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Chapter 15: Problem 2169

Maxwell's modified form of Ampere's circuital law is (A) \(\oint \mathrm{B}^{-} \cdot \mathrm{dS}^{-}\) (B) \(\phi \mathrm{B}^{-} \cdot \mathrm{dS}^{-}=\mu_{\mathrm{o}} \mathrm{i}\) (C) $\oint \mathrm{B}^{-} \cdot \mathrm{d} \ell^{-}=\mu_{\mathrm{o}} \mathrm{i}+\mu_{0} \in_{0}\left(\mathrm{~d} \Phi_{\mathrm{E}} / \mathrm{dt}\right)$ (D) $\oint \mathrm{B}^{-} \cdot \mathrm{d} \mathcal{\ell}^{-}=\mu_{0} \mathrm{i}+\left(1 / \in_{0}\right)\left(\mathrm{d}_{\mathrm{q}} / \mathrm{dt}\right)$

Short Answer

Expert verified
The correct representation of Maxwell's modified form of Ampere's circuital law is: \[\oint \mathbf{B} \cdot d\boldsymbol{\ell} = \mu_0 I_\text{enclosed} + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}\] Thus, the correct choice is (C).

Step by step solution

01

Recall Ampere's Original Circuital Law

Ampere's circuital law states that the line integral of the magnetic field, B, around a closed curve is proportional to the net current enclosed by the curve. Mathematically, it is given as: \[\oint \mathbf{B} \cdot d\boldsymbol{\ell} = \mu_0 I_\text{enclosed}\]
02

Understand Maxwell's Modification

Maxwell added displacement current to Ampere's circuital law, which accounts for the changing electric field in a capacitor. This addition generalizes the law to cover both static and dynamic cases.
03

Write Maxwell's Modified Form of Ampere's Circuital Law

With Maxwell's modification, the Ampere's circuital law becomes: \[\oint \mathbf{B} \cdot d\boldsymbol{\ell} = \mu_0 I_\text{enclosed} + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}\] where \(\Phi_E\) is the electric flux, \(\epsilon_0\) is the vacuum permittivity, and \(\mu_0\) is the vacuum permeability.
04

Compare the Options

Comparing the step 3 equation with the presented options, we notice that option (C) matches the correct representation of Maxwell's modified form of Ampere's circuital law: \[\oint \mathrm{B}^{-} \cdot \mathrm{d} \ell^{-}=\mu_{\mathrm{o}}\mathrm{i}+\mu_{0} \in_{0}\left(\mathrm{~d} \phi_{\mathrm{E}} /\mathrm{dt}\right)\] So, the correct choice is (C).

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