Question

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

Answer: Only (2) Gauss's Law for Magnetic Fields

Step-by-step solution

Review the Maxwell's equations

Recall the four Maxwell's equations in their integral form are: (1) Gauss's Law for Electric Fields: \(\oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{enc}}{\epsilon_0}\) (2) Gauss's Law for Magnetic Fields: \(\oint \mathbf{B} \cdot d\mathbf{A} = 0\) (3) Faraday's Law of Induction: \(\oint \mathbf{E} \cdot d\mathbf{l} = - \frac{d\Phi_B}{dt}\) (4) Maxwell-Ampere Law: \(\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 \cdot (I_{enc}+\epsilon_0 \cdot \frac{d\Phi_E}{dt})\)

Analyze the implications of magnetic monopoles on Maxwell's equations

If magnetic monopoles existed, there would be isolated magnetic charges that would contribute to the magnetic field. We should examine each of the Maxwell's equations to understand how the existence of magnetic monopoles would alter them. (1) Gauss's Law for Electric Fields already involves electric charges, so it wouldn't be affected by the existence of magnetic monopoles. (2) Gauss's Law for Magnetic Fields assumes that there are no magnetic monopoles, as it states that the magnetic field does not contain any isolated charges. If magnetic monopoles existed, this equation would be altered as there would be a net magnetic charge enclosed. (3) Faraday's Law of Induction is related to the change of the magnetic field over time and would not be directly affected by the presence of magnetic monopoles. (4) The Maxwell-Ampere Law deals with the current enclosed by a loop and how it contributes to the magnetic field, it does not directly involve magnetic monopoles.

Identify the affected equation(s) and select the correct answer

From Step 2, we can conclude that the presence of magnetic monopoles only affects Gauss's Law for Magnetic Fields (equation (2)). Therefore, the correct answer is: a) only (2)

Key concepts

Gauss's Law for Magnetic Fields

Gauss's Law for Magnetic Fields is one of the four fundamental equations known as Maxwell's equations that form the foundation of classical electromagnetism. This specific law states that the net magnetic flux through a closed surface is always zero. In mathematical terms, this is formulated as \[\oint \mathbf{B} \cdot d\mathbf{A} = 0\]where \( \mathbf{B} \) is the magnetic field, and \( d\mathbf{A} \) is an infinitesimal vector area on the closed surface. This equation implies that unlike electric charges, magnetic pole charges (north and south) are always found in pairs so that their effects cancel out when considering any closed surface.
If hypothetical magnetic monopoles were to exist, this law would be directly altered because it would introduce a net magnetic 'charge' enclosed by the surface, thereby challenging the zero-net-flux principle currently observed and accepted in magnetostatics.

Magnetic Monopoles

The concept of magnetic monopoles refers to hypothetical particles that would carry a singular, isolated north or south magnetic charge. In contrast to a bar magnet, which has both a north and south pole, a magnetic monopole would have just one kind of magnetic pole. Their existence would revolutionize our understanding of magnetic fields because, according to current knowledge codified in Gauss's Law for Magnetic Fields, magnetic poles always come in 'dipole' pairs.
While there has been no experimental evidence confirming the existence of magnetic monopoles, their theoretical implications are significant. If discovered, they would not only amend Gauss's Law for Magnetic Fields but could also provide a symmetric counterpart to electric charges, potentially leading to an elegant unification of electricity and magnetism within the framework of classical physics.

Faraday's Law of Induction

Faraday's Law of Induction describes how a changing magnetic field within a closed loop induces an electromotive force (EMF) around the loop. This is elegantly expressed by the equation:\[\oint \mathbf{E} \cdot d\mathbf{l} = - \frac{d\Phi_B}{dt}\]The integral on the left represents the electric field along a closed loop, and \(\frac{d\Phi_B}{dt}\) on the right side denotes the time rate of change of the magnetic flux through the loop. The negative sign corresponds to Lenz's Law, which states that the induced EMF gives rise to a current whose magnetic field opposes the change in flux. In the context of magnetic monopoles, Faraday's Law as presently understood would not need revision since it primarily concerns how magnetic fields change with time, rather than their origins from 'charges'.

Gauss's Law for Electric Fields

Gauss's Law for Electric Fields is another crucial component of Maxwell's equations, which relates the electric field emanating from charges to the amount of charge enclosed by a surface. Mathematically, Gauss's Law for Electric Fields is written as:\[\oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{enc}}{\epsilon_0}\]Here, \(\oint \mathbf{E} \cdot d\mathbf{A}\) represents the electric flux through a closed surface, \(Q_{enc}\) is the total electric charge inside the surface, and \(\epsilon_0\) is the vacuum permittivity. This law asserts that electric charges are sources or sinks of electric fields, leading to a non-zero net electric flux in contrast to magnetic fields where the net flux is zero. The existence of magnetic monopoles would not impact this law, as it deals explicitly with electric charges, highlighting the asymmetry between electric and magnetic phenomena in our current understanding.