Question

Suppose the divergence of magnetic field \(\vec{𝐵}\) is nonzero and is given as \(∇⃗.\vec{𝐵} = \mu_0\rho_m\), where 𝜇₀ is the permeability of vacuum and 𝜌_m is the magnetic charge density. If the corresponding magnetic current density is \(\vec{J}_𝑚\), then the curl \(∇⃗ × \vec{𝐸}\) of the electric field \(\vec{𝐸}\) is

• A. \(\vec{J}_m-\frac{∂\vec{B}}{∂t}\)
• B. \(\mu_0\vec{J}_m-\frac{∂\vec{B}}{∂t}\)
• C. \(-\vec{J}_m-\frac{∂\vec{B}}{∂t}\)
• D. \(-\mu_0\vec{J}_m-\frac{∂\vec{B}}{∂t}\)

Answer 1

The Correct Option is D

To solve this problem, we need to consider the modified Maxwell's equations that account for the possibility of magnetic monopoles. The divergence of the magnetic field in this case is given by:

\(∇⃗.\vec{𝐵} = \mu_0 \rho_m\) 

where \(\mu_0\) is the permeability of vacuum and \(\rho_m\) is the magnetic charge density.

According to the modified Maxwell's equations, the curl of the electric field \((\vec{E})\) in the presence of a magnetic current density \((\vec{J}_m)\) is given by:

\(∇⃗ × \vec{𝐸} = -\frac{∂\vec{B}}{∂t} - \mu_0 \vec{J}_m\)

This equation indicates that the change in the magnetic field over time and the magnetic current density both contribute to the curl of the electric field.

Now, let us match this derived expression with the provided options:

• \(\vec{J}_m-\frac{∂\vec{B}}{∂t}\) - does not match because there's no negative sign with the current density.
• \(\mu_0\vec{J}_m-\frac{∂\vec{B}}{∂t}\) - not correct due to incorrect signs and factors.
• \(-\vec{J}_m-\frac{∂\vec{B}}{∂t}\) - This option also has an incorrect factor.
• \(-\mu_0\vec{J}_m-\frac{∂\vec{B}}{∂t}\) - Correct match as it incorporates both the time derivative of the magnetic field and the magnetic current density with the correct factor and signs.

Thus, the correct answer is:

\(-\mu_0 \vec{J}_m - \frac{∂\vec{B}}{∂t}\)