Question

A wire of radius $R$ carries current $i$. The current density is given by $J=J_0(1-r/R),$ where $r$ is measured from the center of the wire and $J_0$ is a constant. Use Ampere's Law to find the magnetic field inside the wire at a distance \(r

Short answer

Question: Find the magnetic field inside a wire with a radius R and a current density equation J = J0 (1 - r/R), where r is the distance from the central axis. Answer: The magnetic field inside the wire at a distance r<R from the central axis is given by the expression: $B={\mu }_0{J}_0(\frac{1}{2}r-\frac{1}{3}\frac{r^3}{R})$

Step-by-step solution

Ampere's Law

Ampere's Law relates the magnetic field and enclosed current as follows: $\oint \overrightarrow{B}\cdot d\overrightarrow{l}={\mu }_0{I}_{enc}$ For a wire of radius R, to find the magnetic field inside the wire at a distance $r<R$ from the central axis, we apply Ampere's law to a circular path of radius $r$ with $B$ and $dl$ being tangential to the circular path: $B\oint dl={\mu }_0{I}_{enc}$; or $B(2\pi r)={\mu }_0{I}_{enc}$ where $I_{enc}$ is the current enclosed by the circular path. Then, the magnetic field is given by: $B=\frac{{\mu }_0{I}_{enc}}{2\pi r}$

Differential Current

In this step, we find the differential current $di$ inside the wire using the current density $J$ equation. The current flowing through a thin cylindrical shell of radius $r$ and thickness $dr$ can be given by multiplying the current density by the area of the shell: $di=J\cdot dA$ where $dA=2\pi rdr$ is the area of the cylindrical shell. The current density $J$ is given by: $J=J_0(1-\frac{r}{R})$ Then, $di=J_0(1-\frac{r}{R})\cdot 2\pi rdr$

Total Enclosed Current

In order to find the total current enclosed within the circular path of radius $r$, we integrate the differential current from $0$ to $r$: $I_{enc}={\int }_0^{r}di={\int }_0^r{J}_0(1-\frac{r^{\prime }}{R})\cdot 2\pi r^{\prime }d{r}^{\prime }$ $I_{enc}=2\pi J_0{\int }_0^r(r^{\prime }-\frac{r^{\prime 2}}{R})d{r}^{\prime }$ After integrating this expression, we have: $I_{enc}=2\pi J_0(\frac{1}{2}{r}^2-\frac{1}{3}\frac{r^3}{R})$

Magnetic Field Inside the Wire

Now, using the equation for the magnetic field from Step 1, and substituting the expression for the enclosed current found in Step 3, we can find the magnetic field inside the wire: $B=\frac{{\mu }_0}{2\pi r}\cdot 2\pi J_0(\frac{1}{2}{r}^2-\frac{1}{3}\frac{r^3}{R})$ Simplifying the equation, we get: $B={\mu }_0{J}_0(\frac{1}{2}r-\frac{1}{3}\frac{r^3}{R})$ This is the expression for the magnetic field inside the wire at a distance $r<R$ from the central axis.

Key concepts

Current Density

Imagine you have a hose with water flowing through it. The amount of water flowing through a specific section of the hose each second is like the electric current in a wire. However, if we want to understand the flow of current in more detail, we'll look at current density, which tells us how much current flows through a specific area. Just like you could measure how many gallons per minute flow through one square inch of the hose, current density measures the current flow through each square meter (or other unit area) of a conductor.

Mathematically, current density is denoted as \textbf{J} and in the case of our textbook problem, it's given by the formula:
$J=J_0(1-\frac{r}{R})$,
where $J_0$ is the maximum current density at the center of the wire ($r=0$), and $r$ is the distance from the center, while $R$ is the total radius. What's interesting here is that this formula shows us the current density decreases as you move away from the center towards the edge of the wire—an insightful detail about how current distribution isn't uniform in all conductors, which could have implications for the wire's electrical behavior. To give an example, if you have a long, cylindrical wire, the current density is like a gradient, denser (more current) in the middle and less dense (less current) towards the outside.

Magnetic Field Inside a Conductor

Now let's talk about the magic that happens around a conductor carrying current: the creation of a magnetic field. A basic principle of electromagnetism is that electric currents produce magnetic fields. If you've ever used a compass, you've seen this magnetic field in action—the Earth's core is essentially a giant conductor that generates a magnetic field!

In our exercise, we are examining the magnetic field inside a circular wire. This is where Ampere's Law comes into play, which states that the magnetic field in space can be calculated by measuring how much electric current is enclosed by a loop. With a formula like:
$B=\frac{{\mu }_0}{2\pi r}{I}_{enc}$,
we can calculate the strength and direction of the magnetic field at any point inside the wire. ${\mu }_0$ is a constant called the permeability of free space, and it's the measure of the ability of a material (in this case, space or air) to support the formation of a magnetic field within itself. The enclosed current, $I_{enc}$, is the total current flowing through the area inside the curve we're considering—in our case, a circle of radius $r$ inside the wire.

This formula shows that closer to the wire's center, you have a stronger magnetic field, and as you move out towards the edge of the wire, the magnetic field gets weaker. This explains why things like transformers and motors, which rely on magnetic fields, are designed very carefully to manage these field strengths.

Cylindrical Coordinates in Electromagnetism

In our daily life, we often describe the location of an object by saying how far left or right, forward or backward, and up or down it is—that's Cartesian coordinates. But for cases like wires, where things are circular, it's easier to use cylindrical coordinates. These are like the coordinates someone on a merry-go-round would use: how far from the center you are (the radius $r$), your angle around the center (the angle $\theta$), and how high up you are (the height $z$).

When we deal with currents in wires or any phenomenon that is circular or cylindrical in nature, cylindrical coordinates become a really powerful tool in our mathematical toolkit. They let us describe physical laws in a way that lines up with the object's shape, which often simplifies our calculations. In electromagnetism, this is incredibly useful—wires are often circular and their magnetic fields curl around them. So, we use these coordinates to not only describe the location of points in relation to a circular shape but also to understand how fields like magnetic fields behave in such an environment.

For example, when we calculate the magnetic field inside our wire using Ampere's Law, we do so at a certain radius ($r$) from the wire's center, which directly relates to our cylindrical coordinate system. This harmonious marriage between the physical world and mathematical description helps students and engineers alike comprehend and apply these concepts to real-world problems.