Question

It would be mathematically possible, for a region with zero current density, to define a scalar magnetic potential analogous to the electrostatic potential: \(V_{B}(\vec{r})=-\int_{\vec{n}_{0}}^{\vec{r}} \vec{B} \cdot d \vec{s},\) or \(\vec{B}(\vec{r})=-\nabla V_{B}(\vec{r}) .\) However, this has not been done. Explain why not.

Short answer

Short Answer: The concept of scalar magnetic potential is not defined because magnetic field lines form closed loops and don't have a clear potential difference like electric field lines, which start and end at charges. This fundamental difference makes the idea of potential less applicable to magnetic fields. Instead, magnetic fields are described using the Vector Potential, which accounts for their behavior more suitably.

Step-by-step solution

Reviewing Electrostatic Potential (Electric potential)

Recall that in electrostatics, the electric potential is defined as: \(V(\vec{r}) = -\int_{\vec{r}_{0}}^{\vec{r}} \vec{E} \cdot d\vec{s}\), where \(\vec{E}\) is the electric field and \(\vec{r}\) is the position vector. The electric potential is a useful concept in electrostatics because it helps us to express the electrostatic or electric field as the gradient of the scalar potential: \(\vec{E}(\vec{r}) = -\nabla V(\vec{r})\).

Diving into Magnetic Fields

Magnetic fields are different from electric fields, as they're associated with moving charges (or currents). The magnetic field is represented by the vector \(\vec{B}\). According to Ampere's Law, the magnetic field produced by a steady current is related to the current density \(\vec{J}\) by the curl of the magnetic field: \(\nabla \times \vec{B} = \mu_{0} \vec{J}\). In regions with zero current density (\(\vec{J} = 0\)), Ampère's law becomes \(\nabla \times \vec{B} = 0\).

Introducing the idea of Scalar Magnetic Potential

The idea of scalar magnetic potential is based on the analogy to the electric potential, as follows: \(V_B(\vec{r}) = -\int_{\vec{n}_{0}}^{\vec{r}} \vec{B} \cdot d\vec{s}\). This would imply that \(\vec{B}(\vec{r}) = -\nabla V_B(\vec{r})\).

Explaining the absence of Scalar Magnetic Potential

The main reason why scalar magnetic potential has not been defined is because of the fundamental difference in the behavior of magnetic field lines compared to electric field lines. While electric field lines originate at positive charges and end at negative charges, giving rise to a clear potential difference, magnetic field lines are closed loops that do not start or end at any point. The concept of potential makes sense for electric fields because the path integral between two points in the electric field is independent of the path taken, caused by the conservative nature of electrostatic forces. This property does not hold for magnetic fields, as the work done on a charged particle by the magnetic field is zero due to the force vector being perpendicular to the displacement vector. Thus, it is not useful to define a magnetic potential analogous to the electric potential. In general, magnetic fields are described using the Vector Potential (represented as \(\vec{A}\)) based on the equation \(\vec{B} = \nabla \times \vec{A}\). Vector potential is a more suitable concept for magnetic fields as it accounts for their behavior in a way scalar magnetic potential cannot. In conclusion, the scalar magnetic potential concept is not used because magnetic fields form closed loops rather than originating and ending at charges, making the idea of potential difference less applicable and not as useful as the concept of a vector potential for analyzing magnetic fields.