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Chapter 31: Problem 7

It is speculated that isolated magnetic "charges" (magnetic monopoles) may exist somewhere in the universe. Which of Maxwell's equations, (1) Gauss's Law for Electric Fields, (2) Gauss's Law for Magnetic Fields, (3) Faraday's Law of Induction, and/or (4) the MaxwellAmpere Law, would be altered by the existence of magnetic monopoles? a) only (2) c) (2) and (3) b) (1) and (2) d) only (3)

Short Answer

Expert verified
Answer: a) only (2)

Step by step solution

01

Analyze Gauss's Law for Magnetic Fields

The Gauss's Law for Magnetic Fields states that the total magnetic flux through a closed surface is always zero. Mathematically, it is given by: \(\oint \vec{B} \cdot d\vec{A} = 0\) This equation implies that there are no isolated magnetic charges (monopoles) because the magnetic field lines form closed loops (i.e., they never begin or end within the volume enclosed by the surface). If magnetic monopoles were to exist, the magnetic flux through a closed surface enclosing a magnetic monopole would not be zero. Thus, the Gauss's Law for Magnetic Fields would be altered.
02

Analyze Gauss's Law for Electric Fields

The Gauss's Law for Electric Fields relates the electric flux through a closed surface to the charge enclosed within the surface. Mathematically, it is given by: \(\oint \vec{E} \cdot d\vec{A} = \frac{Q}{\varepsilon_0}\) Since this equation deals with electric fields and charges, it would not be affected by the existence of magnetic monopoles.
03

Analyze Faraday's Law of Induction

Faraday's Law of Induction states that a time-varying magnetic field induces an electromotive force (EMF) in a closed loop. Mathematically, it is given by: \(\oint \vec{E} \cdot d\vec{l} = -\frac{d\phi_B}{dt}\) This equation relates the electric field to changes in the magnetic field but does not involve magnetic charges or monopoles. Thus, Faraday's Law of Induction would not be affected by the existence of magnetic monopoles.
04

Analyze Maxwell-Ampere Law

The Maxwell-Ampere Law relates the magnetic field around a closed loop to the electric current and time-varying electric field passing through the loop. Mathematically, it is given by: \(\oint \vec{B} \cdot d\vec{l} = \mu_0(I + \varepsilon_0 \frac{d\phi_E}{dt})\) This equation deals with the relationship between electric and magnetic fields, but does not involve magnetic charges or monopoles. Thus, the Maxwell-Ampere Law would not be affected by the existence of magnetic monopoles.
05

Choose the correct answer

Upon analyzing all of Maxwell's equations, we found that only Gauss's Law for Magnetic Fields would be altered by the existence of magnetic monopoles. Hence, the correct answer is: a) only (2)

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