Classical Electrodynamics · Problem 5.5

Maxwell Equations, Macroscopic Electromagnetism, Conservation Laws — John David Jackson

Physics — E&M · solenoid-field-components

Problem

(a) Use the results of Problems 5.4 and 5.3 to find the axial and radial components of magnetic induction in the central region (∣z∣≪L/2|z| \ll L/2) of a long uniform solenoid of radius aa and ends at z=±L/2z = \pm L/2, including the value of BzB_z just inside the coil (ρ=a−\rho = a^-). (b) Use Amp\`ere's law to show that the longitudinal magnetic induction just outside the coil is approximately Bz(ρ=a+,z)≈−(2μ0NIa2L2)(1+12z2L2−9a2L2+⋯ )B_z(\rho = a^+, z) \approx -\left(\frac{2\mu_0 N I a^2}{L^2}\right)\left(1 + \frac{12z^2}{L^2} - \frac{9a^2}{L^2} + \cdots\right) For L≫aL \gg a, the field outside is negligible compared to inside. How does this axial component compare in size to the azimuthal component of Problem 5.2b? (c) Show that at the end of the solenoid the magnetic induction near the axis has components Bz≃μ0NI2,Bρ≃±μ0NI4(ρa)B_z \simeq \frac{\mu_0 NI}{2}, \qquad B_\rho \simeq \pm\frac{\mu_0 NI}{4}\left(\frac{\rho}{a}\right)

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