Magnetic Vector Potential

Sample Solution

Magnetic Vector Potential and Current Loop

Magnetic Vector Potential (A):

The magnetic vector potential (A) is a mathematical construct used to describe the magnetic field (B) in a region. It's a vector field, meaning it has both magnitude and direction at each point in space. The magnetic field is related to the vector potential through the following relationship:

• B = curl(A)

Here, "curl" is the mathematical curl operator that takes a vector field and gives another vector field representing the circulation or rotational nature of the original field.

Magnetic Vector Potential of a Current Loop:

Consider a current loop carrying a current I. The magnetic vector potential due to this loop at a point P can be derived using the Biot-Savart Law, which states that the magnetic field element dB at point P due to a small current element dl is proportional to the magnitude of the current element (I), the vector direction of the current element (dl), and inversely proportional to the square of the distance (r) between the current element and the point P.

The Biot-Savart Law can be expressed mathematically as:

• dB = (μ₀/4π) * (I * dl x r) / r²

where:

• μ₀ is the permeability of free space (constant value)

We can integrate this expression over the entire current loop to find the total magnetic vector potential (A) at point P due to the entire loop. This integration is often performed using vector calculus techniques. The resulting formula depends on the specific geometry of the loop and the location of point P.

For a simple circular loop of radius R lying in the xy-plane and centered at the origin, the magnetic vector potential at a point P on the z-axis (z-distance h above the center) is:

• A(z) = (μ₀ * I * R²) / (2 * (z² + R²)^(1/2))

Electric Field from Scalar and Vector Potential:

While the magnetic vector potential describes the magnetic field, a scalar potential (φ) can be used to describe the electric field (E) under specific conditions. In situations where charges are stationary (electrostatics), the electric field can be related to the scalar potential through the following:

• E = -grad(φ)

Full Answer Section

• E = -grad(φ)

where "grad" is the gradient operator.

However, the relationship between the electric field and the potentials becomes more complex when dealing with time-varying electromagnetic fields. In such cases, a more comprehensive framework like the full Maxwell equations is needed to describe both the electric and magnetic fields and their interactions.

Therefore, finding the electric field directly from the scalar and vector potential in the presence of a current loop (which implies moving charges) is not a straightforward process. A more complete analysis using Maxwell's equations would be necessary.