Question

L=λ/60C=1 3. Consider an infinitesimal electric dipole of length ℓ=λ/(50+10c), which is placed symmetrically at the origin and oriented along z-axis in air. The antenna is assumed to be modeled by a constant current I0​=c( A).I0​=1 A

Find the vector potential. Substitute all parameters.

Answer 1

The vector potential for the given scenario is: A = (μ0 / 4π) * c * ln(|z|)

Step-by-step explanation:

To find the vector potential for the given scenario, we can use the formula for the vector potential of a current-carrying wire segment. The vector potential (A) is given by:

A = (μ₀ / 4π) * ∫(I * dl) / r

where μ₀ is the permeability of free space, I is the current, dl is an infinitesimal length element along the wire, and r is the distance from the wire segment to the point where we want to calculate the vector potential.

In this case, the wire segment is an infinitesimal electric dipole of length ℓ = λ / (50 + 10c) and the current is I₀ = c A.

To calculate the vector potential, we need to integrate along the wire segment. Since the wire is placed symmetrically at the origin and oriented along the z-axis in air, the integration will only involve the z-component.

Substituting the given parameters into the formula, we have:

A = (μ0 / 4π) * ∫(c * dl) / r

To proceed, we need to determine the expression for dl and r in terms of the given parameters.

Since the wire is infinitesimally small, dl can be written as dl = dz, where dz is an infinitesimal length element along the z-axis.

Similarly, the distance r from the wire segment to the point where we want to calculate the vector potential can be written as r = √(x² + y² + z²), where x, y, and z are the Cartesian coordinates of the point.

Now, we can substitute these expressions into the formula and simplify:

A = (μ0 / 4π) * ∫(c * dz) / √(x² + y² + z²)

Since the wire segment is placed symmetrically at the origin, we can assume that x = y = 0.

Therefore, the formula simplifies to:

A = (μ0 / 4π) * ∫(c * dz) / |z|

To integrate this expression, we need to consider two cases:

• Case 1: z > 0

In this case, the integral becomes:

A = (μ0 / 4π) * ∫(c * dz) / z

Integrating with respect to z, we get:

A = (μ0 / 4π) * c * ln(z)

• Case 2: z < 0

In this case, the integral becomes:

A = (μ0 / 4π) * ∫(-c * dz) / (-z)

Integrating with respect to z, we get:

A = (μ0 / 4π) * c * ln(-z)

Therefore, the vector potential for the given scenario is:

A = (μ0 / 4π) * c * ln(|z|)