\)**: Represents the magnetic field vector. - **\(I\)**: Current, a scalar quantity. - **\(\rho\)**: Radial distance from the wire. - **\(\mathbf{a}_\phi\)**, **\(\mathbf{a}_\rho\)**, and **\(\mathbf{a}_z\)**: Unit vectors in cylindrical coordinates, representing the azimuthal, radial, and axial directions, respectively. - **\(a\)**: Upper and lower limits of integration. - **\(z\)**: Axial position variable. #### Integral Term: This portion of the expression accounts for the contribution to the magnetic field from a differential current element along the z-axis. The integral evaluates the contribution from \(z = -a\) to \(z = a\). It uses cross-product and vector subtraction within the integrand to reflect the vector nature of the magnetic field due to differential current elements. This expression commonly arises in the context of electromagnetic theory, especially in applications involving the calculation of magnetic fields around conductors with current.
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