Magnetic Vector Potential and conductor

[sikrut]

Homework Statement

Show that, inside a straight current-carrying conductor of radius R, the vector potential is:
$$ \vec{A} = \frac{\mu_{0}I}{4\pi}(1-\frac{s^2}{R^2}) $$

so that $\vec{A}$ is set equal to zero at s = R

Homework Equations

$\vec{A} = \frac{\mu_{0}}{4\pi}\int\frac{\vec{I}}{|r'-r|} dl'$

The Attempt at a Solution

I'm really having a hard time even setting it up.

 

[Simon Bridge]

Start by drawing a picture of a current carrying wire with a significant radius.
How does the current vary with radius?

 

[Simon Bridge]

Hint: cylindrical coordinates.
Some people like to turn the line integral into an area integral too ... there are lots of approaches.
For marking - it is usually best to use the method covered in class.

 

[vanhees71]

Another hint: The Biot-Savart Law is not very efficient in many problems. It's easier to use the local form of the (magnetostatic) Maxwell Equations:
$$\vec{\nabla} \times \vec{B}=\mu_0 \vec{j}, \quad \vec{\nabla} \cdot \vec{B}=0.$$
The second equation ("no monopoles") is already solved by the introduction of the vector potential, cf.
$$\vec{B}=\vec{\nabla} \times \vec{A}.$$
So, just take the appropriate derivatives (in cylindrical coordinates), and check that you get the right current density.

 

[HalfLostAndSearching]

As a scientist, it is important to understand the fundamental principles and equations that govern a system. In this case, we are dealing with the concept of magnetic vector potential and its relationship to a current-carrying conductor. The homework statement presents a specific scenario where we are asked to show that the vector potential inside a straight current-carrying conductor of radius R can be expressed as: $\vec{A} = \frac{\mu_{0}I}{4\pi}(1-\frac{s^2}{R^2})$, where s represents the distance from the center of the conductor to a point inside the conductor.

To begin, let us first define the magnetic vector potential. It is a mathematical quantity that is used to describe the magnetic field in terms of a vector field. It is defined as: $\vec{A} = \frac{\mu_{0}}{4\pi}\int\frac{\vec{J}}{|r'-r|} dl'$, where $\vec{J}$ represents the current density, $\mu_{0}$ is the permeability of free space, r is the position vector of the point where the magnetic field is being evaluated, and r' is the position vector of the current element dl'. This equation essentially tells us that the magnetic vector potential at a point is equal to the integral of the current density over the entire length of the conductor, taking into account the distance between the current element and the point of evaluation.

Now, let us apply this equation to the specific scenario given in the homework statement. We are dealing with a straight current-carrying conductor of radius R, which means that the current density is constant throughout the conductor. Using this information, we can rewrite the equation as: $\vec{A} = \frac{\mu_{0}I}{4\pi}\int\frac{\vec{dl'}}{|r'-r|}$. We can further simplify this by considering that the conductor is symmetric, and thus the current elements are equidistant from the point of evaluation. This allows us to write the equation as: $\vec{A} = \frac{\mu_{0}I}{4\pi}\int\frac{\vec{dl'}}{s}$, where s represents the distance from the center of the conductor to the point of evaluation.

Now, we need to consider the limits of integration for this integral. Since we are